Modern Physics

MODERN PHYSICS S E C O N D E D IT I O N tegr.'. Useful 'n ( ) J , , (/I.". ') J , \ ,,0'"' '' "' - ," s,n.� I...

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MODERN

PHYSICS S E C O N D

E D IT I O N

tegr.'. Useful 'n

( ) J , , (/I.". ') J ,

\

,,0'"'

'' "' -

," s,n.�

I.

, - -I.- ,\(\ �/l1f l

(h -

I.

"

4

(1\ -

-

.00

e-a\
When the orbit of the planet Uranus was found to devime hl'-' '1"'-"\\ )( ' I' as 1"'-""' all) I t " Ir,lIn 111..' log lit III m/!li l\"l!il'(" �\ 111 lUll " II'i kn!1lh "' �IUC3otJUltatoh woO III 1he- An'IC' .... alRI"' .prlle, hl lh... ell\" ..:I ." hI th..' 1\\\1 PIt'\I\II!' l'il ..'d' 11 I' • ,Ol'C' nm,w��' III dl '\UIi '\'i kn�lh l'Ulltt.ldl\lfl '" an \11'11..·.11 Ilhl'lllll l'.IU'l·d t>� !.kl .., '\ III h,hl Ira\('hn, Itl the \>h"" ",., IHIIll Ih\.' 1Ih1\ 1I1g \1"'ll'.;I The dl,-d " n.-al Wr{lOSt" ' lhe pl.lII\. " \11 kn�lh f ":\. tlntlng lu HI.t>, \\ hll hI li..!, II \I In ,\nna', Jr ..ulC, \fII'.1 ami hl.'! """1.1111 \Ill � l". h h.lpren II) ". ;lh�n\.·d \\ Llh .1Il (oJ \11 Itl.: ra-,'lOg pl..n\' _" /"1'\"1 Idl rll" \dm� t,,"I'. th"'n III Ih"'ll 1fi.lllw. "nlHI !Ukl .\",) "III t"t' a U"'lan..'\.· a" ..n It'\\ lhull l ..

1",,\

tune

'�Y'" 2.' Wh('ther Iln(' Ilhl�·t Ii" uhl\lc ,l1\llllll'r l.kJ"l:'lIIh I'll Ihe:' 1.t>'>('1 \\'1'" II"l1\e ul rl'lc.-cn..·('

1he I�ith Ilr an 'It'lje(1 In .. lrame Ihl,.ugh .... hlt,;h Ih.: \lhll.'\:1 1Il,1\,', . ' lI11llll('r Ih:!n Il'ri kn,lh IIlth.: Imllle 111 " h....h 1 1 I' III re I .-\�aLn, lIiJ I h.lt 1\ rc,-!uu!.'d I' rel.llt\'.: nHltllm If Anna \lR h!.'f 11",,;;\r .... '·1\· h'lld In, u pl.lnL. til kn)!th r� II \.\ llulJ ,x'\.·up;.- a di,tanl.'e h:" than f i.lI.:n)rdill� hI \1t'l$'f"\ I.'r-- In

"vh', Irame ,II n:ll'rcn..:e. tlc�au,� Allllil', plJnk mUH', rdativ!.'

\111 Pllrlcd

10 8"b

We nt'" ,,:ull'ldl'r " "Iue,uun thai In \lnt" hlrtn $r anulhl'r " II ;':OIlHUIlII '1IUn.'l' \11 ;.:,m,lt'rn.tltl\n lor thl' l"ol'glOnll1!; "udenl (It "pcl:lal rda! l\- I I)' I n '0 J"ing. .... e "'ill t.e�1R tll '>('1.' till' inll'rJcpc.·l1lknl:c \11 th..' l'on,cqucm:c, \Vhal ap�ar III he paruJu,l" arc UI,O"II) JU�I nample, of applyi ng. (lIll' l'Un\equcn\:l!

01 'pe"'IJI �lall\'lI) and IglI\lnn� tht" olher,_ They arc a pa\,.'!..agl' ,",cal, and wc mu,1 buy till' ",hole p.ll"I.agl', Anna hi.!' dc\clopcd a re"olutlollJry new plane capahlc Ill' 1.I11aulL ng 'pt.·cll, near Ihal 01 I Ighl It i .. -lO Ol long \.\hen par\'cll lln thc runway. Bob ha, a 2() III

(b) A,"eonlm� 10 Bob

Illng iAlrplane hang.u- With \Ipcn dtxlrn ay .. at ea....h e nd. ;1\ LIl hg.un: 2.K(ill. ,-\nna. a yuung and lbnng Iype. t.lke.. oil. "I.'ce:!cmlc.. It) high ..peed. then '''(lOP' throu�h the hangar AI a high enough "�cd. Anna' .. plane fil'i entirely ""tthin lhc hangar all al on..:'(' a;':l,.'llnJlOg to Bob. (lCclIp)ing only 20 In .., ,huwn

tn Figure 2Jl(bl. In other ",onh. Btlh ..cc, the lail (II Anna's plane ill (JOe dour ....ay al pre"'J�ly nOlln on hi .. W.lkh. \\hiJe Bob Jr. ,.:C\ the p lanc ' .. no..,c al thc olherdoot'\l,'uy al�o pred,ely at nlxlO. Nnw ('()n�idt:r Anna... pt'n,pe,,·!lvc. Anna •

" at rt\t rtlati\'c to her plane and thu, sec .. It'> kngth 01.-" 4() m, She \I.'l''' the

lei A':l'NUla III Bob, anJ thc-),' \hllUldn'I, I"" thC'� d(J 'lnke 0 \lmultaneou,I}' 11\ A.n",,·,

!t

a.;>,:\",'\,bn): II' Ann.

(e) R\,b ,,«, IhI! (I!nler oJ Anna\ ,hip mme t4n.:� ovcr \peed, II> thus

Jt

The lime- according 10 Ollll, ctl\·

�, -

·W

It,n To find the rtadtng!. B\,b and ht� a"I�Ia.nh \� on Ann3'� doc..... al l - 10 n". � c\.lUld u� (1-1�b) Ilgaln, bul lhe fa...ter ....a)' il> Ilm� cldallon. Bob mU\1 "'et' Annl'� l clock &l.h.nclng hall J.� f�hI a� hi, ov.n (")'.. - 1) For c\cr) 20 n� (In hiS dt,lI\.·b. u.:h of Anna', mu�' til! '<en 10 atha.nl,:e 10 n�

rhul....

tel FI�un'

�.lb

figure 2.16 ..ho.... , thl!

ontrachon and IImc dilaoon ob,,·ious accordm,!! 10.) make, Icngth ...80b. but it also sh()\1.� Anna ob..enlng lhe� eifr:c!'). Al the p:1\Smg of docks

labelw � e\ent I and ('\enl 2, ob ..r:ncn In Anna'� frame at t' - -60 ns and ,. • - 20 ns-that is. �O n, apan In thclr frume--look al the samr clock In

1.l n. a.- 1\SIUIIf ..... . fTamt .... 1ft at ad\� 001)' � III Thi,,.,, tame II..... en" verify tu'Dt Wlaooa. 11M InIJ htf � 'M.1UW 1WWf .,..u. the ..... of ¥tew. UtI: tA,It,,",cllxh m 1kJ,tI. (ramt, fur '\Muan thal lklb' �1oI.-b MY um�Bt.:hroluftd I '.'flIh) and ". llbttn� al lht tr&mt " (} lutan' m .... fr.mt and onh h.t.1f hruhlp' Ii'n.Ilh.,." SICC' Ihr \'t'f) IC'QI.h of a"b'

".... .."

r11P ', *

--

....... .. .... _

- -I• ...,.

Wp. II " ('\wn.:k'd 10..1 half. k"onl

"',Ull" 1 1 7 stK,\Io Ii Anna' �'I)t1lrl('mtnlllf) ' Ie" llf lhe C',C'n'Ii. in tixample

� " (\mlJl'Vln� \Io llh .'I,Urc' ';:. I ft. \10(' � Ih�u. iL... I\l)(C\i (,lU'hcr. ill ('\lcr)' C\lC'nl whcrt" ,,,\1 �'I\""'L' p,u' \1\\)( ,,,hl the 1,lur 10it'lc1C1J1. Annu and Bllh 1ljR'C on th..� �·k...... � · tt:1ilhn�\ Tht' IInl'lll.lnam f'llim 1� th" It j!i. the fa....,·l lh•• !i.imul, 1010('11) \11 ('H·Ot... I' n\lt ;,Itoo..." IUll' that �1I\I\Io'" t',Kh lltoo�er"('r hI "'C(' the IlthC'r'!'I d\... �, runmn.:, ... 11'\10\'1 Ih...n hI' l'f h\'r 1\\\ 1'1 and Ihl' lIther'" di�tall\:(,� ,,:un· tru....tl'd h�ul't·, � 1(1 ,lIld ..1 1 7 r('H' al nthl"r IIlh�re'tlll� pt)inh. hlr in�llIn('C', "'l'n!, � .lIld , '''' .... or III II I.hll('r\·111 oil/a 10 Ihe 1\\11 fnun('�. SeC' Appcontl j .'( B . III Ih... '11Uallllll In E:\umpit!' 1 .J too)' a mul't" ltrurhku) f" t hu1hcI h .. ....·u""1n

dPph'lJ\:h

L" lng I"knlh:ill p.I"" IIl� ...hlp... "�alli. t'I�Un: :!. 1 8 dltrilie!i. Ein"lC'in'.� �C'l.'.

�;I-i

all dm:..·lItm, A"h\IU�h ll\lh �el!'''' the pul...e n';.lI,:h the end), \11' AIIIllI'" moving lie

the pul,(' 'prealilng 'ylmnetncally I'mi'll hl� (lfl!!in NtlW ('lm·

'l'C'"

...uJer :\no,,', � I(,\\' JudglRg b) the 110\'"

\\e ...t'e In hl'r frilllll'. ...he ,ee� thc

pul\.(' n!.l....h Iht!' end, \11 her ,hip 'IIHuh.ll1ellu,ly .loJ Ihe

puh(' 'pre.)l,hn}1

1 0 at 40 n,. 1:;1I,:h \C'e"

.. ...

'Y mmetnl·ally at the 'amI! 'peed Inml Ihe ('enter of hi� nr

her o\\n ...hlp.

� I -10 II 'i1> Evrnl 2

EXAMPLE 2 . 5 Pe\lpk lIn l.lrth 1..011..... thai It \,ll..e, IIghl 40 )etln. to rc.h:h Plllnel

h,h J"'( been tW,,1m (';\11 ...he get

It'

PI",n"t \ b) Ihe I ll11e

""h,lt 'flCcJ .... rt"qUlrcJ ' CN(I/j' The tJI\lan\.'l: light trn..el... year. llnd the �ymhlll l\

I

I). [t happ..-n' 10 he 9.5 X

the \:onverlon here.)

X

from Ear\h. Anna

'h" 1\ 10 yeah ()Id') II \\1, 1Il

a year i, culled u light·

IO I� Ill, hU1 11 i, l'u\ier nOI 10 u,"

SOt n N It might

Event .1 Ih,II IO t>e 40 I)' lrom Earth by the time \he i,

10, Annn wlluld have

10 tra\'d 1'00'ler Ihan c But PI;!nel X I... .w Iy a....ay uccoruing 10 an Enr," (Ib'erver··· Bob, 01 (OUI"<e To Anna In her "'PJ(e�hlp moving at \. rel;lIiw 10 Eanh, the \«,Ill

Earth-Planet

X

" obJet alsu l"'!nmr: that Bob agc... 1 8 year,. If Bnb iii 1 00 when ('arl get, to Earth and , l< · ..:d 1 X year... dUring the night, thcn Bob mU!t>t be M2 year, old. uccording

l

.

l. when Carl pa"..c'> Planet X Thb i... truth

t t

m

"u

Carl's frame'

) return, Anna must accelerate: that i,>. she must c.:hange from one inertial another. (It make� no difference whether her ,�peed eould be kept con" I \1\ inertial frame ha� COn�tant I'e/ad,)·.) So far. we have not considered IS ollght affect Anna·... perceptions, but Carl doe!iiin 't accelerate. so we :'J

v nn hi.. observation... Figure 2.20 demonstrates the main point. Anna i'om one frame. where Bob is 18 years old, to another, where Carl

u: H l"

1

.:r that Bob I ... 82 years old. Although on the return leg of the journey. does determme that Bob is aging more slowly than herself, it is 100 late, )

•

'0; hu�e forward leap in age (according to Anna) has settled the mIder. Ju they reani.1e. l� � 2 1 Illu\lrales, Bob will be much older than Anna when

I v.o questions may still haunt the reader. ( 1 ) How is it pouible tbIl whee wbIIe CIII A'., lnd Carl are both at Planet X. Anna says Dob is 18 yeon old. lIIpJ4* says Bob " 82 years old? (2) Can Carl ",U Anna about Bob', fuIuIe?

A_. n" tOO ycar"lI gld IIut JutIf IooIc at you

yCHI·... GIlly 601

All tgg tnMl!, Bob What 0 differetlCe o little C'lCceleratign mokes!

Bob sends a light signal (rom Earth toward Planet X so that it arrives at the precise inSlanl Anna and Carl pass Planel X. If according to Bob il lakes light -1-0 years ro reach Planer X and it lakes Anna 50 years. then Bob had better send rhis signal when he i� 10 year� old. Let the light signal be an image of Bob blowing oul candle� al his 10th birthday party. When this signal reaches

Planer X. both Anna and Carl will �ee an image of Bob as a JO year old. Each

will ha ve infercepred all earlier signals. �o they will have knowledge of Bob's

prior activities. but none beyond thaI. The answer to question 2 is. No, because neilher has acruaJly seen any of Bob's life aJ!er his 10th birthday. However. we don't answer the question "'What is Bob's age?"' by saying,

"'The age he appears 10 be in the light signal that just arrived." Neither Anna nor Carl will say that Bob is 10 years old when they reach Plane! X, because each realizes thaI Bob wiJi have aged while the ligh! traveled. But Anna and Cart are

scrupulous record keepers. From their observations. they can calculate what Bob's age must be when they pass Plane! X, and the conclusions they reach dif fer markedly. Le! us investigate this and. in so doing, answer question I .

Figure 2.22 shows how each of the three observers sees the situation as Bob's light signal leaves Earth. As we showed earlier, Anna sees the distance befween Earlh and Planet X as 24 Iy. She sees me light signal and Planet X

both moving. and rhe separation bet ween them is decreasing at a rate of 1.8e. (This value is a relative veloci ty according to a third party, Anna. She doesn't

Figure 2.22 A lighl beam traveling from Earth to Plane! X according to three differ· ent observers.

4Dly

D.8e Anna

Bob

(a) According 10 Bob

fo Ihe righf at c C

E==> Anna

J.

24 1y

D.Sc

q Carl

and Planet X mo\·jng to the left at OJk _0 they approach each other at 1.8e·

Annil .se(';s the lighl �jgnal

mewing

o

0.80@ ,I

(b) According to Anna C.trl see� borh the light signal and Planet X moving with a fe/alive \'clm:iIY ofonl) 0.2('

W fhe right.

c

241y

6)

0.80

(c) According to Carl

VI (Y, - l)rne' � Iq �VI 10-27 kg)(3 X 10' mI,)' � (1.6 X 10-" Cl l�vl = � V � 470 MV KE �

( 1 .5 - 1)(1.67 X

and EoersY

39

40 Chepter 2

Special Relauvlly

1

h the same speed. 0.74 , al kinetic energy fonnula wit Had we used the classic energy increases vel)' ra · Ic y :::: 291 MY. Kinetic V J, d aine obt e we would hav er potential difference is required an ches e, and a much larg as speed approa ally. would be expected clas"ic

��

Let us now apply the

relativistic conservation laws.

EXA M P L E 2 . 1 0

with object 2. of mass 12m 0 and heading ea�t at O.Se, collj�es Object I ,. of mass 9mo ether. Find the mass and speed of the tog k stic twO The c. 0.6 al t and moving wes y mass umt.) . an arb"Ilfar (NOle: fIIo IS ' ' resulting combined abJect. SOLUTION

as east. The reader is encouraged to verify that if Let u� define the positive x direction re the collision would fonnula, the objects' momenta befo we were to use the classical t would be at h l t would be O--that is. t e fina objec al be equal and opposite. so the to ch would whi 10 me that mass is constant, RelativisticaJly. we must not assu lision e the col aus is Bec c t. stan m con is energy assume that the system's intemal we should expect the internal en rgy so . lost be will rgy ene tic kine pletely inelastic, entum-conservatiOn speed unknown. the single mom to increase. With both mass and erful new 1001' pow a e hav e the problem. but we equation is insufficient to solv rgy must be �o ene l tota and possessed by an object. expression for the total energy d object, and we bine a subscript f for the final com served. In what follows we use :::: � and 'Yo.lIe = 1make use of the fact that 'Yo.&

=1. :

:

a would imply that the As noted. the classical fonnul g. We see that it will be movin : Energy(,),1/me2) conserved

5 91110r,

-

)

'Y

final object should be at rest.

'",c2 + 'Y.!lII,-e2 = 'Y. lllrc?-

UI

+ 45 12m0c·' -

=:

' "" "'re("r!C )'

J _r=� , /J V

r

=

-

g the momenlUm equation by the

(wo unknown!>. Dividin We have twO equations in energy equation gives

110, The initial mass W3'> 2 1 mo' so ation yields t1If = 29.851 equ er eith in ing ert ins Re doer. indeed increa\c. the masslinlemal energy

2.7 Momentum and Energy

A

calculation of total kinetic energy before and after the collision In

Example

KEf

2. 10 veri fies that kinetic energy has decreased. Using equation (2·26).

-

- KEI = (-Yo. 1e-

1 ) 29. S5"'02 -

= -S.S5m c2 o

The kinetic energy change is exactly

[B - 1 )9moc2 + (� -

- 6m2 (i.e.. 2 1 - 29. 85

1 ) 1 2m02]

=

-S.S5).

Kinetic energy is convened into increased thermal energy. which regi!o.ters as increased mass. In fact. this relationship must hold for any number of particles colliding or breaking apart (see Exercise

96).

If the total energy is conserved

as particles interact, then the changes in internal energy and kinetic energy must be opposite.

(2-27) EXAMPLE 2 . 1 1 The nucleus of a beryllium atom has a mass of 8.003 1 1 1 u. where u is an atomic mass unit: 1.66 X 10-27 kg. This nucleus is known to spontaneously fission (break up) into two identical pieces. each of mass 4.001506 u. Assuming lhe nucleus to be initially at rest, at what speed will its fission fragments move, and how much kinetic energy is released? (Nore: The masses are given to fairly high precision because. as we see. the initial and final differ only slightly.) ;OlUTtOr

Relativistically or classically. if a stationary object breaks into two equal fragments. momentum conservation requires that the fragments move oppositely al equal speed. We could use equation (2·27) to find KEf and then the speed. but let us sim· ply apply energy conservation. Usi ng a Be subscript for the beryllium and and 2 for the fragments, we have

I

y � u

8.0031 1 1 u Se � ;;i; :;� ;;;: ;"--; � I .000012 2(4.00 1 506 u) 2m m

-:-;;= � ;= """ = 1 .0000 12 VI ( "Ie)' I

-

= II = 0.005c

For me energy released. equation (2·27) gives aKE � -amc'

�

=

-(2 X 4.001506 U 1 .5

X

- 8.003 1 1 1 u)(I.66 X

10- 14 J = 93

10-21 kg/ule'

keY

This is in good agreement with the ex.perimemally measured \-·alue.

It may puzzle the reader that none of the recent ex.amples has involved a second reference frame-a pattern has been broken. But views from different frames are nor so much the mterest here as is the fact thal momentum and energy in an\' frame depend on ma�s and speed in wa}" different from the

41

42

Chapt.r 2 Specill Re1Btivily classical The fascinating point is that it is the claim that conservation laws must be the same in all frames that led to this discovery. Before we move on, we note a general fonnula that relates the energy, momentum, and mass of a particle. It is left as an exercise to show that the speed u can be eliminated between

E = l'ume2 and p = l',,"U, yielding

(2-28)

Because this relationship is independent of the speed, it is often very conve· nienl, and we will use it at various points in the text.

The Particle Accelerator Einstein's famous mass-energy equivalence is central to the purpose of

particle accelerators. These immense scientific apparatus speed up particles to nearly the speed of light. They are the tools of high-energy physics, which seeks to identify the fundamental building blocks of the universe. The electron is the most familiar fundamental particle, but many others have been found. Still others are, at present, mere theoretical predictions. Often, these undiscov ered particles are expected to be fairly heavy. But how do we create mass? As Example

2. 1 0 demonstrated. Einstein'S mass-energy relationship is the key. If

particles collide inelastically, kinetic energy decreases and mass/internal energy increases. In high-energy physics, we smash familiar particles like pro· tons and electrons together at very high speeds to make available a large initial kinetic energy from which to create massive new particles. The greater the mass expected of a particle, the greater must be the initial kinetic energy. It is instructive to contrast two common types of particle accelerator. In a stationary-target accelerator, a moving particle is smashed into a stationary particle. In a collider (short for colliding beams apparatus), the two particles smashed togelher are bolh moving toward a head·on collision. To produce the maximum mass with the minimum input of energy, the collider is preferable. Why? Consider two identical particles. If both are moving at speed u in oppo site directions, the total momentum is zero. Therefore. if they stick together, the final particle must be at rest.

All the initial kinetic energy is converted to mass.

[f, on the other hand. one particle is moving and one is stationary, the initial total momentum is nonzero. so lhe final particle must be moving. Some kinetic energy is necessarily left over after the collision, not available to become mass. Although the simple example of a completely inelastic collision is not realistic. the underlying principle holds no matter what may result from the collision. In a collider. the final kinetic energy is a minimum and the final mass a maximum. Exercise 93 compares the two accelerator types quantitatively.

Massless Particles Experimental evidence indicates that there are particles for which

E = pc. When

these particles interact with other objects. the momentum and energy they carry are found to obey this relationship within experimental uncertainty. Comparing with

(2-28), there is only one possible conclusion: They have no mass.

E = pc

(m

=

0)

2.8 Ol:nl:ral RdlllVif)' and . Ftnl Look II COUIIC/Iop'

Bul shouldn't particles that move about and transfer momentum to other things have mass? There is no theoretical requirement. Thai applying F = rna

is problematic does not prove that m = 0 is impossible. It may "iimply mdicate that this classical law is inapplicable to massless particles. The topiC of Chapter 3 is the most well· known massless particle: the particle of light. or

photon. No experiment has ever demonstrated a mass for the pholon. If it IS not zero, it is immeasurably small. In faci. the assumption that it is zero is cen·

tral to our accepled theories of electromagnetic radiation.

Special relativity tells us some important things about the photon (and about

other massless particles). First, having no mass, photons have no internal energy. Their energy is all kinetic. Second, if m is zero. the only way p = 'YrI"U and E = 'YllmCl can be nonzero is for the speed u to be c, in which case both formulas are

undefined prodUClS of infinity and zero. It is true that all photons move at c. from

the instant of their creation to the instant of their disappearance. However. the expressions for and E are indetenninate. So what does determine the momen

p

(Urn

and energy of a photon? The answer awaits in Chapter 3.

2.8

General Relativity and a First Look at Co!.mology

It might be said that special relativity begins with Einstein's deceptively sim

ple pos(Ulate that the speed of light is the same in all frames. General relativ

ity, a more general theory (sensibly), begins with another "simple" postulate:

Inertial mass and gravitational mass are the same. I n introductory physics, we

learn Newton's universal law of gravitation. The force between objects of

mass

m

and M separated by

tational constant. Taking

m

r is F = GMmIFl, where G is the universal gravi

to be an object near Earth's surface, with M and

being Earth's mass and radius, the force on m is GM""",

rl""'"

_

r

9.8 ml'2 X m

m -

in introductory mechanics. we also learn the second law of motion. The accel·

eration of an object of mass m is proportional to the net force on the object and

inversely proportional to m.

a = F�

I

'm

On the face of it, these two properties of the mass

m

are entirely dIfferent.

There is no fundamental reason why the property governing how h:ud gravity pulls on an object should have anything to do with the property governmg the

Accordingly, it property. in the fir\t

object's reluctance to accelerate when a net force IS applied. might be safer to use an

equation and an

m' i

mg,

sigmfying a gravitational

for an inertial propeny,

10

the second.

43

44

Chaphr 2

Special Rtililivily

RgUN 2.29 The floor pu!ohe!o upward

a = Fnet

on Bob. who is at re'it on Earth. The

floor mUM al'iO push " upwanJ" on

I -

m,

Anna. whose frame at:("elerate� al K in Now consider what happens when an object is dropped from shoulder

gravity-free 'ipace.

height. In this case. the net force is simply the gravitational force. Thus,

a�

,nil,.FN '"' III," -

/II.�

{9.8 mis'

If fIIg and 1M, were truly different properties. there is no reason why we could 1 . 1 fIIg and another with fIli = 0.9m , in which g case they would accelerate at different rates. This is certainly not what we

not have one object with "',

=

expect. In fact, the equivalence of mg and "', has been experimentally verified to better than I part in

1012. Although we tend to take it for granted, their equal

ity has an important consequence: It should be impossible to determine Frame 8

Frame A

whether we are in an inertial frame permeated by a uniform gravitational field or in a frame in which there is no field but which accelerates at a constant rate. Suppose Bob stands in a closet on Earth, frame

B, as depicted in Figure

2.29. He is in a frame of reference that is inel1ial and in which there is a uni form gravitational field of g = 9.8

mls2 downward. (This is a good approxi

mation. although Earth rotates and the field at its surface is not perfectly uniform.) Anna is in an identical closet, frame

A, but out in space. far from

any gra\'itational fields. By means of a rocket engine. Anna's closet is acceler ating in a straight line at 9.8

mls2• For Bob to remain stationary. the Ooor must

push upward on his feet with a force whose magnitude equals the downward force,

FN =

1Mg

x

9.8

mls2.

For Anna to accelerate along with her rocket

powered closet. the floor must push "upward" on her feet with a force sufficient to give her an acceleration of 9.8 Figure 2.30 In both the inertittl frame with gravity and the accelerating frame without. a downward force appears

to act.

"' , X 9.8

mls2.

If I1lg and

til,

mls2. By the second law of mmion, this is FN =

are equal. the forces are equal and would pro

vide no clue to distinguish whether an observer is in frame A or frame

B. The

normal force is only the simplest indicator. The fact is that no mechanical experiment could distingui1>h the frames. In the linearly accelerating frame A. all things appear to be affected by a downward force just m; they are in frame

B.

The floor must push "up" on objects, and "dropped" objects appear to accelerate downward (becau�e, once let go, they do not accelerate along with the frame). All effects could be attributed to an "inertial force" of -mj3 opposite the acceleration. as depicted In Figure 2.30. Provided that mj equals 1IIg. this force would mimic a gravitational force mgg in all respects. By the same token. no mechanical experiment would

be

able to distin

guish a frame that is accelerating in free fall 10 a uniform gravitational field from one that is inertial and without a gravttational field. As Figure 2.31 shows. an observer in the inertial frame would see all objects floating or mov ing at constant velocity, because no forces act. An observer in the free-falling frame would also see objects 'Oeemingly moving at constant velocity (though all would actually be accelerating with the frame). because the gravitational Frame B

Frame A

force mgg is exactly canceled by the inertial force -mia.

2.K Genttal Relativity

Uld. Fant Look at Cosmo1o&Y 45

Figure 2.31 Thing� seem Ihe same in a frame thai is free falling due 10 gravilY as in a fr.lme flrulling without gr.lvity.

m,g g 1-i'-".:t.I T/ m. _»I

�a

=

•

g

00;

0 = 0

Engine

off

Free-fulling frame

The Principle

�if----\'

tnenial frume

of Equivalence

Einstein's customary leap forward was to postulate that

all physical phenom

ena, not just mechanical ones, occur identically in a frame accelerating in gravitational free fall as in an inertial frame without gravity. No experiment could distinguish the frames. Accordingly, he generalized the concept of an

inenial frame by defining a locally inenial frame: one that is falling freely in a

gravitational field. (It is "local" because it must be small enough that nonuni formities in the field are negligible, giving the same acceleration for objects within it. Also note (hat this definition includes ordinary gravity-free inertial frames as a special case.) We may now state Einstein's fundamental postulate of general relativity, known as the principle of equivalence. The form of each physical law is the same in all locally inenial frames. This postulate is only the basis of general relativity. Just as the Lorentz trans formation equations follow from the postulates of special relativity, a mathemati cal framework follows from lhis postulate. Unfortunately, general relativity theory is too sophisticated to discuss quantitatively here. It involves the mathe matics of tensors and differential geometry. Nevertheless. some of it>; astonishing predictions can be understood qualitatively just from the principle of equivalence. Three have anracted particular attention: ( I ) gravitational redshift. (2) the deflec tion of light by the sun. and (3) the precession of the perihelion of Mercury.

Gravitational Redshift and Time D ilation According to the principle of equivalence. light emitted at one point in a gravI tational field will have a different frequency i f observed at a different point. We see this by analyzing not a fixed light source and observer i n a gravitational field

g, but the equivalent case of a source and ob�erver in a frame w i thout

Principle of equiv:Jence

46 ctr.pter 2

Special ReJatlVlI)'

gravity but accelerating at g. In Figure 2.32(a), a source in an accelerati n frame emits a wave front when the frame has zero speed. Anna, a di stanc e "above" the source, observes the wave front afrer a time HIc. But by this ti gHIc. Thus, the light is observe . me she is moving "upward" at speed d 10. a frame that moves at velocity gH/C away from the frame in which it was red. According ro Anna, the Jjght will be redshifred. Provided that g and

�

II =

�rnu� �

e will be smaJl, and we may apply the binomiaJ apprOXimati o n 0 !he Doppler (onnul. (2-18). not roo large,

v

= /source

fobs

c

v

= fsource

+

c

� fsource \.1

_

=

�

� )l � v) -I

!'o"�e

v - -

1 + -

c

-

I

c

v

) �\.1 - 2 �v)

2�

� - �)

Figure 2.32 In (a), a wave front travels from source to

observer in

an

I

accelerating frame without gravily. Because the

observer's velocity at reception of the front differs from the

source's velocity at emission, there is a Doppler shift in

frequency. !hus. there must be a frequency difference in

(b), the eqwvaJent stationary frame with gravity.

v v

! J H

=0

= at = gcH

h:

Observer

1=0

t

a=g

'i'

Source

(a) Accelerating frame

I

'i:!!

'i'

!!. c

l/

Observer

H

J

(b) EqUivalent frame

2JI Gl'neral Relativity and a Fi....1 Look al Cosmology 47 In (he last step, we have discarded the \.2ICl term. Thus (2-29)

Einstein's poMulate is that a gravity-free fm.me accelerating at g is equiva lent to a fixed frame in a gravitational field g. Thus, equation (2-29) must also apply in the frame as shown in Figure 2.32(b). As light moves upward. its fre quency must become smaller and its wavelength longer. as i l lustrated In Figure 2.33. We reach a lruly fascinating conclusion: This is a time-dilation effect, but it has nothing to do with relative motion between the source and observer frames. Consider 600 nm light, which has a frequency of S X 1014 Hz. As one second passes at the source, S X 1014 wave fronts are emitted. It might be said that S X 1014 distinct events have passed in the life of the source. But in a given second, Bob receives fewer wave fronts, because he observes a smaller frequency. To witness ail S X 1014 events in the source's life, Bob has to wait more than one second. Relalive to Bob, the source. deeper in the gravi tational field, is aging slowly. In fact, the time passing at the lower point must be less than the lime passing at the higher point by the same factor that appears in equation (2-29).

(2-30)

Does time really pass morc slowly on Earth's surface than at some altitude above? The weakness of most gravitational fields (gHlc2 « I ) makes gravi tational time dilation a small effect. But loday's high-precision clocks. whose time bases are atomic oscillations of ex:tremely short period, can reveal it.

R E A L - W O R L D E X A M P L E R E LATIVITY A N D T H E G L O B A L

POSIT I O N I N G S Y S T E M

Signals from GPS satellites can locate objects on Earth's surface easily within a few meters, but this requires great precision in timekeeping. Each satellite sends a time signal. and an error of just 10 ns would translate, at light's great speed, to a distance error of 3 m. This requirement has thrust into the mainstream of modem technology a topic previously of concern to only a few physicists and astronomers. Special relativity tells us that ;} Salellite moving relative to Earth's surface will run slowly. General relalivity tells us that a clock runs slower when it is deeper in a gravitational field. so gravitational time dilation will cause a satellite's clock to run faster than a clock at ground level. Equation (2-30) gives the simplest case of a unifonn field. In Earth's nonunifonn field. the corresponding relationship (see Exercise 98) i s

'" (:,., [1 satellite

_

-'-

(

GM

c1 rEanh

_

(2-3 1 )

Figure 2.33 A light beam's wavelength grows longer and its frequency decreases 35 it rhes in 3 gravitational field.

_ 1I 1o ..... _ Md , . ... di _ _ ... mw' o( ... Eonh. 1bio ia _ ..... .. .. i'W 12-30) .... II IDIP lint _. The ractor gH in (2-30)

.... . ,.• • _ � enC. dlft'erenl

pOInls lIn a rol:;llin,g gillbe move: ;.I1 diller

ent speed\, and a r«c,..cr mlghl also be In motion. H()"�er. c\en Eanh',

equlluriaJ �pta1 1 mud saaeJlilc', peed

lmaller than the ulc:itile'" !ilpc'C'd, ,,0 u"m8 Ju"l lhc "ill ,IV(' Ui a eood appro:umahon. Thus, (l)r J d\)!.;k. Udmg on

boMrd I.hr satcUlle, we haw

\

(I

(3.9 10')2 "

I

.1 X

10"

.lf t-Mth

iilf

(I

1 . 7 ")( 1 0

+ X 5 X 10

10)

1 . °.l1

,aldlth:

") .l 'sellite

ft.,.- the gnmtJlionai del'l. wt'" CklJ f:;.:uth ... rJdUh I�) (he- limit "Irltm/t', and n(.)le thai GMt:;.,h = (6 b1 x JO I I N 1I1'··lg l 2)(51.)� x I� kg} .. X 1014 m'/"l

4.0

•

Thus,

I

I

I

I

.ltOCC-lIllrl l

�L�6 X )C)

The gr.mtallOrw.I e.t1"iX'( rruk.es the- soilcIlifc;'s time ,lightly latfer. 11:- dlX-'" run,

,,�'elI-d�pendenl effn·1 make... ml.' ,,;ud· hit" !!; ciul..-t run .fio)fe'T, hut b) Wl (''"tn timer ilI1l0unt. Adding the frJ(.:tional change\ fur each. yoc ha\C a tiny bit latta Ihan a surfxc dl"'�. The

:\.26 ")( 1 0

j,1�lIjle

I

i

10)

:0: -4.4 ')(

(b) Inserting 1:1 discrtpancy

i

of 10 ns.

I()

l(

H)

"�I

.l'Wltelhle

10

.lts.:Jreun(: = 23 ,

10

10

...

X.5 X

10 I J

"C i« tlwt In . "!) tbi.'(1 l11l,lt' cftOU¥h t m ..... VI\lt,Ikt k wnu\aW 1I.1 maLe me ')'SIt'm

.......

In prw:'",� ..k-=b � GPS ub;llllC) � presct 10 . rreqUC'QC)' thai ..",'OOftl!l fur th: lra\ IbIklfl4l t'ffo.'t n:l&tl\C 10 Evth's ,urf;ac(". *' II dc� ,ll\J:o 00 1M "n('l�u ,wtllt rMhus V�k""'I\C's aod ek:" "'IOO\ 01 re..""'CI\'nS \'&I), AAd ",,""\lUnhnt. 1\If thew trT.I! �u'C l'Cal·timc CD.:ubtio� In lhc ��h ins umt. f�1d !M\ffi('huv. wMp§ hnlC ml('("\.als. C\� when (hen' l!o nmti(m. \\h) lhoulJn'( II "art' '�l-e inh:na1s') Intk"l:d, lint of 1he lCntt'i of s.entruJ rctati\'I1� IS that a IlU�\l\l' h('u\('nly body warp)! sl""'-"Cotime nearby, Kcpn:!'OC'nting w:trprtl sp.tn'·tim\! 11\ Ihn'(.' Junensill1l i\ Jillkuh I! I.. t'ASIl'r in t"'ll JimclhIOO'i, 11\ \\'hkh :oop'KC l'i !In.'a, hgun.' 2. .l.l 5011l)ws a 11I.1.\M\(' If

J.

¥r.\\ Ilatiunal

no reblive

h(a\ around the clu,>ter Figure

Special Relativity Stands Up to tke Test Cla.,sical

from one frame 10 another mo\ing relative 10 the fir\t at about 60 I..m/s. AI such a low �pced. the effeci 01 the frame motion wa" expected to be ..,mall. but they ruled OUI eflech almo�t 5 orders of magnitude '>maller. Even the icon E =

lI/e2 receives regular attention, A team of ,cienti�" from

uni\'er,itie� and lab,> in ,e\'eral countrie' recently mea,url!d the ma'>s change .....hen a nur.:leu� captures a neutron to become a different nudeu� and Ihen emi" electromagnetic rJdiation (NlIIlIrt>. 22-19 I)e"emhcr 2005. 1096-1 097). The emitted electromagneti" energy was found to equal the "hange III IIIC2 of the neutron Jnd nuclei "IIhm 0.00004";( Eimtein\ cJaim� seem 10 be holding up quite well Gravitational lenses: Applying the Bending of light

n

If il gra\Ilational field "an bend Ji!!hl. then light ,>hould b.

T '

to. ween it and Eanh,

no

I-

, (

1•

.tblt" to tIa\'cl from source to ot-o,en'cr by III/litiph' palh cUound a lIla�si\'e nhJe,,1. Indt:ed thi, happen" and II i,

I

kUlI'" n a, A;ra\,llatLOnal lcn�jng_ Figure 2 .40 " a superb

l

�xamplc Tal.cn b) the Hubble Space Tcle,copt'_ it sho..... fIlultlrle Im.1ges (I' the ,arne gala,y. The massive

. One- application is in \artin� lIU h

krJling massive objects

.htft u':.uclatcd with lhe expan'>lon of the unh

r. :lie,e disturbances are

With lar�e fI:dshifb can be '>cen len\cd by obj speed is c. 65. You fire a light ...ignal at 60" nonh of wC"..t. la) Find the velocity component.. of thi., .,ignal according to an observer moving eastward relative to you at half the speed of light. From them. determine the magnitude and direction of the light signal's velocity according 10 this other observer. (b) Find the componenh according to a different observer. moving westward relative to you at half the speed of light. 66. At' = O. a bright beacon at the origin flashes. sending light uniformly in all directions. Anna is moving at speed I' in the +x direction relative to the beacon and passes through the origin at I = O. (a) Show that according to Anna. the only light with a positive x'-component is that which in the beacon's reference frame is within an angle 9 = cos- I(v/e) of the + x-axis. (b) What are the limits of 8 as )I approaches 0 and as it approaches c:'! (c) The phenomenon is called the head light effect. Why'! 67. Using equations (2-20), show that

Section 2.7

By applying the relativistic velocity transfonnation to the left side of equation (2-23) and using the algebraic identity derived in Exercise 67, verify equation (2-23). 69. What is the ratio of the relativistically correct expres sion for momentum to the classical expression? Under what condition does the deviation become significant? 70. What are the momentum. energy, and kinetic energy of a proton moving at a.8e? 71. What would be the internal energy, kinetic energy, and total energy of a I kg block moving at 0.8e? 72. By how much (in picograms) does the mass of I mol of ice at ooe differ from lhat of 1 mol of water at O°C? 73, A spring has a force con< < I +

(c) Deduce gIrl from Newton'5 universal law of gravita tion, then ilr�ue that cquation (2-.1 1 1 follows from the

2 wi le'

(b) By construction. v cannot ex.ceed ... for il il did. "

re�ull.Ju�t a\ (2-.l0J duc... fwm (2-29).

.•

clock would run faster, oUlpacing the ground clock. Were il to move at

H(r) -

ground level. its time would faJl behind a stationary

= k(t

1').

No matter what k and b might be, the object would be al ground level at , =

and ils derivative I'U) into the given integrals, then

(b) Write a program. in which k and b can be easily

eltceeds lhal on Earth's surface. Were it merely 10 float

clock. Consider motion described by

Hrr)

carry oul the indefinite integration to obtain two func

0 and I = I . so we can vary these

-

on anomer, and [hat will also print oul the totaJ vaJue of ..11 when I = I . Use 9.8 mls2 for g. (c) Plot all

_iT

functions trom I = cases: (k, b) =

nly (" th("rmal motion ..1" .:hilfli�. In wall,

T

Figure 3.2 bpcriment �how� fhal (IS frequen.:y increases.

the blackbody

spectral energy dt'nsjly reache, a m31ti mum. then falls off C/as,ical wave theory predicts a divergence.

'" :s I �I !

CI:u..\K":tI wave Theory

bprnmenT...J data. and Planck's lheory

The quantum age dawned wilh the work of Max Planck in the year 1 900. Planck wa" trying 10 find a theory thill would explain blackbody radiation. All malerials emil electromagnetic radialion. because mey contain charged panicles Ihal jiggle around. and an accelerating charge radiates electromag_ neric energy. The amounr of energy radiared depends on [he average energy of rhe motion. which. in tum. depends on the lemperature. For example, coals radiale invisible infrared energy even when cold, but when heated. they emit more radiation. much of it in the red end of the .�pectrum. They visibly glow "red hot" Most materials, however, also reflect electromagnetic energy. A blackbody is defined to be any object from which eleclromagnetic radiation emanales solely due to the thermal motion of its charges. Any radiation that strikes it must be absorbed rather than reflected, hence the name. (The term must not be taken too literally. The Sun's surface, from which reflection is insignificant. is a blackbody.) While coal is a good approximation, fabricating a true blackbody might .�eem problematic. Imagine. however. an objeci with an interior cavity and a small hole connecling it to the exterior, depicled in Figure 3. 1 . Any radiation enlering the hole would refleci from the cavilY's inner surface many times, losing energy to the object al each refleclion. Essentially none would reflect back directly through the hole. On the olher hand. all areas of the inner surface contain charges in themlal motion. constantly absorbing electromagnetic energy and reradiating it as they jiggle around. They will furthennore be in equilibrium with the electromagnetic energy in the cavily-the charges and the radiation will have the same temperature, T The portion of the radiation leaking out of the small hole will be characteristic of this temperature, so the hole behaves as a blackbody of temperature T. Experiment demonstrate!. that the energy emitted by a blackbody, or equivalently a cavity. is small at low frequency, reaches a maximum, then falls again toward zero thereafter. This is illustrated by the experimental curve in Figure 3.2, which plots electromagnetic energy dU per frequency range df. known as spectral energy density. Classical theory, on the other hand. differed. If the electromagnetic radiation in a cavity behaves strictly as sinusoidally oscillating waves of arbitrary amplitude, the average energy ofa wave of any given frequency ...hould be kaT. (Appendix C discusses Ihis point and the other delails of classical wave theory and Planck's work thai we merely outline here.) MUltiplying this by a factor that accounts for the number of different waves per frequency range dfin volume V, the classical prediclion for spectral energy densilY i� dU - = kBT x -I 81TV ">

df

e'

Spectral energy density via cla-'i'�ical wave theory

Something i� certainly wrong here, for as Figure 3.2 shows, this parabolic function diverges as f increa...e ... without bound. If true, all malerials would radiate infinite power.

.\,21bc PhocoeJeI:IriC '"

Planck found that he could match the experimental dala with a I•.'Urious assumption: The energy at frequency f is somehow restricted to E = "hi, where " is an integer and h is a constant. The specitic error in cla.'.;sical wave thi..,(lry i!io in the average energy of a given wave. which b obtained b)' !Ote-grating O\'cr an a, is a characteristic of the particular metal. Table 3.1 lists some values (subject to variation. depending on impurities and other factors). If light were lOtrictly a wave, this effect should have several specific traits. First, if light of one wavelength is able to eject electrons, then light of any wavelength should be able to do it. Independent of the wavelength. the mte at which energy arrives (the intensity)--and therefore the rate at which electrons

Figurw 3.3

The photoelectric effect:

Light liberating an electron from a

metal surface.

•S(5

71

iIIiI.!'t�

.. . ...... t n " . ..

... .

-

2.3

7

...... P+niac. PIrtideI

Eo- Seooad.

if ... _.,. is low, then """" Ibough ....trons mighl sIiIl be .jecIed, a _

r

...... time lq should arise. Because • wave is diffuse. considerable

tUne

miPt be needed far enough energy to accumulate in me electron's vicinity.

1.9

2.2

•

_ .clOd couJd be ...... orbilnriJy buJe simply by _ing

• • oVl

..

.

(See &erciIe 16.) F'maUy. at any liven frequency. if the intensity is increueca.

die � electroas should be more energetic. A stronger electric field

sbouId produce a larJer acceleralion.

Imagine the experimenter's surprise when weak light of 500 om wave.. length ejects electrons from sodium. with no lime lag. while light of 600

3.7

wavelength cannot, even al

Drn

mon)' lim�s me intensity. Moreover. the energy of

the electrons liberated by the 500 om light is completely independent of the

4.3

intensity. Classically. this cannot be explained! In 1905. Alben Einstein proposed the following explanation: The light is

4.4

behaving as a collection of particles. called photons. each with energy given by

4.>

E = hI

Energy of it photon

(3-2)

h

where is Planck's conSIant. A given electron is ejected by a single photon. with the photon ttansferring all its energy to the electron and then disappearing_ multiple photons very rarely gang up on one electron. If the light's frequency is too low, such that the photon energy hJis le!'>s than the work function c/J. then there

is simply insufficient energy in any given photon to free an electron.

So none DrF

frrt>d. no moNer hoM' high 1M inlemily: no mailer how abundant the photons. (The phalon energy becomes internal energy or reflected light.) However, If the frequency is high enough. such that hJ > q,. then electron.., can be ejected. The

kinetic energy given to the electron would then be the difference between the

photon's energy and the energy q, required to free the electron from the metal.

(3-3) The subscript " max" arises because

cP is the energy needed 1O free the least

strongly bound electrons. Others may al ...o be freed. but less of the photon \ energy would then be left for kmetic energy.

Einstein's interpretation of the photoelectric effect explains not only the observation that a cenain minimum frequency i ... reqUired but also the other classically unexpected result\. I f a single photon-a particle

of concentrated

energy rather than a diffuse wave--doe\ have enough energy. ejection should be immediate. with no time lag. AI'iio. the electron's kinetic energy .,houJd depend only on the energy of the single photon--the frequency-not on how many !OIrike the metal per unit time (the IOtem.lty). In all respects. Einstein's explanation agrees with the experimental evidence. and the achievement earned him the

192/ Nobel Prize In phy\ics.

EXAMPLE 3 . 1 Light of 380 nm wavelength i\ directed at a metal electrode. To determine the energy of electron!> ejected. an oppo\ing elcctrO'ltalic potential difference I!. e"'lablJ"hed belween it and another electrode.

a,

"hown in Figure 3.4. The current of photoelec..

Iron, from one to the other i\ \topped completely when the potential difference i.,

3,2 n. �

1 . 1 0 V. D�tenninc (a) the work function of the metal and (bl the maximum-wave len�th light that ('an ejt.'Ct electron... from thi... metal. SOLVnON

(a) In the region between the electrode.... the electron... lose kinetic energ)' a' they � (1.6 X 10 1'Jl Cl gain potemial energy, II a potential energy difference 01" ( \ , 1 0 V) "'-' \ .76 X 10-111 J = 1.10 eV i, the most they ('an ...unnount. their

qV

.. ..... 'It

...... 1.. Cunonl_ .... 01..,..-. _ _ lbo _ilia potenlial encfl)' diffmmcc equIs 1be muimum kinetic eneru of the photoelectrons.

\..inetic energy k;\nng the fiN electrode mu,t be no larger than 1.10 eV. The potential diff�rence that barely ,tops !.he now is known a... the stopping po,rn. tiat Using equalion (3-3).

1 .76 X 10 1 9 J = (6.63 X 10-'� J . s)

�

q, = 3.47 X 10 IIl J

=

(

' X IO' m "

)

_1 -_

_ _ _

----=- __

380 X 10-Q m

2.l7eV

.- '"

(b) I f the wavelength o f the light were increased to A', the frequency-and thus

the photon energy-wou ld decrease. The limit for ejecting electron ... is when

an incoming photon has only enough energy to free an electron from the metal. with none lell for kinetic energy. Agai n u..ing equation (3-3), 0 = lif' - q, = (6.63 X lO-H J , s )

(

3 X 103 m1s

A

'

-

.

'

� A' = 573 nm

)

- 3.47 x lO-wJ

Wavelengths longer than 573 nm have insufficient energy per photon, so no

photoe lectrons are produced. The maximum wavelength for which electrons are freed is called the threshold wavelength, and the corresponding minimum

frequency is the threshold frequency.

The central point in Einstein'S explanation of the photoelectric effect is that electromagnetic radiation appears to be behaving as a collection of parricles. each with a discrete energy. Something that is discrete, as opposed to continu ous, is said to be

quantized. In the photoelectric effect. the energy in light is

quantized. EXAMPLE 3 . 2 How many photons per second emanate from a 1 0 mW 633 nm laser? SOLUTION

For each photon, E =

c

hf = h-A =

(6.63 X IO-" J · ,)

(

3 X

10' ml'

)

633 x I 0 9 m

=

3.14 X 10-I'J

To find number of particles per unit time, we divide energy per unit time by energy per particle:

number of particles time Clearly. photons are rather appear COnll PUOUS.

=

10

10-3 J/ ' -,-"" :-:-'= 3 . 1 8 X 1016 partide!J� 3.14 X 10 I' J/ partic\e X

-

-

"small " and it is easy to see how a light beam CQuid ,

"

t

+

71 CMptet 3 W.ves and Particles I: Elcctromagnetic Radiation Behaving as P'.utides FJgurw 3.5 In a nighl vision device, a light image becomes an image of free electrons. amplified in a multichannel plate and then reveaJed on a screen.

The photoelectric effect has long been used in simple Jight sensors, where light intensity registers as a photocurrenl, bUI it is also used in more sophisti�

cated ways. Let us take a look al one. REAL-WORLD EXAMPLE N I G H T V I S I O N

Incoming

light

By "replacing'" a photon with an electron. whose charge makes i l easier to "amplify." the photoelectric effect i!. a common fronl end on optical imaging systems. One example is the night vision device (NVD). A typical NVD is shown schematically in

Ob}ccti\'e len�

Figure 3.5. An objective lens focuses an optical image onto a thin piece of material. called a photocathode. where the photoelectric effect transforms it into an image of freed electrons. Naturally. the dimmer the light. the fewer the photoelectrons. Amplification is accomplished via a microchannel plate. This element has hundreds of thousands of channels per square centimeter. An electron entering a channel at

:;;::".,�!t". Ptu.!oeh....ul!n� ........ . /.k. •, pn'lloccd ...

.. :; • •

,

•

�."

..

., ••. .

.... �

;H'C

Pl that the deutemo, h.we equal I'..

Thu,

FlgUN 3 . 1 ' In

pair production.

I
D.

the bottom one as a wave. because A

A simple application of the criterion to electromagnetic radiation is Ute ca!:!> along the patient's body, look.ing at each ..lice. a three-dimensional image of in learning what is inside things without breaking them den),.;ty versu" p6,ilion emerge�. Figure :\. 1 5 "hOWl; (a) the open, probing everything from human bodies to construction basic layout of a PET machine and (M an actual image. materials to superconductors. While the century-old tried and-true method of Section 3.3 is still the leader, the conven Figure 3.15 Positron emis�ion tomogrolphy_ (a) A tracer emits a tional X-ray machine, with its hot filament sealed in a tube, is rather unwieldy. and the tube's lifetime is often short. The positron that annihilate, with a nearby electron, yielding two pho tons that place the annihilation along a line between the deleCtors. i huge and growing field of nanotechnology-applications n which some crucial element is measured in nanometers Many lines combine to produce a two-dimensional image of a slice. (b) Multiple slices produce a three-dimens.ional image-in may provide a new method. In work conducted at the this ca-.e, "hawing high tracer uptake in the hver and kidneys. University of North Carolina, carbon nanotubes are allied (a) with the quantum-mechanical effect of '-field emission" to produce a beam of X-rays strong enough to replace the Two phNl)n.. from an electron conventional X-ray machine in several uses. A nanotube is a regular meshwork of carbon atoms. forming a cylinder of only aboUi I om radius, and is closely related to many other all-carbon structures discovered in lhe 1980s (see Chapter 10). These continue to surprise us with new, remarkable properties and are a very hoi IOpic of physics research. In the X-ray source application, bundles of naootubes are deposited in a lhin layer on a metal disk. Electrons are coaxed from the layer toward a target not by heating a filament. which wastes power and produces electrical noise, but by field emission, a room-temperature way of producing a flow of electrons that relies on the quantum-mechanical effect of tunneling (discussed in Chapter 6). The resolution of the new tech nique is ell.cellent, and another potential advantage is faster response time for tracking moving objects. (See Vue et aI., Applied Physics Letters, 8 July 2002.)

r.�!!t;t·./

...., po,;lwn aIlmhilaliun are

.

(b)

Medical Imaging with Positrons As noted in Section

3.5, once a positron is produced, it soon engages in pair annihilation, simultaneously yielding two photons of a characteristic energy (see Exercise 42). This trait is ell.ploited in an increasingly common medical imaging procedure known as positron emission tomography (PET).

dch:l.:lcd �Imu\l.m�·ous.ly.

Chapter 3 Summary

fundamental daim� as possible. explain to your friend what evidence Ih.i.� pro\'ides for the particle nature of light.

3. You are conducting a photoelectric effect experiment by \hining hght of 500 nm wavelength at a piece of metal and detennining the �topping potential. If. unbeknownst

Electromagnetic radialion behaves i n some situations as a col Itttion of panicles-photons-having the paniclelike pmper tiC:. Appealing to � few 92

6.

it? (Imagine the two objects are hard spheres.) (c) Is it reasonuble to suppose that we could know this'l Explain. An isolated atom can emit a photon. and the atom's

aspects suggest a panicle nature? 10. A cohere", beam of light slrikes a single slit and pro duces a spread-out diffraction pattern beyond. The number of photons detected per unit time al a detector in the very center of the paltern is X. Now two more slits are opened nearby, the same width as the original, equally spaced on either side of it, and equally well

from I� blackbody, t: is not lhe c:onec1 speed. For ,..u.. ation moving uniformly in all directionl, the averap nm'ponl'''' of ,'elociIY in a Biven diftlctioo is

llluminah:d b)' the beam_ How many ph-Nons will he detected per unit time at the center d�tector now',' Why?

Exercises

Section 3.2

Section 3.1

11. For small ;:, ('- i� approximately I + ;:. (al U\e this to

show thai Planc\"\ spequare mitlimeter per o;«ond. (a) Whal is thr inrens.ity of the light at the cenler of the �reen'l (b) A second ml is now added \'ery do:;e to the fina, How many pholons

will be (!eleCled per "IU3re millimeter pet � at the center of lhc scrtt'n now'1 .5. Electromagnetic "waves" strike a single slil of 1 IU'I width. Determme the onRulorfull width (angle from first minimum on one side of the ceniCT to firsl mini mum on the other) in degree� of the central diffTaction maximum if the waves are (a) visible light of wave length 500 nm and (b) X-rays of wavelength 0.05 nm. (e) Which more clearly demonstrates a wave nature? '6. A bedrock topic in quantum mechanics is the uncer tainty principle. It is discussed mostly for massive objectS in Chapler 4, but the idea also applies to light; Increasing certainty in knowledge of photon poSition implies increasing uncertainty in knowledge of its momentum, and vice versa. A single-slit pattern that is developed (like the double-slit pattern of Section 3.6) onc photOn at a time provides a good example. Depicted in the accompanying figure. the patlern shows that pho tons emerging from a narrow slit are spread allover. a photon's x-component of momentum can be any value over a broad range and is thus uncertain. On the other hand. the x·coordinate ofposition of an emerging photon covers a fairly small range, for w is small. Using the single-slit diffrnction fonnula nA = w sin 8, show that the range of likely values of P:.:' which is roughly P sin 8. is inversely proportional to the range w of likely position values. Thus, an inherent wave nature implies that the precisions with which the particle properties of position and momentum can be known are inversely proportional.

Comprehensive Exercises n. A. phocoo has die lime ID('!IIIeID I lm . .. ..... ..,.

ina at Ir/J mls. (a) Determiae me pboIoa'a � (b) WhaI is the nbo oflhe ItiDetic � ol me rwo?

(NOk.' A. phocon il all kinotk 0ftCtIY.)

48. A. photon and an objed of maJI III haYe tho MmCI

momentum p.

(al A!>suminS that the massive object is mavio. slowly. so that nonrelativistic fonnulas are valid. find in terms of m. p. and c the ratio of the mauiw object's kinetic energy to the photon's kinetic energy. and arsue that it is small. (b) Find the same ratio found in part (a). but usins rel ativistically correct formulas for the massive object. (Notr: £2 "" p1c2 + m2c" may be helpfut .) (c) Show thaI the low·speed limit of the ratio of part (bl agrees with part (a) and thai the high-speed limit is I . (d) Show that at \'f'n' ruSh speed. the kinetic eneqy of a massive objecl approaches pc.

49. Radiant energy from the Sun arrives at Eanh with an intensity of 1.5 kW/m2. Making the rough approxima

lion thaI all photons are absorbed. find (a) the radiatiOll pressure and (b) the total fon;:e experienced by Eanh due to this " solar wind." SO. A flashlight beam produce� 2.5 W of electromagnetic radiation in a narrow beam Although the light it pr0duces is white (all visible wavelengths). make the sim plifying assumption that the wavelength is 550 nm. the middle of the visible spectrum. (a, How many pboIons per second emanate from the flashlight? (b) What fon;:e would the beam exert on a ''perfect'' mirror (i.e one that reflects all light completely)'? 51. The average intensity of an electromagnetic wave is whcre Eo is the amplitude of the electric-field ponion of the wavc. Find a general expressioD for die .

.•

L"" posiuve P, p..

=0

�etftij.

L""

negative P,

Single·slit n i tensity

Ught wave

'"

photon flux) (measured in photonsls · mI) in ECmlS of Eo and wavelength A. 52, Show that the laws of momentum and energy conser-. varion forbid the complete absorption of a pbocon by • free electron. (Notr: This is not the photoeIcctric effect. In the photoelectric effect. the electron is not fn:e; the I!II!DI participates in momentum and encrgy CODgeI'YItion.) 53. An clectron moving to the left at O.Be collides 1riIb ... incoming photon movin. to the right. Aftr:r die coUi sion. the clcctron is mavinS to the riP at 0.& aDd aD outgoing photon moves to the left. What .. die wne- length of thc incoming pboloo? 54. An object moving to the rigbl at O.Be illIb'Ddt bad-cJ8 by a photon of wavelengd:l ,\ moving to tbe aft. The

96

a..peer 3

Wa\ICI and Particles I: Eleccromagnetic Radiation Behaving a icles .. Part

objcct absorbs lhc photon (Le., the photon disap�) and is af'lcrward moving to the right at o.&-. (a) Deter· mine the ratio of the object"s ma.o;s after the collision to

its mass before the coUision. (No": The object s i not a "fundamental particle," and il� ma.o;s is therefore subject 10 change.) (b) Does k.inetic energy inl,.Tease or decrease'!

55, Photons from space are bombarding your laboratory

and smashing massive objects to pieces! Your detectors indicate that two fragments. each of mass mo' depart

(;;uch a collision moving at 0.& at 60" to the photon's original direnion of motion. In terms of mo' what are the energy of the cosmic-ray photon and the mass M of

the particle being struck (assumed initially stationaryl'.'

calculus. For inslance, wa\'e." on a string obey the "wave equation," a PattiII differemia' equation in posillon and rime, and Ilght obeys MaxweWs «:qua. lions. similarly inml\'ing (:alculu.. wilh bolh position and lime variables. 1be dislinclion belween particle and \\'3VC nalure..; also govern� ":hal questions we ask. For particles. it is oflen. "Where is il going? When will II gel there?' For wave.". \\Ie ask. "Whal i..; il\ amplitude? Whm is ils wavelength? How '�PJead our is it? Where is il zero"''' We consider the.�e questions �oon, and afterv.'ard we introduce lhe SchrOdinger equarion. the equarion obeyed by marrer waves. Firs!' however, is the Big Quesrion: Just what j� a mailer wave. and how do we know il eveQ exists? Nothing illustrates the poinr better than the doubJe·slir experiment. Figure •. 1 'nlensily palfems when an electron beam �Irikes various �lils,

4. 1 A Double-Slit Experiment

a'

Imagine a beam of monoenergetic electronsl siriking a barrier with a slil beyond which is a screen rhal regislers each electron's arrival by prodUcing Ihf{'ltlgh 1I \\oIJ... llil small flash. When the slil is "wide." as shown ar the lOp of Figure 4./. the beam passes straight through and produces--clectron by electron-a stripe on the screen essentially the same width as rhe slit. Bul with a narrow slit, we find electrons registering sporadically OI'er fhe entire screen. Although this spread_ ing alone is hard to reconcile with the notion of electrons as strictly panicle\ if we add a second slit, the conclusion is inescapable. Suppose, then, that we add a second narrow slit. Again, with either Slit "". ",",I, open alone. electrons are detected sooner or later at all points on the screen. 11.[[1"\\ shl. But when both are open together, we see certain places, where electrons had tk'('llon, OlITI\C bee" detected with either slit open separately. where electrons are now never "'t ry one JII Uh, Ihl.' Krrtn, Thty detected. Opening a second " door" decreases to zero the number of electrons uon'j J \ " !U .lilly arriving per unit time at specific. regularly spaced locations on the Screen f'Illllls even at such low intenSity that rhey must pass through one at a time! This is impossible to explain if electrons are simply particles passing through one slit Single narrow Sltl 0 .... hUI whell ' or the other. A particle passing through one slit would not suddenly have rea ,hIs JI"l: npen. son to avoid specific locations on the screen just because another slit had been f\ thefl' lin' \ momentum. in no conceivable e xperiment would something so big as an airplane e,'er beha\'c as a wave,

In ordinary �ituations, the wavelengths of matter wave!. are short enough to ensure particlelike behavior. because Planck's conslant is so small. How. ever. as an object's momentum apprOtnM..'1Uft' \�f the aklm5 ....,... 01 on the surfik."e. Let us taU • lool at tht a"\."C'kratina p:ltential, in lht� apph�·.rims.

an

....pptvi"9 the f'hy"lclo

2

Often referred to

as

(4-14)

the Heisenberg uncertainty principle, for i ts discov

erer Werner Heisenberg (Nobel Prize 1 932), it is a shocking revelation. There

is a theoretical limir on the preci sion with Which some familiar quantities can

be known �imuhaneously. If we know a particle's position t'XDCtly. we can .1p. = 00). if momentum is know nothing aboul ils momentum (,1x =

0 :::::)

k.nown e1l3clly, position is completely unknown. The plane wave is II. good

example of the latter case. This fundamental matter wave ha.s a wa"'clcOJth perfectly regular throughoul space, gi\'ing il a perflXtly p�i� momentum, but It represents a particle equally likely to be found anywhere. A property in which there is no uncertainty is said to be momentum IS well defined (j,p( (6.x

::::

00).

-=

well defined.

For the plane W8\'e,

0), bUI position couldn't be mort' undefined

Don't be troubled by the inequality in {.t�I ·n-there is no unl:ertainty

about the uncertainty principle. The � rellecls the simple fael that then.: i)o. a particular wave shupe, called a

Gaussian,

also known as a bell I:urve, for

which the product of lIncertainties is a minimum. Figure 4.14 shows a Gauss ian wave form, a constnnt C times a " Gaussian factor" constant. It is mallimum at x

=

e-(,J1d

where

e

i)o. a

0, falls off toward 0 symmetrically as

.\'

becomes large, and the rate of fall-off depends on e. If e is large, the wave form is broad, falling off very slowly; whereas if e is smail, the W Wa\le�

I

I

This value happens to equal the experimentally veritied minimum energy. and the radiu� is also the correct most probable radius at which 10 find the electron (which doesn 't rest on the proton. whose rlldiu� is 10.000 times smaller). That they agree so closely is an accident-we have made many approximations-but it is no acci. i a dent that they are of the correct order of magnitude. The uncertainly principle s powerful tool.

The Uncertainty Principle in Three Dimensions The quaJitative idea behind the uncertainty principle is [he same in m ultiple dimensions as in one. The more compact the wave along a given axis, the less well we can specify the wavelength and therefore the momentum component along rhot axis. The result is a logical generalization of [he one-dimensional

result:

Note that the dimensions are independent. The single-slil pattern of Figure bears this out. Passing through the slil, narrow along x only. produces a large uncertainty in px, indicated by the subsequent detections being spread

4.12

over a large region of the screen. In the y-direction, the aperture is wide. so less is known about this component of position, and there is correspondingly little spreading of the pattern in that dimension. Thus, apy and Il.x can be small

simultaneously.

The Energy-Time Uncertainty Principle The momentum-position uncertainty relation is, a[ heart, a mathematical rela tionship. A width in space is inversely proportional [0 a "width" in the spatial frequency physics,

k

p =

= 21Ti"A (see Section 4.7). It is the fundamental wave-particle

lik, that takes it the final step. The same math relates a width in

lime to a width in the femporal frequency (j) = 21T1T. With E =

nw, the corre

sponding physical consequence is

Energy.·Ime unc)rti inty prinCiple

(4·1 5)

How do we interpret this? If a stale, or even a particle, exists for only a limited span of time. its energy is uncertain. One example is the fleeting life of certain exotic subatomic particles. Their lifetimes can be quite Short-less

than 10-20 seconds-and this leads to considerable uncertainty in their mass! energy. Another example is the state temporarily occupied by an electron as it jumps down through energy levels in an atom. Because the state is occupied for a finite time interval related to

At, its energy is uncertam by an amount 6.£ inversely

at, which in turn gives rise to an uncertainty in the energy of the

photon produced when the electron drops down. This effect contributes to the broadening of atomic spectral lines (see Exercise

72).

4. 5 The Not-Unseen Observer Let us spend a little time ..ummanljng the limitations thai quantum mechanics

quantify

the effcch of places on our knowledge. Although we won't begin to external forces until Chapler 5. if the forces are known. the SchrOdinger equa tion may. in principle, be solved for the wave function of a massive ohject, which contains all information that can be known. But this isn't everything we might expect classically. The uncertainty prinCiple. for IOstance, ..ays that a wave funclion of simultaneously precise momentum and position is a theOl-eti cal impossibility. It follows that any experiment or measurement thai precisely

determines position must resull in a state 10 which nothing is known about the momentum and vice versa. Suppose we carry oul an experiment on a particle, experiment A. applying external forces in such a way as to determine both its po�ition and its momen· tum as precisely as possible, such that .6.x 6.p =

h/2. Assume. for the sake of A ' We have found

discussion. that ax is 1 00 lAm. and call the wave function '"

the wave function. but we aren't satisfied. for we haven't really "found" the particle-its ·'Iocation." All we have is this mysterious probability amplitude. We conduct another experiment, experiment B. in which the particle reg· iSlers its presence at a detector at a definite location. We rejoice-we have found the particle. However, there are no "point detectors." If the detector', width is smaller than the

100

lAm position uncertainty in \If

A' then we have

indeed narrowed down the possible locations, but we haven't established a

location with complete certainty. Yes. we have reduced the uncertainty in position, but if this is so. experiment B has changed the wave function. At the very least, it has increased !J.p. If we repeated this pair of experiments many times--experiment A to establish lhe initial wave function \If

A and experiment B to "find" the particle

experiment B would find it at various locations within the 1 00 11m uncertainty

A' and the number detected at a given location would be proportional to j\f! 12. In essence, we would simply verify that I'" I:! is propor A A

of wave function '"

tional to the probability of finding the panicle after experiment A. But because

experiment B changes the wave function, we can't "watch"-repeatedly find

A'

the same particle while preserving a single wave function 'I' The double-slit experiment, depicted in Figure 4.17, is a good example of these ideas. In effect, the slits are an experiment A, establishing an initial wave function

"'A beyond them, and experiment B is the detection of a particle at Figure 4.17 Eltperimem A eSlablishes

'ITA' which repealed eltperiments 8

Eltt""" beam

1111 W

verify.

112

Cftapter .

Wavea and Panicles II: Maner Behaving II!; Wavc�

the screen. By sending in a beam of particles one at a time, we are carrying out experiment A then experiment B repeatedly. Where "'A is large, experiment B registers particles in abundance; where 'IfA is zero, experiment B registers no

particles. We cannot conduct an intermediate experiment, determining which

slit a given particle passes through, and yet hope to observe the interference pattern exhibited by "'A' for this intermediate experiment would itself alter the wave fUDction. (A recent confirmation is discussed in Progress and AppHca� tions.) To observe interference at the screen. we must allow each particle's wave function to pass through both slits simultaneouslY--llS k

Ju..t a pure plane wave A.' The Alk) pll\t !>ay", eill&:t1y the: ..arne thing. It "a)" that the ""urn" in l4-21) I" 1u"t a ,ingle plane wa\'c of wa\'c num· ber All. At the other e,lrenle. when t: i, "cr)' ...mall. tJI(t) i... compa.":l. but AIll is corre�pondingly \'cl)' broad, We conduJe that making a wa..e fU1\(;;lilln 4tl \l more compact ma),.e, ih nmge of wa\eh:ngth.. les.. l'Omplll:l and ..ice \CnMI, As 10 the phy..ic,. we l..now that r :'> Irk. '0 a function iCon)!.i)!.ting 0\· a tun�e e numberimp\e probabiht)'

-..../2

The imaginary plll1 dis;tppears. the integral of an odd function (sine) over an interval symmetric about the origin. Carrying out the remaining integral,

AI' ) =

C sin(+kw/2) - sin(-klV/2) -

21T

k

=

C�in(k\V/2) 'IT

k

(4-25)

amplitude of a single slit.

, ,0',,,,,_-\,,,

\

c

w

Figure 4.28 plots A(k), and its appearance is conspicuous. Evidently, there are certain wave numbers. when A(k) = 0, not "present'· at all in !jJ(.l). Wave number is related to momentum, and in the case we consider here, it is the x-component of momentum Figure 4.28 The Fourier tran.,form the "sideways" momentum. With certain values missing, there should be certain A{k) of the single-\lll·(knl\cr ('\po.'nn1ent " hcam III c\e.:t!'l.ln.. ol�..:e\erule,l lhl\.\\Igh :'i-l V ,.. Ihn.-..:teU l1nnllllll), 111 11 !lId.el \uriIll:C. IInu ..\n>ng retl«:1111n "

q,III '\0 U"Ulg the Bragg 101". 'h,,", thal th,.. Iml'he.. a "pallllg f} III md .. cI ..):: ' f ::.J

rrer'l !l ( Id

,·e .w. "

) 1,

,

I.'t'

.J,m,j·

"

... ....;.J:

I'

,

•

,

"

.

('I'

,

,

r

1z2 r1 '!' (..l.. I)

!Jr.

• 35.

",'

'l'lX, t) ]s bv ,Jdimtion "" I t) I 'Ii2(X, I How i� the- cumplt!x appro'".:h l:h"lsin In �eci n -L3 mor.; conwnienl than the a.lternative pmed hert! J In Set·tinn 4.3, WI! claim that in analyzing ek:(;tromdg netl!': wavc!-" we (;oulrJ hanrJle the field� E and B \Ahere

f

36,

37,

G · dl

'0J

= !... c a,

G ' dA

waves would have to obey lhIS' COil). Electromagnetic 0 . , oes thiS change of approach make plex equatIon. and/or 8 complex? (Remember how a complex n £ umber is defined.) An electron moves along the x-axis with a well.deli .fIed momentum of 5 X 10-25 kg·mls. Write an ev"preSSIOD " the matter wave associated with lh'IS eee. descnbmg I tron. Include numerical values where appropriate, A free particle is represented by the plane wave funcll, flJl W(x. r) =: A exp[i( 1.58 X 1 0 1 2 x - 7.91 x 1 016,)' where $1 Ul11ls are. understood. What are the po.' .... minimum kinetic energy.

47. An electron in an atom can "jump down" from a higher energy level to a lower one, then to a lower one still. The energy the atom thus loses at each jump goes to a photon. Typically. an electron might occupy a level for

a nanosccond. Whot uncertainty in the electron·s energy does this imply? 48. The rfJ is a subatomic particle of neeting existence,

Data tables don·t usually quote its l ifeti me. Rather, they

quote a "width," meaning energy uncenainty, of about

ISO MeV. Roughly what is its lifetime?

49. A crack between two walls is 10 cm wide. What is the

angular width of the central diffraction maximum when (a) an electron moving at SO mls passes through? (b) A basebnll of mass 0.145 kg and speed 50 mls passes

taint)' I'or a typical malTOscopic object is generally ao much smaller than its actual physical dimensions that. applying Ihe uncertainty principle would be absurd. Here: we gain ",-nne idcu of how small an objccl would ha\'e to be �f(lre quantum mechanics might rear ilS head, The den�ity \If aluminum 2.1 X HP !tgIml. is

.

typical 01" M)lid� and liquilh around us. Suppose we could nam}"" dl'IWn the velocity of an aluminum sphere 10 ..... 'Ihin an uncertainty of I � per decade, How small ....ould . il ha\e U1 be f\u Ib po�ition un..enainty to be at. lea,t a� largo: a... of ih radiu�'?

-I,!"k

52. A particle i, conne..:ted to a spring and undergoes

one-dimen'ional m(ltioR.

(a) Write an e"pr.:s...ion for the total lkinetic plus

potential) energy of the panicle in terms of ilS pl)�ition .l, i ts rna,s m. iu. momentum force constant I(' of the sprin g .

p, and die

lb) Now treat the particle a., a wave. Assume Ihat the:

product of the uncenainties in position and momen

tum i, governed by an uncertainI)' relation

� fl. Aho a.\!'ume that because x Is. 00

through? (c) In each case, an uncertainty in momentum

6.pSl"

is introduced by the "cxperiment" (i.e., passing through

a'\lcragc, 0, the uncertainI)' � is roughly equal to a

the slit), Specifically, what aspect of the momentum

becomcs uncertain, and how does this uncertninty com pare with the initial momentum of each?

,

typical "alue of �l"\. Similarly assume that ap Eliminate

p in fa

'\lor

;a

\PI.

of x in lhe energy e:xpftSSioo.

(c) Find the minimum possible energy for the wrIe.

50. If things really do have a dual wave-particle nature, then if the wave spreads, the probability of finding the particle should

:::::

spread proportionally, independent of

the degree of spreading, mass, speed. and even Planck's

constant. Imagine that a beam of particles of mass m and speed l', moving in the x direction. passes through a sin

gle slit of width w. Show that the angle 6t at which the first diffraction minimum would be found (nA = w sin (In'

53. The energy

01' a particle of mass m bound by an UIIUIWIl

�pring io;, pl/2m + b:c.

(a) Clas...ical ly. it can have zero energy. Quantum

m.:chanicall)'. however, though both x aDd p .e "oo avcr.lge" lero. it. � \:.

ted photon doe'i thi, impl)i'? INUIt":

{�

t:1\ce-··;,e., the energy of the photon-but he� it m..an'i the wu-t"fla;rll\' in that energy difference.) {b) Tn what •..tnge in wa\t'length, does this correspond'� {A... m,ted in Exerci-.e 25"7, the uncertainty principle is one C\)ntrihutor III the hroadening of 'ipectral lines.l (c) Obtain a general fonnula rel ating loA to �,.

lal s S

lal > ,

73.

where C is a constant. (a) Find and plot versus f3 the Fourier transform A(/1) of this function. (b) The functionft, ) might represent a pulse occupying = position) or finite lime (cr either rmite distance = time). Comment on the wave number speclrum f i cr is position and on the frequency speclrum if is time. SpedficaJly address the dependence of the width of the spectrum on 8.

a (a

a

69. A signal is descri bed by the

function

D(t} = Ce-l d/T (a) Calculate the Fourier transform A(w). Sketch and interpret your result . (b) How arc D(l) and A(w) affected by a change in 1'7 70. Consider the following function:

-00
+�__"'_ U ___'_,_--c t,,"''-idd('n region.

_ _

-A

-

is

I"

0

Position

x

+A

141 Chapter 5 Bound Sa.us: SI1DP� Ca,-.es

mass reaches the poin! x = +A. the energy is all potentjaJ and the kinetic energy is O. This is a turning point. The mass is stationary, but there is a force

on it to the left. so it must return toward the origin. It cannot proceed to ValU es If it did. the potential energy would, of course. be larger,

ofx larger than

+A.

but since the total must be con�lant. the kinetic energy would have to be nega. tive. Because for any given E. the mass is confined 10 values of x " inside" the

pOiential energy plot. the shaded region outside is known as the classically

forbidden region. Figure Atom

I

5.3

shows a different system in which bound states are possible. 2 moves in response to their shared

is fixed at the origin. while alom

eleclrostatic interaction. We won't study the origin of this potential energy. We simply note that atoms, due to their dispersed orbiting electrons and compact positive nuclei, do not attract or repel as simple point charges. The potential energy is positive and large for small intera{Omic separation x, and has a mini. mum value that is negative. then asymptotically approaches tive side as x --...+

00.

At small values of x, atom

0

from the nega.

2 is strongly repelled by atom I

(F = -dUldx > 0), and aI large x, it is attracted (-dU/dx < 0). Accordingly, a bound, oscillalOry state-a diatomic molecule-is possible, but it depends on the total energy. If the tOlal energy is negalive,2 labeled Ebound' atom 2 will be

bound. osciJJating belween turning points X(l and xb. [f the total energy is posi. Eu bouml' atom 2 has only a single turning point. Were atom 2 n

tive. labeled

moving IOward atom I . it would momentarily stop at this point, then mOve away. never to return. Although Ihe potential energy does increase as x

becomes very large, so that alom

2 slows due 10 the attractive force as it moves

off, there is no outer turning point where the potential energy rises to meet the total energy. This situalion differs from the simple mass on a spring, for which there is no total energy large enough for the maSS 10 be unbound, because the

Figure 5.3 Energy versus position for the interatomic force

between a large atom fixed at the origin and a small one free to move. Atom J

o

'''''''

Turning / point

o

j 01-1-;""'-----1,--;''----+ I�-----Turning \:/poinlS-......, vio l-h � ;.,

�'iY ma} be n��ali'e due!u the u,ua.! fn:c<Jom 10 choooc lero poIenual en�rg}·. We ,'1Ioos" tlllO' ';lII\f), it 11\ the' I'Cgi\ln bt·twall the wail)o and Ihclt t'lisurilitt thal thc O\'C'r all ..... a\(' lun.. titln .' l.'ontlnul'U' In th\' (\'t:iul\ hclwee wbich I$. never O. WbeD Re � I) is 0. _ '9\x. ., .. whereabouts is a fact in the real situations to which quantum mechanic! i!> muilllUlTl. ItId \'lr;e wna. We miPl va.Iia • applied. We cannot, for example, say exactly what an electron orbiting a lluantllln-� aanding __ &'1 . .. _ lpinning ;d)out!be,,;.....u. ill die Re-1II1 ,.... nucleus is doing. Given these unavoidable limitations in the micro�opic

ignored in our solution of the SchrOdinger equation. The assumed simplicity

all

""

114

CNtpt.r 5

Bound Stales: Simple Casn

world. the predictions of quantum mechanics have succeeded. While those of classical physics have failed. Sight unseen. the "particle" is a wave. The standing wave of minimum energy is known as the ground state 8fId it.. energy as the ground-state energy. It is in the ground state that quantum. mechanical behavior deviates mosl from the classical expectation. Maq Important, the kinetic energy is nol O. A bound particle can�ot be stationary. As we learned in Chapler 4. this would violate the uncertamty principle, for having a position uncertainty comparable to L and a momentum identiCally 0 is impossible. Another deviation from the classical is that in its ground stale the particle is most likely to be found near the center. Classical1y. if we Were t� "rum on the lighls" suddenly and catch the particle somewhere in the course of its constant-speed back-and-forth motion. it is equally likely to be found anywhere within the well. On the other hand, the correspondence prinCiple (Section 3.4) says there should be a limit in which the particle behaves classi. cally. This limit is at the "other end"-large n. Larger n correspond to shoner wavelengths (An = Wn), and the shorter the wavelength, the more paniclelike and classical the behavior should be. As we see in Figure 5.8, the larger the n the more evenJy the probability of finding the particle is spread aver well-the classical expectation. Before we apply what we have learned, we reiterate that the wave func_ tion tells us probabilities of finding the particle at various locations. The inte_ gral of the probability density (probllength) over aU space is the total

th�

probability of I . So to detennine the probability of finding the particle in Some restricted region of interest, we simply integrate over that region.

EXAMPLE 5 . 1 An electron is confined in an infinite well. in the ground stale with an energy ofO.JOey. (a) What is the well's length? (b) What is the probability that the electron would be found in the left-hand third of the wel1? (c) Whal would be its next higher allowed energy? (d) If the electron's confines were roomier, L = 1.0 mm, while its energy remained 0.10 eV, what would be the probability of finding it in the well's left-hand third, and what would be the minimum possible fractional increase in its energy? , ,

(a) In the ground state, n = I . Inserting values in energy expression (5-16), we

obtain 0.10

X

1.6 X IO-L9 =

] 21T2(J .055

J

2(9.1 I

X

X JO-34 J ' sF

10 3 1 kg)L2

(b) The probability density is f¢(x)J2.

, (fL

I¢n (x)1

�

"'''

)

- sinL L

'

=" L = 1 .94 nm

2

"."x

L

L

� - sin'-

As noted above. we sum lbi!. over the region of interest, x = 0 10 x = U3.

probability =

J

pcobability length

dx =

J/� (x)/ n

'

dx =

J o

IJJ

2

-

L

sin-'

(",TX) -

L

dx

'., c.. I , _10 10 . ....-'""' - -

I'"

•

o

I 2n1f -- Sln�3 2mI' 3 I

- _.

�[� 6

L

-

L

';0( 2o"/3) 4,1'11"

J

Thus. for an electron in the ,. - I !>tate, , , I 1 21r probability - - - - !Oin:\

2.".

3

3&

I

3

- 0.137

�

0.196

The probability IS les!. than the classical exp«tation of one-third, and it a,n:<s with the probability density plOh of Figure 5.8. In the ground stale, 1I/I(.t)l! is largest In the center and smaller near the walls. In fact. the probat'lility of finding the electron in the Cf'ntu third must be I - 2 X 0.196 " 0.609.

(c) The energies are proportional 10 ,,2, '0 the n

ground-..tate energy.

...

2 stale is 0.40 eY. four time!'. the

ed) To evaluate the integral. we need fl. 0. 10 X 1 '6 X 10

19 J

n1112( 1 .055 X 10'34 J ' S)2 "'"

2(9. 1 1 X 1 0 lI kg) never inliOltc,

de's tntrgy. Showing it invol\les some algebraic: gymn.....

_ .... .

__ ..... c-.

... .. .. _ _ _ _ _ A ..c.e .. • 7 '._ .. .. .... _ _ .... ·C'-AI. •

••

•

-lh '*

0rIphUaf: .1 0fW w.y W 1ft' how lIIqUIuI oo � 'l-!l) Impl.a

.... y quantl, ..,lOn It I, �tl all. an PCTl"IlC' 10 �)W thai aflC'r

lAIfttlhJ thco delllnlllo", of .t and ell from eqll:lIIOn, (� I " and

('\

U,

r"o«'U Al,1

� ,-..;

,

:U l

,

C·

I ,W

lL .

I.

Il.

lL .

•• ••

•

I. l :\,

)

••

20) and "")'1"1 out \Orne "pot'll&. '0\(' lI:.:an .. ka.s1 I3OLtIC'

Vo an oq....Uon (5.�l). ""

..

... ... 0( _ __ .. .. .... ..... ...... .. . ,S-ZJI IkJkb for 0lIl) � .. � , u ....,..

....., .. II " .. .....,. a.a&MI C CUttk. and . t• • baIIt ... ..,. ... AI. ...t rnmns.. a bat. w� hAve Z ,.UL

:.:=��

...... 1 . ..... .. .... _ "' C,.2l»· ·· t "' ''' .. ... _ _ _ 100 . _ ... .

'T

. ". or

,

(:1) ' ,

.. _ !, -l, t>.

-

1.t.'1 u.. no\\ I.."ompare the tiMe and infinite \\-ell .... Figure 5.15 shows allowed \\il\C fun..:ti\'n .. . md pmNblhl)' lkn"ltie.. Ill( Nlth well ... " here. as usual. rach thlli.;wmal a\'I" j... at a hl'lght proponlonal to that ..tatc·... energy. Becaut;e aU 3t� ..ituatlon... imllh� ...tanding W3\e.... certain characteristics are the oound- ..t -.am.:: Ene�) i ... qu�tlleJ, .tnt! there i.. 3 ground ...tate of minimum tOIa! energy tn \\hl�h the �ine(K energy I'" nm 0--3 "fundamental" ..tanding wave wilt! some m;t'\imum \\avelcngth linllled h� the \\ell'... \\Itlm. But what of the differences'!

t64

aa.pew 5

Bound Srares: Simple Cases

according [0

(5-24), this in tum is Uo

-

E, or Ihe magnilude of the ("pOi", .

kinetic energy Thus. no experimen! could be certain Ihal the particle il finda:

. negative kinetic energy.

..

Funhermore. we must be careful in interpreting the penetration of classically forbidden region. We cannol conclude that a panicle is passing oulSide the well and then going back inside. Once oUlSide. where !be potential energy doesn', change and the force is thus 0, what reason ....0Uld . . have to tum around? More 10 Ihe point. we can't watch a quantum-mec

consra:: �

particle undislurbed anyway. We must be content with Ihe meager kn OWIM _ Ihal there is a probability of finding the particle outside the well ifan atte � were made to find il there. If no attempt is made, we simply have an

turbed standing wave that happens to extend beyond the waJls.

un:

Matter wave penetration into the classically forbidden region has an el twmagnetic analog. When light waves undergo IOtal internal reflection atec. interface between two media, no sinusoidal wave propagates into the medium. All energy is reflected. But there is an oscillating electromagne ti field thaI decays exponentially with distance from the interface. FUnhennorec

see�:

if th� seco.nd medium is thin e�oug� that the �eld doesn't decay too

IllUch

. . . WI thin, a 51gmficant amount of hght Intensity WIll get through. If light wave:l can do this. �I shouldn" s�rprise us that mauer waves might be able to escape

. through a thlfl wall . TIm IS the phenomenon of quantum-mechanical tun nel. ing. which is discussed in Chapter 6. Related to penetration of the clas�ically forbidden region s i another

important distinction between the finite and infinite wells. As Figure

5.15

shows. the energy levels in a finite well are lower than those of an infinite weI! of equal width. This is true simply because the wave functions in the fiDilt well do not stop abruptly at the walh. We noted previously that the penetrati on depth is greatest for the highest energy. and to meet �moothly such a gradUally sloping exponenlJal outside. the wave functions inside must be nearly horiZOn_

tal al the wa/l. As we �e, this implie� a longer wavelength in�ide the well than

for the corresponding infinite well function, and so a smaJler

k and E. At tht other extreme, in the ground state. the penetration of the finite well's clasSi_ cally forbidden region is small enough that the wave function inside mu�t already be nearly 0 at the wall. �o i1'. wavelength is only slightly longer than in

the infinite well ground state. and j", energy only �lightly lower.

Figure 5.16 C1a,�ical oscillator: ma.��

on a �pring_

5.7 Case

3:

The Simple Harmonic Oscillator

Of our simple case�. the harmonic o�ilJator i5 the mmt realistic. Once again, the potential energy define, the problem. The harmonic o..cillator potential

i energy s

U(x) = � d. where 1(i, the �pring con�tant. as dq>icted in Figure 5.16. U(x) i� a continuou, function. Moreover. II

This case is more realistic becau\C

is a good approJl:imation of the actual potential energy whenever particl�

undergo small oscillations about an equthbnum pm.ilion

• -

In the immediate vicinity of a local mimmum. all contJOuou� potential energy functions "look'"

parabolic; mal ilj, there IS a parabola that clO1C!'ly iipprmimates the function about

its local minimum. The bond between twO atom, in a diatomic molecule j, a

5.1 c... 3:Tbe

good example. Figure

5.17 shows the potential energy sbared by two atomJ m

a diatomic molecule. plotted versus their separation x (discussed t'w1heI- in Chapter 10). The force is strongly repulsive (large slope) for small separations

and weakly attractive (small slope) at large separations. but in the immediate vicinity of the equilibrium separation xu' U(x) is nearly parabollc. 1berefore. the lowest energy vibrations about the stable equili.brium point will be essen· tially those of a simple harmonic oscillator. Inserting the harmonic oscillator potential energy. the time·independen.t SchrOdinger equation is I - h' d'�(x) + KX'.p(x) 2m di' 2 -

�

E.p(x)

(5-25)

This is one of the few cases that can be solved exactly�without numerical methods-for the wave functions and corresponding quantized energies. Unfor tunately, it is still rather complicated. We leave the ex.act, methodical solution to a higher·level course, hut the arguments that energy should be quantized

are

similar to those for the finite well. In that case, requiring smoothness a( the

E. we !JI(x) exponentials that diverge as \xl - For only certain. discrete values of E does !JI(x) have a simple decay to 0 outside. The

points where

U(x)

jump1. would demand that. for almost all values of

include in the wave function

00.

Figure 5.17 For small oscillations about a point of stable equilibrium, all potential energies resemble a parabola-a !.imple harmonic oscillator. The trw interatomic pOll!l1tial ..... I� �tf(lngly repUI'l\e btl!cp) at 'ma l1 .�

U(xl

\\

-.... \ '0-+ o r+-T --------� .--------\ \ \ mH\lmUn, �«n \ , . high.:r end�. ,I "

•

so a

�:HSl.\

the 1....0 .

•

""II,., I the mHlImum a!

'(I"

the true

potllti 1 agn.'I:' ..iln the modd P :thcT

lat�r_

polentiJ.1 ,)f a h.uTI'Ionl.

...... . . '... _ .k l "C O "" . .. ... ,.

.. -01 . ... .... .... C...tiaJl .. ... Jio, .; _ . ... bere. excepI dw tUrtina .-ilh. nice. smooth po IInI pIoce. .... ___ ..wuon. "'" _Iy ...-10. Stili, _ ... ' c:erIIID £ wIues IS the C1IF\'aR1ft' outside JUSI ",chi so lhal tMy settle do.u • •

for.':

bodI ea1Iedlei.lnd these IR" the QUIIIbl.cd mer;ics. (See Section S_IO lhal demonstrates tMse behaViors,) AhhouJh we woo'l solve the SchrOdinger equation (or lhe OSCiUator � we pin bil lIl(Jfe e.periente w1lh it by lun"ng (he, q U�llon around_ .... Ibu soIvua, (SolS) (or \It(z), suppose we rlOd a fundlun tit a certain (1lI'II " aI8IIOII of . simple compuICf I«hnique

�

wort

. HO'N f... c.an we go with II"

EXAMPLE 5.3

'lllution or;:

mau m . one "lan l"O:nlercd at the onC Dcu-nlllrlC 1M en�'l!Y and the properly nllrlT\alJl�J "3\(111a.ll1f

A," ltll:n;:nc" to..,wllf\l llw cla 1I..11Imll. the

At '...It C'nc.." \en ')� A.. .' + l. I J

I

�hl

-'-"

l

)' 1),;.:.... �h1 hlshn' 11!C'''' t1\1" lUmina """l'Iit IC'1.tr.;1l1C' \ \.

..

\"'tl�l

( ) ..•Vi

Itl (ab'''' - '.)1 ,..,..

•

lo

I\ UIlJ!.JlIt funcu.;m txi\I,.

" Wh"'�

� ....

lAWn lAWu !AaIo

l .... � 11"'0

·1

"ntttI ll b ...,... • • O . .. _ ... ..... 1' ... ...' I . 10 _ ...... .... . '

o

..., ....... .. .... � .., .. - l· - 2�-·.. -··

dae.n',WMtIlM The proNtnbty \ mpll at lhe tCntC'1 In tbc Jround lblc.

(.j

(ooU'&ry 11) lhe c:la.\.�I�·a.I U""" (b)

..ypoUiM ..... ...... ..... ..... ... .-.:r o f dte ..... ......

'"" hacDoe .

f, . . .

n

...

•• _

. cs

,

... _ ..... c-

HAL·WO.LD EXAMItLE THERMAL BEHAVIOR OF A

I

DIATOMIC GAS .. ........, � ..

..... ... �

M

-

learn '" ditl"cmll kincb 0( ps.es luw�

� .- hne OM nlw, eumtiaJJy illck Pi.... II

me ..... .we � JUft MImI W du�.cr around a IN' different: vaIaea. n..

....... .,. IbaI IheIe vaJun � 10 tram;lauonal moboo 01 .. • whole. IOI8bOII 01 _ mokc_, Md VJtotk.. due to the iDIu __ .....N ... RJr Ihe COIIIIOOD duIomic pscs Nt' 0:- and H!, the- klual \alucs . roo. apa .... .,ree quilr Mil with tM arpmrntJ--bul onl) If the atom" don'l vibnIe. .. n.r.-swa 10 this cl&ukally puuli", lilUMKIfI j, thai lW.:IUalllr C'ncrglC'\ � qaaa.. OI'CCK:.I

....-air •

tiad. In r:oIli.... � and "Itntitlfllli C'lX'rgy �'"llI'I be e\.:hanged, and f i theat weft III('ft enc:rv in ttans�1IJ Chan in \,Ibtati..... wC' ml}:hl e�p..ible to veflfy an a':crage po'>itlon m any ca...e. The expectation value is

the ,."Iue ",e \\-ould obtain If we were to begin with a particle in ...tale A, do an experimenl lo " find'· the partJ(;ie.

begin lI�ujtl with a particle in ..tale A, fmd it,

and ..0 on. repeating the identical experiment many lime... and averaging the location'>. Note th..t the expectation value of -"C 1\ a numerical \'alue, not an explicit function of -"C, hecau..e the integral ha\ been taken over that variahle. By the \ame logic. the ex.pectation value of the \Quare of the lXNtion IS

, 1 .

/ "\"'(.\)\'

all 'pact

nu. la n.mll:ally equUlD tJ-ld) &em �J.

IL,

WId! I �!be pnIIrIt ...., Q __ ...., .

IN aacrk probtbI.Ilty deaIay D\Qt. �. fonh--o

By the same definition, if we lake a difference between these two, divide by

.ix. then lei .1x approach 0, we have the derivative of the derivative-the

second derivative.

tP/

lim

.1..-0

- =

dr

(/(X_:-�·1 I(X)) .lx

lim ----

.1.!. After Ihal. equation (5- 3 5) simply !'Ioay'" that to find I/J al the point x + �. the computer need only k.now its value ... at two earlier points, I/J(x) and "'Ix ..u). But i t can then find I/J at the point x + 2..u. becauhould change only a little from one point to another. a C the mitial slope just right M) that "'(xl die!>

nicely to 0 at one end. but thi" use� up the arbitrariness of the ..lope. and ",(x) still diverges at the other end. We conclude that this E is "wrong" and move on to another, At only certain values of E. we find that for an} initial ",(x) at the "tarting point (unles... the \tarting point is by chance actually a node). we can choose a slope there Categorizing wave functions as eigenfunction.� of operalOrs has many applica_ tions. II is widely used in more advanced analysis, such a.s matrix quantum mechanics. Probably its mOM useful application, though. is the one we have highlighted: It is the si mplest way of detennining whether a function has a well-defined value of a given observable.

......- ... ..1" t',

,•

• RoaRasl AND APPLICAnONI Quantum Wells as light Sources With advances In 10 choose . frequency anywhere In the visib&o ... _ abo a unall and cffiClenl PKu,c wcll wiled lO '" iKIua.ia& fabricating mlcrOM:Op" �tructures in recent decades, the UloC of hghl 1hat 1'10 anticipated In � IIDII c0m ph),iC� of quantum well� ha.. anumed a major role in modem technology, The ","ell con,i'h 01 e\eClroMatic puter dc!>iln, bamer. in �emiconductor cl')'�tab (or. In \ome easel>, Fi� S.21 1Iohowt. two umpks produc:cd by a lC*m . carbon nanotube.., diS\:;u��ed In Section 10.10), and the Sancha Nauonal � tt.l lc::atu.re In � MId usual caplive� are the electron and Its �emlconductor alter .-.ec:mingly OPPOSite applil;aOOo or quamum doU: cfti,;ienl ego, the hole. If the particle'.. quarter. confme it in only gener-dtion of normal ",",liar h&M. Althoup. III more cfficiad. one dlmenl>10n, leaving It ef>senllally free in two other,., we than the old �tandard Intandacent. tudaY'l 1luocaccnt bulb. be"dcs being rather bulILy, 1\ mll typiuJ.1y \eu than 50% effi· call it a quantum well If the partide IS confined in two dimen�lon\ and free in one, II is in a quantum wire, cient. Sohd-'ta� lighl 1>OW\.·u--1acILina naawcd cbambeR. When confined in all dimensions. it i5 In a quantum dot. tran�formc:r.., and hut fllamenl\-are already ut.ed in many An three are being studied for a host of applications, but low·power appli.;allon!l. but they may one day n:plKe the the dot is particularly attractive because, meeling standing �tandard Iightbulb a, the tran\1',tor did 11.\ vacuum tube pn:de. ce��r. Coupled With the efllClenl LED, the quantum dol is. wave conditionpetlal molecules aU' trolled, for they also depend on the simple georneD)' of the introduced-a ","apture probe'" and a " reporter probe" ....ells. As we find for a particle in a box, a ..maller L implies higher energies. Many early difficulties in producing unifonn which, when encountenng a DNA molttule. fonn a three� molecule �and""'lch. The quanlum dot i'!i the crucial dots have been ov'ercome. A major advance was the develop indicator that the landwich has fonned and thus that ment of a "self-assembly" method in the 1990s. in which DNA is pre-.ent 11 if> clothed in speCial molcy mOl)' tit nkulatC\l frum thr- Yra� function "Ia +tu; + Gt! n, Write the �moothne�s conditions.

Lt

A.

��

(b) In Section 5.6, the smoothne�.. condition.. were combined to eliminate

B, and G in favor of C

In

r·"{�ll \ . ( ,)

e x

the remaining equation, C canceled. leavi ng an equation involving only .l and (}, solvable for only

Ie)

cenain value� of E. Why can't thi.., be done here1 Our solution

physical ly ? (d) Show Ihal

is smooth. Whot is ..till wrong with il

D: F�

� (B - ; A )

and that

omething as small as an electron, we must apply quantum mechanics. In the region to the left of the step, x < O. we win have a right-moving incident wave, and we alloW for a left-moving reflected wave. Each solution of ( 6- 1 ) is stH! a solution if it includes an arbitrary multiplicative constant, which will eventually tell us how they compare; so for the incident wave, we use Ae+ilr, and for the reflected wave, Be-ib:. In the region to the right of the step, x > 0, where U = UO' the SchrMinger equation is

,p",(x)

2m(£

O = --C*Ck' A*A k 1 ",lf.i. 0,

Agure 6.'

Reflection and lraruomi!lihOfl

probabilities for a potential �tcp.

LO r---r---'--'_--�

I

O.

o. 0.4 0.2 ()

1

I�

then going. back through to x < O. for there i� no force beyond the step lhIt could reflect anything. We ..imply accept that as long as no attempt is made 10 lind a particle. we merely have an undisturbed wave that. though it reflects completely, happen.. to peneuate the classically forbidden step. The "penetra tion depth" Ie; the "arne a.. in Section 5.6, where the wave function outside the finite well wa.' abo ..imply a dying e;(ponenlial. I S = - = a

,

EILu

-;= �== h

V2m(Uo - E)

The clo.. er E i... to Va' the greater is 8, and the �Iower the decay of the wave function. In any case, the wave function is very small for x » 6. As always, eleclromagnetic waves show analogous behavior. Just as a Vo > E step completely reflects massive particles, a smooth metal surface completely reflects light-making it a good mirror. The light wave doesn't propagate through the meta!, and no photons are transmitted, but there is an electromagnetic field within the metal, oscillating with time and decaying ex.ponentially with depth. For an electromagnetic wave, the penetration depth ie; often called skin depth. Figure 6.4 shows reflection and transmission probabilities plotted versus EIUo for a potential step. If E is less than Vo' the wave is totally reflected. When E exceeds Ua' the reflection probability falls rapidly with increasing E.

6.2 The Potential Barrier and Tunneling Tunneling is one of the most important and startling ideas in quantum mechanics. The simplest situation is a potential barrier, a potential energy jump that is only temporary. If a particle's energy is less than the barrier's "height," it should not get through---classically.

E > Uo

Let us first consider the case where E is greater than the barrier height Uo. As shown in Figure 6.5, at.x ::::; 0, the potential energy jumps up, giving the pani. cle a "kick" in the backward direction, and at .x ::::; L, it drops back to 0, giving it a forward kick.

U(x) Figure 6.5 A clas�ically surmountable potentia) barrier.

=

{�O

x < O, x > L O < x < L

Classically. the particle shouldn't reflect; it should merely slow down between .x ::::; 0 and .x = L, then return to its previous speed. Let us see what quantum mechanics has to say. In the region x < 0, we should again have our incident and reflected waves. Between x = 0 and x = L, the situation is ex:actly as it was to the right of the potential step when E was greater than Vo' Thus, SchrOdinger equation (6-3) applies just as before, giving the same mathematical solutions: e+ik'� and

t--"" ,l. In the Cllse of me simple step, we threw out the h:tl.-mo....ing t

a '. (or beyoml \ ... 0 able to renect a wa.. e. Now th�� is-the nothing was there pOCential drop at x ... L Therefnre we cannot jUstifulbl)' thn\\\' l\ut lhi" solu tion. Finally, In the region ,\ '> I.. where the potential �nerg) is a�ain O. the SchrOdinger equation i'io the �ame as fl1r \ < O. a... Ilrt tts mathemattcal solu tions, t'*/lr and t-Ill . But here W I! d11 1hrow one out, r'- loll , for O\\thing mo..�s to the left in this reg\On. Altogether. we h:\\'e

tII1 not energy quantization. 'ncident particles may be sent in with any energy, bur resonant (complete) tran.')mL�sion occurs 31 only certain energie�. A familiar behavior in light (yel again!) is completely analogous. Nonreflec. tiYe coalings exploit the same wave properties. If !.he wid!.h and refractiYe index are cho.�en properly, a thin film will pass lighr withoul renection. In both applications, rl!llections within the film ("over" the barrier) preci!;tly cancel lhe wave reflected from the first interface. As always, it is important 10 remember that wa\'es. nol particles. interfere. A particle-photon or massive--encoullIering an obstacle may later be detected as a parficfe. having been either reflected or transmiued. bUI the process inyolves a probability that come!> only from the It'(f\'e function. E
C', there ,:- no 'real m�lll\C'ntum in....ide ' Ium::llon.. 110 IlIlt rerott'>C'nt nght- llml le{t.mo\,ing ['W1' the b;lrrier. ilnd the '''''0 .:il''' Still, bllth are necl1et.l tll cn..ure ..moIJthne..... It i.. lert [I" an c'erriloc to ohtain thc 'mllo,lthne" I;onditilln... fnlm whkh the rdle..:tlon and lnln"ni�"'lm rorohahilitie... lE).C'rri-.e 1� :-.how thllt find ", eo they l;;Jn ;\l;tu311) he dcdul;cd by a ..impic f('pi;Jl;l'menU R

T

'

t

..inh� \/�m{ Uo

,inh)1 \/2m(Vo

/:')L/IIj

+

''

E) VIII

�(EIVo)l1 - E.'!Vu)

4(£/ Vo)(\

�Vi,,�un

"i nh

E, Vo) E)i�hT+ 4(£!Uo)(l

FIgure 6.1 Wa\'e 'un.;ti�\n� (or pani. in gcneral. nOtl/,ero--a particle can escape through a barrier thJ.t it cle .. �'I dillerent energie.. inddent fnun can't �urmount cla....ically. H ilc'ld, il "tunnels" through. and the principles of quantum mechanics demand that there be such a possibility. The �olulion to the left of the barrier is ..orne t.� -----�c--"'"'--c��_.,,-� combination of incidenl and reflected wa\'es. of po!'oili\'e and negali\'e momentum. At :r: = O. it smoothly joins a function inside the barrier that tends to die lc.J ------'c--�--�� off (the C in the eKponentially increasing C,e-+a..'being u!>ually quite ..mall). And at.� "" L. thi.; "moothly joins a transmined wave of positive momentum. There ;!. f' l -'""'- no physically acceptable solulion thai is identically zero beyond the barrier. Figure 6.7 show!. wave functions (real part) for particles of different enert:, _ .... gies incident on a barrier from the left. Note that the wavelengths decrease a!> kinetic energy increa''t.\11o: I\'PIl\""'\"

,

A calculation of the transmission probability for our multifarious potcn is tOO involved to present here. But c:akulated decay rates (dt!c;.,ys energy tial per second) agree well with experimentally determined values. \n simplitit!d form, the calculation is as fonows: number of decays

number of times a strike). barrier

time

time X transmission probability

one strike

= -,-,:- X -

time to cross diameter

T

I

=

diameter of nucleus/speed

=

v -- T

X T

2rnuc

Of course, once the alpha particle escapes, alpha decay of the original nucleus is no longer possible, because the element has changed (e.g., uranium becomes thorium). So refen·jng to a number per unit time for a �ingle nucleus is nonsense. But with the huge number of nuclei in even a macroscopically small sample of radioactive material, simply multiplying by the total number yields a proper average decay rate. As noted, agreement with ex.periment is very good, but we must not overlook the simple fact that the probabilistic

nature of alpha decay is evidence of its quantum-mechanical basis. Watching individual nuclei. we would find great variation in the time required for decay, and this is exactly what we expect if the decay is governed by quantum mechanical probabilities. Example 6.2 provided the key to understanding another classical mystery of alpha decay. Most alpha particles have energies E in the 4-9 MeV range, yet their decay rates differ by more than 20 orders of magnitude. as shown in Table 6. 1 . The probability of the alpha particle tunneling in a single encounter with the nuclear surface is very small, and as we found in Example 6.2, when

this is true, transmission probabilities are extremely sensitive to the value of E.

210 ehllpter 6 Unbound States: SlepS. Thnneling. and Particle-Wave Propagation

TABLE 6.1

Energies and decay times in 0: decay

.emitting nuclew

a-particle energy (MeV)

ct;;;

o Mean time t

;&

Po-2 1 2

8.8

4.4 x 10-7 ,«

Rn-220

6.3

79 secon�

Ra-224

5.7

5.3 days

Ra-226

4.8

2300 yurs

U-238

4.3

6.5 x 109 years

-

-

In conclusion, according to classical physics. alpha decay-in the unex.. pected low energy of alpha particles. in its probabilistic nature, and in the vast range of decay rales-would be completely baffling. Its explanation is a tri umph of quantum mechanics.

FigUnl 6.13

Tunnel diode.

AV = 0. / := 0

+

ill'

.-

',

_

I(� - f-,

I

Net electron flow

The Tunnel Diode A tunnel diode is an electronic circuit element whose response to applied volt. ages is unusual and very fasl. In a narrow region between the device's two ends (leads), there is a change in [he maleriai's physical properties thai pre venlS simple conduction of electrons from one end 10 the other. In essence, the electrons a[ the ends are separaled by an electrostatic potential barrier they can't classically surmount, crudely depicted in Figure 6.13. By design. how. ever, the barrier is fairly thin, and significant IUnneling occurs. With no applied voltage, it occurs equally in both directions-there is no net flow. When a potential difference is applied. the situation becomes asymmetric. Right and left tunneling rates differ, and a net currenl flows. The tunnel diode may seem to have liule utility, as an applied VOltage will induce current flow in many materials. But one of its distinctive features is how the current varies with voltage. It doesn't steadily increase as applied voltage is increased: at some points, it decreases. Moreover, changing the applied voltage changes the transmission rates almost instantly, and quiCk response is very desirable at high frequencies. The more common devices thaI control current via changing voltage rely on relatively slow thermal diffusion of the charge carriers. (See Section 10.8 for further discussion of diodes.) Although its early promise as a high-frequency switch in integrated circuits ha,s dimmed. the tunnel diode has found use in a varielY of modem electronic circuits. A more complex device. the resonant tunneling diode, is discussed in Progre"is and Applications. SQUIDS A tunnel junction. two conductors separated by an insulating barrier. is a k.ey element in many electronic device�, and when Ihe insulator separates two superconductors (see Chapter 10), it is known as a Josephson junction.

fl.) A.lpha Decay atId Otbcr A.ppIiatioea

through which electrons tunnel in pair!'. The Josephson junction is at the hean. of devices known as SQUID� (superconducting quantum interference devices). The insulator serves as i.\ weak link between the charges' motions on the two sides, coupling them together as would a weak spring connecting two pendula.

111

FIIiIU" 6.14 "The effect of Oat: osdlla tor on another depends on the pbMe

relationship.

As illustrated in Figure 6.14, with this kind of coupling, energy/current flow is intimately dependent on the phase relationship between the things that oscil late on the two sides-pendula or charge-camer wave functions-and there may be constructive or destructive interference. In the case of charges �pa rated by the weak link in a SQUID, the phase relationship is also a very sensi tive function of any nearby magnetic field. Depicted in Figure 6.15, a SQUID is. in essence. a loop with two Josephson junctions. Interference caused by

1.

No '_ No mcrzy tRMfer

..,

Weak link

'I;'I;VVVI/VVIi,VI/V1.fU

small changes in the magnetic flux passing through it produces easily detected changes in the current I. SQUIDs can detect extremely small magnetic ftelds. such as those produced by the human heart and brain.

Energy tran�rer

-

FOlXe --_

Field Emission As we know. to remove an electron from a given kind of metal requires a

certain minimum amount of energy: the work function cP. In effect. the metal's electrons reside in a potential well, due to their attraction to the

positive ions. and they are " reflected" at its waHs by a potential step they cannot surmount. as shown in Figure 6.1 6(a). Owing to the random thermal

Figure 6.15 Elements of a SQUID sllpercondllcting quantum interference device.

distribution of speeds, at any given temperature, a small fraction of the electrons have sufficient kinetic energy to escape the metaL Those that do escape participate in an equilibrium exchange of electrons with the sur rounding space. Heating a metal filament to enhance this effect. known as

Figure 6.16 Electrons in a metal (a) behave as though in a finite well. An electric field (b) alters the "wall," so that tunneling may occur.

o:or \

Metal

....

Electn;n� bound by pt) enlial �tep

a � j

+ Vo \ 1 -:0]

\f

° 0°

Cathode

\ Anode

Tunneling through a potentia. barrier !

la)

(b)

Insulating junctions

(tunneling)

212

Chapter 6

Unbound States: Steps, Tunneling. and Particle.Wa�e Propagation

Figur. 6.17 One pixel of a field emis·

thermionic emission. has long been used as a source for electron such as those in the conventional cathode�ray tubes used in televisi

sion display. Applying a positive bia ..

X.ray machines.

turns on any of the three different col· on of subpixel.

Indi�idual pi:\el

---

--�- -

Rod

r

pho'>phllf

phosphor

Blue pho�phor

�

ons

tid

But there is another way to coax electrons from their potential WeU .u positive electrode is brought nearby. the potential step may effectively � changed to a �tential barrier" In �igur� 6.16(b), the positive ele�trode 1IlOdi. . fies the potenllal energy function seen by the electrons-Iowenng it OU ts

ide (Remember: Negatively charged electrons are at low PO tential energy where the potential is high.) Now electrons moving too slowly to

the metal.

s mount the work function barrier may tunnel. The technique can be u�1Jr· t generate an electron beam and is known as field emission, after the ele(; o trj field in the region of changing potential energy between metal and electrodec In many applications. it is preferable to thermionic emission, for he aling ·

metal filament not only wastes power but often produces considerable ele ctri� cal noise. Field emission is the heart of an alternative kind of flat-screen display i

2.5

active development, the aptly named field emission display (FED), depicted

mm

t �

Figure 6.17. Its potential advantages �ver the tr�ditio�al �iquid crystal di�Pla}

commonly used in laptop computers mclude Wider vlewmg angle and quicktr response.

Microtip\ ,

Column line

The Scanning Tunneling Microscope Figure- 6.18 Sc.:hematic diagram of an STM in operation. ,

IIL-,

Scan

. _'

'

Barrier width (variable)

Tip

•

by an STM.

to better use than the

scanning tunneling microscope (STM),

'

whose initial Rohrer). In thl\ design won the 1986 Nobel Prize in physics (Binnig and device, a slender metal tip is positioned near the sample under study.

Becau�

tip and sample are not actually in contact, the free electrons in each cannol pass between them in the usual classical way-a potential barrier intervene� . However, in the STM. the separation is made small enough that significam quantum-mechanical tunneling occurs. Variation in the tip-sample separation

smaller than typical alOmic dimensions translates to easily measurable

changes in tunneling current. Thus. as the tip of a tunneling microscope i$ scanned laterally over a sample's surface. as shown in Figure 6.18. it is able

Sample

FigUf1!! 6.19 A mi" ing atom.

Tunneling probability is very sensitive to barrier width, and nothing puts thl

as ..een

to "see'· individual atoms. Furthermore, calling the tunneling direction thr and the scan direction the x. by repeating the scan with the tip displaCed

;::

�lightly in the y-direction. an entire "topological map" of the sample's SUr.

face can be generated. Phenomenal results have been obtained. Figure 6.19

�hows a lone atom missing from a pattern of iodine atoms adsorbed on a pial. inurn surface. The hexagonal geometry that characterizes the iodine bondtng IS qUite clear.

ogy.

The STM ranks as one of the most indispensable tools in modem technol.

Its uses are already legion.

includmg studies of the geometry and compo.

sition of an endless list of �urfaces; locating Important biologicaJ molecular groups such as the fundamental building blocks of DNA: mapping micro.

scopic vortices in superconductors; nudging atoms from one point on a Sllr. face to another. And no end is in sight.

REAL-WORLD EXAMPLE S E E I N G SMALL DETAILS WITH AN STM The metal tip of an STM has a work. function, and the sam� is true of samples we might study with an STM. The two work functions are not, in general. equal. and in !.he STM. a potential difference is applied between tip and sample. giving a .-.Iopc: to the potential energy function felt by electrons moving between them. Ne\icrthele$". by trenting the space between tip and sample a!', a simple rectangular barrier who� "heigh'" above the electron energy is typical of a work function, we can understand the tunneling curren! should vary in a detectable way. even for the: e'l�mel}'

why small variations in tip-sample separations "seen" by an STM.

Applying the Physics Electrons in an STM tunnel between sample and tip. Using a Iypical metallic work

function of 4 eV for Uo to change to produce a

-

E, about how much would the tip-sample separation have

\0% decrease

in tunneling current? Use the wide-barrier

approximation.

In equation

(6-18), only L changes. Thus,

The tunneling current will vary as the tunneling probability varies. Setting the ratio

to 0.9 and taking logs on both sides gives

'" ( 0.9)

�

::::::) AL = 5 Atomic radii

10 31 kg)(4 X 1.6 X 1 0 1.055 X 10 34 J . S 10-12 m = 0.005 nm

-2t>L X

\12(9. 1 1

X

1 9 J)

are on the order of O. J nm. In practice, the resolution of even the

best STM doesn't exceed about 0.01 nm. but our estimate certainly suggests that this remarkuble device should be able to "see" atoms. As an alternative to translating a varying current

to an image of varying height, STMs often sense the tip location

needed to keep the tunneling current CO/lSlallf, and the motion of the tip thus tracks

the height of Ihe sample.

6.4

Particle-Wave Propagation

Thus far, our study of unbound states has concentrated on the effect of simple

forces on plane waves. A plane wave is not the most realistic malter wave for a single particle. To represent a reasonably compact moving particle. a tra\:eling else wave pulse is much better, It is broad in one region and essentially where. But the behavior of a pulse is considerably more complicated. even

0

with 110 steps or barriers, so we will have plenty to keep us busy as we restrict our attention to pulses moving in vacuum or in homogeneous media. A wave pulse may be tre:lted as a sum of plane waves, and to understand the behavior of the whole, we consider the behavior of its pans. Each constituent

214 Ot.pter 6

Unbound Stales: Steps, Thnneling. and Particle-Wave Propagation

plane wave moves at its own speed, or phase velocity, giving rise to two important consequences: ( 1 ) The speed of the region where the probability t: density is largest-the speed of the particle, or group velocity-may dd plane waves; and (2) an mitia distinctly from the speeds of the constituent l well-localized wave pulse will spread out with time, known as dispersion. its constituents get progressively out of step.

� �

Phase and Group Velocities Figur. 6.20 Wave group: The crests

and envelope move at different speeds.

Figure 6.20 shows a traveling wave pulse, which we associate with a traveling particle, at several successive times. Regarded as a sum of plane waves, We refer to it as a wave group. The light red arrows indicate the motion of a panic ular crest. The thin red outline traces the "envelope" of the wave group, a shape that moves with the group and defines the maximum possible displacement al each point. The dark red arrows indicate the motion of the envelope, which clearly does not move at the same speed as the crests. The speed of the enVe_ lope is known as the group velocity. If we are interested in knowing When the particle is likely to arrive at some destination, it is the speed of the envelope, where the probability is large-group velocity-that is of interest. The speed at which an individual crest moves-phase velocity-is rather unimponant. Nev_ ertheless, an understanding of the group's motion comes only through study of its parts. Let us begin with the simplest possible wave group.

A Sim ple Wave Group

Consider a wave group consisting ofjust two plane waves of equal amplitude A. Using the form we first met in equation (4- 10),

(Note: Despite the i and the use of '1', the main ideas in this section apply to aU wave phenomena, including electromagnetic radiation.) We wish our group to have reasonably well-defined wave number and frequency, so we choose k and w values that deviate by only very small amounts dk and dw above and below central values ko and wOo

k, WI

�

=

ko + dk Wo + dw

Thus,

,/

Factoring out common terms leaves

"" ko - dk W2 = w0 - dw �

Now, using the Euler formula, we obtain a fonn es..�ntially identkal to thai of a classical beat frequency. The sum of two sinusoidals b«umes " pmdm.:t of two who�e wave numbers and frequencies are va!'>1ly different.

F&gu.... 6.2' . ..., ,

We have a complex exponential that moves at speed wdko' the phase \elocity central value!\ ko and wOo modulated by a cosine function appropriate to the dwldk. A!\ !\hown ,In Figure 6.21, it is the latter tern, th,\t speed at moves that defines the envelope-the group velocity is dwldk. We see this more clearly bility density. by computing the proba 'I'·(x. I)'I'(X,

t) '= 4A2 cos2{(dkh

-

(dw)ll

(6.20)

The phase velocity disappears. Because a particle must move as the probabil.

ity of finding it moves, we expect the speed of the particle to be best repre·

\'l!F,

dwldk. Our surprising conclusion is that the sented by the speed of depend on the actual central values Wo and ko' doesn't group/particle velocity bul rather on how w varies wilh k from one constituent wave to another.

Although this simple wave group exhibits an important feature-that the

e�d �wldk- it isn't very particlelike. it is peri. probability densit� moves at sp : odic, as is any jillile sum of penodlc funcl1ons. So let us now tum to a more

particlelike case in which the probability density isn't spread out all over space.

A Particlelike Wave

The most general way of expressing a wave group is +00

'V (x,

t) �

100

(6·21 )

A(k)ei{h-w,) dk

This is a sum of plane waves of all different wave numbers, each multiplied by

its particular amplitude A(k), and each including its time dependence. (It is equa·

rion (4·2 1 ) with time dependence added.) Position x and lime

I are the usual

independent parameters, and we will address A(k) soon, but where do we get w'?

This is a central question, and the answer is that it can be considered a function

of k, for each phenomenon has built into it a relationship between w and k, or

equivalently between E and p. Consider two familiar cases: electromagnetic (EM) waves and matter waves. From equations (4-7), we

see that for plane·wave

solutions to Maxwell's equations in vacuum, w = ck. equivalent to E = cpo For plane-wave solutions to the free·particle SchrOdinger equation, equation (4. 1 1a)

showS that w = hk2/2m, equivalent to E = p2/2m. A relationship expressing frequency as a function of wave number is known as a dispersion relation. EM wave dispersion relation: w(k)

Matter wave dispersion relation: w(k)

=

ck

� -

2m

(6·22)

(6·23)

Slmple wave IJOUp.

216 Chaptw 6

Unbound Slain: Sleps. Tlllmding. and Partide-Wa\'e Prttpalafion

Before looking closer at the group behavior, consider the phase velociry in these [WO cases. By detinilion. phase velocity is just the standard fonnula for the velocity of a plane wave.

I'

pha\C'

W = A( = . k

(6-24)

Thus.

EM waves: I'phase

Mauer waves: \I

ph!N:�

= W

k

w k

= C

(6-25)

lik

= -= -

(6-26)

2m

Whereas electromagnetic plane waves. unsurprisingly, all move through vacuum at the same speed

c. matter plane waves of di fferent wave number

move at different speeds. Here is where the distinction between pbase and group velocities arises. Our main object of study is a wave group Whose In the case of malter wave numbers cover a range of values centered on

ko-

waves, the phase velocity corresponding to this central wave number would

Vo

= po/2m = vol2, which doesn't equal the velocity of the par ticle. However, the phase velocity is not the important one. (This is quite a be flko/2m

relief, for as we soon see. it can exceed

cO The wave function (6-19) for the

two-wave group bears this out. Its complex exponential moves at the central, or average, phase velocity

wr/ko•

but (hal part of the wave function disap. pears in the probability density, which is what really corresponds to the par· ticle's mol ion and moves at a different velocity. Does this also hold in a more general group? As written, wave group (6-2 J ) is general, but almost always we are inter ested in a wave function of a particular shape, and one of the most conunonly considered shapes is a Gaussian wave packel-a single bump. At

t=

0, we

desire our wave function to be of the form

(6-27)

Note that this is a right-moving plane wave multiplied by a Gaussian bump that falls off away from the origin. Because of the Gaussian factor, it is nOI infinitely broad (� *-

)

XI ,

�o its momentum is not perfectly well defined.

However, the oscillatory complex exponential gives it an approximate wave number of and thus momentum of Mo. In fact, its real part, involving cos

ko kox. would resemble the top/starting wave

in Figure 6.20. The way we

ensure that integral (6-2 1 ) agrees with (6-27) at 1 of

= 0 is by the proper choice

A(k). Appendix F, applying aspects of Fourier analysis covered in Section

4.7. discusses what choice is needed, then tackle� the integral. Here we simply study the result. For the wave function given at

1 =

0 by (6-27),

equation t6-2 1 ) gives the wave function at arbitrary lime r. and ils IU\lbabH it)' deosity IS

1"'(\, 1)1' where �' -

(b-2Rl

dw(k) \ dk '0

--

and

Given the many factors in (6-28). it is worthwhile noting tiNt that it is cor rect at t = O. It reduces to

which is indeed the complex. square of (6-27). So what doe!". it do gresses? Taking it one piece at a time, consider a ca�e where D

==

as ,

pro

O.

(6-30) This is just a pulse "sliding" along the x-axis at speed s. and given the defmi

tion of s in (6-29). we see that this is the same result as for OUf earlier two wave group.

Note that nonzero D wouldn't change this conclusion. As we soon find out, D governs not the speed of the probability density, but how much it spread!;

in time.

Let us calculate the group velocity for the familiar cases. From equations

(6-22) and (6-23),

EM waves: v group

Matter waves:

\

� dw(k) � !!.. � dk dk Ck k C

� dw(k) � !!..hk' group dk dk 2m

v

\

(6-32)

o

flko

'0

� -

The group velocity of an electromagnetic pulse in vacuum is

III

c.

(6-33)

because all of

its constituent waves share that phase velocity-they must move as one. Equa tion (6-26) told us that a matter wave's constituent plane waves move at differ ent speeds, but group velocity (6-33) is just what we expect it to be: Maim = Po/Ill = vO' the velocity of the particle. No matter what may be the

Z1. CIwptar 6

Unbound Scala: Steps, 1\mncling. and PmtJde-Wave PropagatIon

Figure 6,zZ Dispenion relations for

EM and matter waves. 'The phase

velocity is the slope of a line from the origin. while the group velocilY is the slope of the: tangenl line. EM wav6

w=d

Mauc:rwavb

w \'pb&M

-

wilt.

/ /�

w �

"iI?l2m

'J ,

•

phase velocity of constituent waves. the region of high probability moves • the speed we expect of the associated particle. How do OUf results relate to Figure 6.20? As noted in connection Wirb equation (6-27), the wave function starts out looking just like the lOp plOi in the figure. As time progressc!\. its little crests do indeed move at a phase veloc. ity different from the speed of the envelope. (The actuaJ 'I'(x. f). which is a bit messier than I"'<x. t)F. is given in Appendix F.) But these features disappear in the probability density (6·28). leaving just the envelope. a smooth pulse mov. ing at the group/panicle velocity. Figure 6.22 illustrates the relationships between the phase and group velocities for our familiar cases of EM waves in vacuum and free-particle mat. ter waves. The two plots are disperSion relations (6-22) and (6-23). The dashed oval on the matter-wave plot represents the range of wave numbers and frequencies in a group--a continuum, but still a restricted range. The slope of the line from the origin to the central wave number is the phase velocity of the plane wave whose wave number is ko. Lines to other points in the range, of course. have different slopes, or different phase velocities, but none of these is the velocity of the probability density-the group/particle velocity. As We have shown, this velocity is the slope of the curve at ko. We found that the phase velocity of the central wave was vr/2 and the group velocity was vo' and the diagram clearly shows that the group velocity exceeds the phase velocity. The electromagnetic plot is comparatively easy to interpret. No specific range is indicated. but it is clear that the phase velocities are all the same and must equal the group velocity. The interested reader is encouraged to take a look at Computational Exercise 59. Using a simple sum of real cosine functions in place of equation (6-2 1 ), it obtains plots. suitable for animation. that resemble those in Figure 6.20 and that clarify phase and group velocities and their dependence on the dispersion relation.

An Electromagnetic Pulse, and a Surprise Let us consider the behavior of an electromagnetic wave pulse traveling through a medium where plane waves of different frequencies move at differ ent speeds. EXAMPLE 6.3 For waves in the region of the electromagnetic spectrum used for GPS (globaJ posi tioning system) Signals. the interaction between the electromagnetic fields and Earth's iono!>phere leads to a refractive index that varies according to

n(w) where

w

=

JI

-

b

w'

is angular frequency and b is a constant. (a) Find the dispersion relation.

I.

(b) For a pulse of central frequency wo' detennine the phase and group velocities. (The

reader may quail al seeing a refractive index less than for it implies a phase veloc· ity greater than c. We confront the seeming violation of special relativity afterward.)

SOl\,;,', )N

(8) By definition. the refractive inde� of 8 materiai is the ratio of the speed of liaht In vacuum to the sp«d of 8 pure electromagnetic plane wave in the malmal that is. to the pha� velocity.

e

=

n � --

\'phase But the speed of a plane wave is also

w k Solving for

�

wlk, so that e

VI - b/w'

w. we obtain the dispersion relation. w(k)

�

Vb

+

(ke)'

(b) Using (6-3 1),

We now reexpress this in tenns of the given

=

wOo Using (6-34). we obtain ko

= � Vw'0 - b e

Thus,

- b/w"ij Finally, evaluating the phase velocity also at the central frequency, we have e \lgroup

= cYI - b/wij

The example seems to make a very unpalatable claim: The pulse's group velocity is okay. but its phase velocity is greater than

c.

We don't have to look

far for this "problem" to recur. When mass/internal energy is taken into account. the same holds true for matter waves, even in vacuum (see Exercise 44).

Is special relativity violated? 1t is true that any individual plane wave may

travel faster than c. But a pure plane wave-of infinite extent in space and infi nite duration in time-cannot transmit

infonnation.

It doesn't vary in any

significant way, in a way capable of conveying information from one place to another. On the contrary, to transmit any intelligence, the wave must be modu lated in some way, perhaps varying amplitude or frequency or simply turning it on and off. When modulated, it is no longer a single plane wave, but becomes a

220

CIu!pter 6

Unbound SlalCs: S�PS. Thnneling. and Panide-Wa\'t Proj)a!!lIlion

combination of plane waves-a wave group. The information travels at the group velocity. which is less than c. Under somewhat exceptional conditions which have understandably garnered considerable attention (see Progress

Applications). even group velocity may exceed

and

c. but this requires further

rethinking of what it means to transmit information. and in no case has a vio lation of special relativity been suggested.

Dispersion Not only can differences in the phase velocities cause a wave group to move al a speed quile different from its constituents. but if can also lead to the phenomenon of dispersion. the spreading of a wave pulse. Dispersion arises whenever the dis

D in equation (6-29}--the second

persion relation is nonlinear; that is. when

derivative of w with respect to k-is nonzero. Probability density (6-30) assumes

D = D, and it describes a Gaussian pulse simply sliding along the x-axis, unde_ fonned. at the group velocity. However, when D is nonzero, we have to go back

to probability density (6-28). In the denominator of the exponentiaJ's argument D causes the moving Gaussian to become broader, ultimately a constant of I . In

the factor multiplying the exponential. it causes the probability density as a whole to decrease. so the pulse flattens out. Thus, the probability of finding the particle

spreads over an ever larger region. while the probability per unit length climin D is caJled the dispersion coefficient.

ishes. Governing this behavior,

Dispersion would occur for the GPS pulse in Example 6.3, because the (6-34) for the medium is nonlinear; for the same reason, it

dispersion relation

occurs for a matter wave

even in

I'GCUWlI. As shown in Exercise 48, probabil_

ily densily (6-28) becomes

1 'I'(x, /)1-,

=

VI

C'

+

" '/'/4m'.'

Note thar because the factor

1I.212/4m2£,4,

portional to

e-4•

[ -(X

exp

2.' ( I +

-

Sf)'

n'/'/4m'.4)

]

(6-35)

which causes the spreading, is pro.

the narrower the pulse's initial width, the more rapidly it

spreads in time. Exercise 49 investigates the phenomenon. It is worth reiterating that our wave group is a solution of the free-particle Schrodinger equation, just as is a single plane wave, but unlike the plane wave, it is a good description of a well-localized particle. Figure

6.23

shows

the time evolution of the probability density of a moving free particle. Disper sion invariably leads to increasing uncertainty in a particle's position.

Figure 6.23 Dispersion causes a matter

wave to spread.

1'1'1'

..

o

'I

.... ... AfPII .... ..,

P ROGR ESS A N D A P P L ICATIONS Resonant Tunneling Diode Photon Detector

,\pplil:ati(ln� of tunneling in modem ek�·tronics are muillplying rapidly 1ll1waday\, One holding great promi�c. tlc\eloped by ;'I:ienti�t!> at To�hiha Re�an:h Europe anti ('ambrid!!e llnivcrp and manipulate microscopic Objects. and the atomic-scale dexlerity of the STM i� increa!>ingly an�wcring the call. Researchers at the University of Berlin ha\e employed an STM to carry out a chcmical reaction with single molecules. As shown in Figure 6.16. an STM is used a!> a source of electrons (a) to loosen an iodine from iodobenzene. leaving a phenyl (b). The iodine i\ coaxed to a convenient. out-of the-way location (c) by van der Waals interactions and chemical force\ between it and the tip. By the same

Figure

6.25 An STM with X-ray vision.

,..... 6.26 Sin,.moIccuk SUIJUY via STM

very Impor1anl 10 the pnx:essing and !>Iorage of quUItUIb information. At the olher e�lreme. \e\-eraJ experiments In Ihe pa.\1 dec;Jde or liO ha\'e demomtrated a group velOcity for hght greater than (. The efleet (l(curs in medIa at frequency ranges chanu:teri/ed by anomalous dispenlon ' in which the refral;ti\o'e mdex. contrary to ilS usual bt'ha\-'lOr. del,;realiC' With frequency. Whenever somcthmg-sul;h a� pha�e velocity-appears to mo\-'c at wperluminal \pc:edli, the first queMIon is always whether it can be u�cd to transmit information In aI/ l'aSe\. the answer hih been no, Some feature of a pulse may \eern to travel through a region faJ;ter than light could through a vacuum; il may even seem 10 emerge before the apparently corresponding feature in the inci. dent pul�e cnlen. BUI it� information i", encoded in corn. plex ways. and even the "front edge" of an incident pul�e

force!.. [he phenyl is slid O\i�r Cd) 10 anolher phenyl. Ihen. with e:U'ilalion pro\o'ided by another ..hot of lunneling elel.'lrons Ce), the IWO ar� "wcJtled" into a biphenyl mole cule

10. (Saw-Wai Hla. el .11

Seplembt'r 25. 2000. )

M.ny Spe.ds of light

.•

Phy�il;al Re\'iew Leners.

carrie\ information about the whole, so Ihe arrival of any feature " not really Ihe question. When anomalous dis per"'lon prevails, II i� Io:ommon to speak of a "signal veloc, ity" dislincl from the group velocity_ and never bas this been found to c,'(l;eed c

The group \'e!ociry of Iigh! has

bet:0lTl( qUite a hOI I{lPU: in m:enl yem. a\ new experimental tcchniques are probing all e\lreme\. In the lale

1 m... light'�

group velocity wa., ,Io\\-ed to les� than I mI� by �ending il

Figu,. 6.27 A /ighl pulse "stopped" and restored.

IOto 8 ruther exolic medIUm. a Bose-Einstein conden\atc

(or BEC. � ChOJp!:cr 9 Pro�rts� and Applicatiom), in which lhe rrfnKIJ\'t' inde'( inl;re.I-\e$ abruptly. Other methods of slowing Iighl pul'o('s 10 petle,trian speeds an:: also being !otudied, Ughl of sUl;h slow group velocily may lead 10 new /ighl swil,he, orother opt�It!(:tronic devices. 1I may also prove applicable 10 nonlinear optic.s. in which a medium doesn'l respond linearly 10 lhe electromagnetic field. This effeel l" exploited. among other lhmgs. to double laser frequencies. Presently, nonlinear optics requires high-power

(. )

mm

'-:::----'

lasel'i. but slow light may provide a low-power alternative. The abilily to slow lighl ha.. recenlly led to a further Iwis!. In \eH!ral experiments carried out at Harvard, MIT,

(.) '----=--'

and the Air For..:e Research Laboratory. using both gaseous and solid media. the group velocilY of a light pulse has been s/o","ed ,really as il inferacts wilh the media, at which POlO! a separate la�er �ignal beginli a lransfonnation of its information from light to atomic lipin �Iates. a\ depicted 10 Figure 6.27. and ending in a final "image" frozen in the "pin slale.\. known a� a polarilon. A later laser signal retrieves the light pulse on demand wilh little di\lortion. Thi� ability [0 stop a lighl pulse may prove "

"

/

U"f f

« ) '----'

10 '--_----'

Chapter 6 Sum mary Just a.'. it has an effect on II massive particle. any change in potential energy ha� an effect on ,II maner w�ve �nction. A ..mall particle subject to II force Will often manIfest Its wave nature. behaving II.. II classical panicle would not. It may be reflected by a force that would not cla!>sically reflect it, even by a force in the particle's direction of motion. Conversely, its wave function can pass mto II region despite a force thai classically etcnmnl' II and [J (the amplitude of the

l . I' eljui\'.tlcnl [{) 'ft' , l ,han(ol nf;cl,I.b)' PUl\'IIJCJ that IA ) ,."" 8" • + d)

Show thul lP'( t I

IjIt_> A' ·

!til

+

A'rA'

t

J(II

14. Verify Ih...t Ihe nan, 11\t.'I\l!! and rell�,�tlon pmoabdJllc, aiwn in cqualltlll II'!,7) udd hI I

IS. C"lt;ulale Ihe rt"IlCClulII pnl",,""JI) hll a � (,\' electnm encountering II ''It'P In whl�'h tht' p"lenI131

lIrvps by 2 c \' 16. A p4J1i�,:le mtWUli In a rt·8 1\lu {If lero fon:e t"IKounle,.... a precipu:e---a sudden drop 10 the jXllenllJI cncrg)

10 an

arbltranly large neSJ1I\1," \;lIUt', What is the pn."abiIHY

dUll i t will "�{l {l\c:'r tht' CtI�C"'1

; f; I' dt" �'fl� h) the waw

17. A beam of patlld� of C'ncrg)' E and ,"{" den! up,m .1 pc.lll"flli�1 Mt'P llf U(I fUnt'li on

(/11M: ( .-)

wa\c 1O'".1e the ,Icpl III the hmll� A

I ,.al'

(.) Detemlinc l'{'ll1pklely Iht' ret1e�'fcd wave and the wa\.'e ,",,,.Ie Iht' ,tep hy enf'()rl'ing Ihe required

if

...

O. and mterpret your re,ult,

...

\

)0 0

0 ;lnd

20. P.Lnldc' �ll energ) 1- an: incidenT from the Icft. ",t\t.·rc

Up)

t:.

O. and ill the orig i n encounter an abrupi dr'lp In 3£. (a) CI"'�lcally,

jltllentlill energy. \\how .

43. Example 6.3 gives me refra(.'uw: index for high.

frequem:y electromagneti(.' radiation passing throop

� in lenDs of. flCtor P ,i...en in Exercise 30. (a, SuppoIC dw in SO11K' system of units, J. Dnd a 3.R'

Earth's iono)ophere. Thc constant b, related (0 the SO called pla.�ma frequem:y, \'anes with aunosphcric c0n

ditions. but a typical value is 8 X IO'� rad1/s1. Giveo .

boIh 2". find two ...alues of 2f that give �sonanf tun·

nel;n., What are lhcsc: dis� in ICnns of wAvelen,gths

of Ill? Is the tenn rt'.JOfUUll 'unn�/jng appropriate']

(b) Show mal lhe ,ondition has no solution if J

= D. and CApJain why this must be so. tc) If a drus;nil parti

d� wanlli to �urmounl a barrier without gaining energy. is adding a s�ond barrier Ii good solution?

Section 6.4 41. The maller·wave di)opersion relation given in equation 16-23) is COn'Cct only ut low spce<J and when

GPS pube (If fl"etjuency 1.5 GHz traveling

lhrough

H !un of ion()oph�re. b)' how much. in melers. would the

wave group and a parlit:ular wave crest be ahead of or behtnd (a' the case may be) a pulse of light passing through me 'arne distance of vacuum?

E p k) values Eo and Po' Although it comprise.. many plane

44. For a general wave puhe. neither nor (I.e.. neither are weI! defined. but they have approx'irnatf w nor waves, the general puhe has an o\"eraU phase velocity corrc�ponding to these values.

mas!Jintemal energy is ignored.

Wt)

Eo/Ii poIh

Eo Po

� - � -- � -

ko

(al Usini! the rel:lIi\lj),tically I.:(lrrect relation)ohip among energy, lIlomentum. and ma�s. ..how that (he COffel.:1 di)'pcrsion relation is

If the puJl.e describes a -'large" massive particle. the uncenainties are reasonably small, and the particle may and momentum be 'laid to have energy Using the

Eo

Po'

relativi,lically correct expressions for energy and (bl Show thai in the limIt 01" I(lw 'peed. (small

p and k)

and ignoring l1la.�oJinternal energy, this expression

momenlum. show that the overall phase velocity is

greater than c and given by

agrees with that of equalion (6-:!31.

42. From e4ulliion (6U

�

-

I!!'rjm nnd the formula l'fWlh:k = plm used 10

relal this to the panidc velocity are relalivislicaJly

incorrect. II might be argued thut we proved what we wished 10 prove by making an even number of mistakes.

(II) Using the relativistically correct dispersion relation given in Exercise 4 1 . show Ihat the group velocity of a wave pulse is uClUulJy given by

NOle thai Ihe phase velocity is greatest for particles whose speed I� lea.')!. 45. For wavelengths less than about I cm, the dispersion relation for waves on the surface of water is

w = V('Y/p)I(3, where 'Y and p are the surface tension

and densily of water. Given 'Y

p=

= 0.072 N/m and

l oJ kglm\ calculate the phase and group velOCities

for a wave of 5 mm wavelenglh.

46. For wa\les on the surface of water, the behavior of long wavelengths is dominated by gravitational effects-a

liquid "seeking its own level." Short wavelengths are

dominated by surface tension effects. Taking both into account. the dbpersion relation is

w = Vgk + ('Y/p)k3,

where yis the surface tension. p is the density of waler,

and g is. of course. the gravitational acceleration.

p = I!k is universally I:orrect, so IIko i� indeed the particle momenlum p.

(0) The fundamen"," relatiomhip

lit is not well defined. but this is its approximate, or cenlral, value.) Making thi� substilUlion in the

expre�\ion for VS >U from part (a). then using the R , relationship between relativistically co momentum

p

rric

and panicle velocity v, show that the

group velocity again is equal to the panicle velocity.

(a) Make a qualitative sketch of group velocity versus wave number. How does it behave for very large

k? For very small k?

(b) Find the minimum possible group velocity and the

r = 0.072 = 9.8 m/s2.)

wavelength at which it occurs. (Use

N/m, p

= 1 0� kg/m), and g

47, For wavelengths greater than about 20 em, the disper sion relation for waves on the surface of water is

... =

...... ..., .. ...... mwmt., tb) Wb.I. "' .... _ wuId ha.... and yeI: bI bouIrd' (e\ Wbu .. .. ......

\. �l. la) - :"�) h'lIo\\ , from (bo21"', th;ll p.... .... · leJ.n1ed in E'anlrok 4.1, In Il GaUM1an fun.:u\ln A, , .,l lhe 1,lnn I$lan WI'''! tun.:h1ln "'",'\lld � prop',rti,'nal l.... � \) ..quareu t'-I "; ! .... ) C\lnlpJ.ring "'" 'Ih the IIme IkpcnOelll G'1U�..lan pmhabihty 01 �uJ.hon ttl-'�). ""'t \et. Ih,ll Ihe un.:el1ainly in pt" llion of the hnle-(','ohing Gall"i;ln wave functllln of II free panicle i, gi'cn b)'

"'-1W¥l. il" \\"I'Io'ft1 eneraY ltale would.. of couno. be \be p:t\lUnd ,tate, bul ,.,ould It be bound'? AuuI'M 1hM• •,

letit tur 110 Yohtle. It OI,..,.-upws. Ita "round stale. wbleh l& much I'Met Ihlm 110, and lbal the burien q\lelity as. \1;ide, Sh\'IW

thllt 0. t\l\I,h avenae time it 'NOUld renWR "'-,und i, klm:" by T (mW'/lOOMLl)a2... . where ,r

-

1.\'�mllLvfa

L

11ml ,.., 11 ,u ut. .. ill f: and in.:reru;e, wiln lime. Suppose the wave I"un.:lion of nn electron is inilially delennineu lO he (I Gau....ian oj 500 nm uncertainty. How long will il lake for Ihe uncertainty in Ihe electron\ po�;tion to rellch 5 m. the lenglh of a typical automobile? 50. Show that the qUite genernl \\ave group gi"en in equation (6.2 1 ) i:. a �olulion of the free-particle SchrOdinger equation. pro\ided that each plane \\ave's wdoes sati,f)' the matter ....ave uispeJ'!;ion relation g;,'en in (6-23).

Com prehensive Exercises Solving the potential barrier �moothness conditions for relationships among the codTitienls A. B, and F, giving the reneetion and tran�mis5ion probabilities. usually involves ralher mes�y algebra. However, there is a spe cial case that can be done fairly easily, though requiring a slight depal1ure from the standard solutions used in the chapler. Suppose the incident particles' energy E is precisely Uo' (a) Write down solulions to the Schro dinger equation in the three region\>. Be especially careful in the region 0 < .f < L. It should have IWO arbitrary constants, and it isn't diffLcult-just different. (b) Obtain the smoothness conditions, and from these, find R and T. (c) Do the results make sense in the limit L _ �? 52. Show that if you attempt to detect a particle while tun neling, your experiment must render its kinetic energy so uncertain that it might well be "over the top." (Hint: Apply the uncertainty principle, and note thaI the parti cle must be localized within the penetration depth.) 53. A particle experiences a potential energy given by U(x) = (.r2 - 3)e-� (in SI units). ea) Make a sketch of Uex), including numerical values at the minima and

51.

F:\cn:i�e 54 �\$'� il n,u�h Iilelime for a trapped panicle t�l e�(a� an cndllsu� by tunneling. {Il) Cunsider .n e\el.;lmn. Gi\'en lhat W � UK) nm, L - 1 Itm, and ", nd «- Uo&.'\liiumption Uo - � cV. fiN \erHy thllo! the: f.. hold�, then ("'aluat!! Ihe hfetimc. lh) Repeat part lo.), but flu a 0.\ p.g panide, \\'ith W '" 1 mm. L =0 I. �, and a harrier heighl lio tnal equal� the energy the panicle would have if ,t� �pc('d were jUlii! I mm per year. 56. Exen:i\oC )$ gi\'h the: ..:undition for resonant tunnehR8 through two bLlll"ief"\. l"oeparuted by 110 I>pace of width 2s, expres,e!.l in term, of 110 fa..:tor 11 given in Eurcise ?>O. Show thaI in the limit in which thebarrier width L _ :xl, thi� .:ondition become\ exactly tnergy·quanlilaliQR con dition (5-22) ,'Of the flRile well. ThUlii. resonant tunneling occurs at the quantized energies of the interve"'R8 well.

�5.

Computational Exercises 57. A computer can solve several equatioRs in several unknown� ea..,ily, and here we study a particular E > Uo barrier problem, where all the values it needs are real, Once the tompUler finds the multiplicative CORSlants of all the function� involved, we can verify equations (6- \3) a� well as 'OCe what is happening when the particles are "over the barrier." Sti\l, it helps to simplify things as much as po��ible. With length. time, and nuw. 1ll our &a po�al, we can choose OUf units so that the panicle mIlS m and the "a1ue of Ii are both I. and the barrier width L is exactly '71". Suppose that in this system of units, the energy E of the incident panicles is I.. t 25. and the barrier height Uo is I., Furthennore, because only ratios 1ft ever

7 Qua ntu m

Th ree D i mensions a n d t h e Hydrogen Atom

Chapter Outline

7.1 The SchrMinger Equation in Three Dimensions 7.2 The 3D Infinite Well 7.3 Energy Quantization and Spectral Lines in Hydrogen 7.4 The Schrodinger Equation for a Central Force 7.S Angular Behavior in a Central Force 7.6 The Hydrogen Atom 7.7 Radial Probability 7.8 Hydrogenlike Atoms IJ 7.9 A Solution Examined IJ 7.10 Photon Emission: Rules and Rates

P

erhaps the most profound failure of classical physics is its inabil ity to explain the simplest possible atom: hydrogen-an electron

orbiting a prolon. According to classical physics. the atom should be unstable. Any lime a charged particle accelerates. it emits electromagnetic radiation. so the continuous centripetal acceleration of the hydrogen atom's orbiting elec tron should cause it to lose energy and spiral into the nucleus. What we observe, however, is that atoms are USUally quite stable, emitting no radiation. They can be induced to radiate electromagnetic energy, but they do so only at certain frequencies. And while easily demonstrated with a simple diffraction grating, this too defies classical explanation. One of the first great triumphs of quantum mechanics was its explanation of these observations. But before we can understand the hydrogen atom quantum mechanically, we must extend the SchrOdinger equation to govern matter waves in real, three-dimensional space.

7 . 1 The Schrodinger Equation in Three Dimensions

to one dimension. the SchrOdinger equation is

a U(x)'l'(x, t) = ih-'l'(x, t) at

231

As we know. this equation

IS

ba."iCd on an

ene� accounting of the t general. a

rur.:.:: I!.I! . ��.... Sec�'" GIIIt

U = E. !qJBliaI coordinates.. U(x. _v, .:}. as Iii me wave fuocllon. 'if! t, Y. to I), In Ihrte dim1,ntly oqall�_ W('re \,h(') I»" h�e. the sollJ·

t,om wou.ld be UponeOllal nther Ihao "nu"",,,jal Ellcmw 17 dl!'mon\lnlC1.. hov,·c'cr, thai C\por1I!'n·

lial� cannot be 0 al both wall\

function IS 0 outside infimte walls, continuity demands that it go to 0 at both

walls. These are exactly the conditions that gave us standing waves in the infinite well-the solutions are essentially identical.I

F{x) =

nx7TX Axsin L ,

G(y) =

ny7TY A sin -Y

Ly

H(z)

=

nz1TZ Az sin T

,

ID

(7-6)

'Tbt only rn.1 difftf'ml.� is thai �ach dirnens.eon is lIadependeni. 10 the COD ,WII' • ond L I1. ·

be 0, f

(1. 1. 1).

The corresponding wave function is . 1 1TX . 1 1TY . I1TZ

sm-- sm-1/I1 1 1(X, y, Z) = A SIO -2 3 1

With three quantum numbers, it isn 't quite so obvious what state would have the next·higher energy. It is left as a quick exercise for the reader to verify that itis (n..., r1y, IIz ) = (I, I , 2) .

Chepter 1

236

Quantum Mecha

TABLE 1.1 States in

the

3D

fr�� / £�,J'.#I, 2"';/) 6

2. I. I 1 . 2. I I. 1 . 2

6

.

11

3. 1 . I 1 . 3. I 1. 1 . 3

11 12

2. 2, 2

14 14 14 14 14 14

\ . 2. 3 2. 1 . 3 1,3,2

2. :l l 3. 1. :!: 3, 2. I

2 , 2:: ., . .:.

.J. I, I

1 , 4. 1 4 1 ..:

1 , 4, 2

2.4. I 4. 1 . 2

4. 2.

1

2. 3. 3 3. 2. 3

3. 3, 2

4. 2. 2 2.4. 2 2. 2.4 1.3.4 3. I . '"

1 . 4. 3 3.4. 1 4, 1 , 3 4.3. I

3. 3 . 3 5. I . I 1 . 5. I I . 1, 5

nil · m

18 18

19

-

19 19 ...:.:,

_ _ _ _ _ _

21 2' 21 21

21 21 22

22

22 2' 2' ,.

25 25 25 25

25

25 27 27

27

27

bers corresponds

10 a

. 51Tx

17r)' h rz sin -s' 10 sin L sin ---=L L = A sin L L x 1 7r)' 57TZ . 1 1T l-rr z 57T)" SIO -- sin--s' A l-rr x 5 sin L L L L sin= A sin L L coordinates.(x, y, Z). mum values at different maxi their have these else. enero-y-IS call d same g the If nothin ce-different wave functions. having The coincidenand energy levels for wh.ch .t .s true are said to -rrbe-h2dege�ial' 11 ) .IS The energy 27( ' /2 degeneracy, l). contex this in enl judgm l 2 ) are mora L 2f. no '/2m 1 2 ( -rr ' plying and sa.d (im -rr2f,'( 2mL ) degenerale. Levels 3(onds funclion. wave said to be 4-folderate single 10 a to be nondegen -each corresp }1TX

37TY

3-rrz

= A SIO-"'5. 1 . 1 L '" 1 . 1 .

:.-.. " ::

_ _ _ _ _ _ _

1 . 2. 4 2. I . 4

•

17 17 7 1:: ..:

_ _ _ _ _ _

3.3. :: '

energies. Of those ers for many allowed numb um quant 2 of sets WS j2rnL2) correspond 10 l 2 ( 7T2f't Table 7.1 sho the energies 3(-rr2fl2j2mL2) and . , 2 2 I ) and (2 I , ( I , only y shown of quantum numbers, respecuvel (- The different sel,' four sets 0 from quan. que lts uni ce, resu /2mL' ) , for instan set of -rr '-h ' e(lch qUOnt 27( y, 11m energy ng Ihe same. energ BUI despile havi ers. numb . tum cflon. The wave funclions are different wave fim

II

-

1.3.3 3. 1. 3

real three-dimensionaI World '" our in ons cati appli any to m e the same energy and the number of equaI-energy An idea crucial s can hav metrY 0f Ihe syste state ' m. that multiple . sym the w.th s ease possi ,Iales incr the box is as symmetric as ble: a cube. Suppose Deg eneracy

Energy (7-7) becomes

9 9 9

, " ,. I

I.

Hydrogen Atom

6

1 2. 2 2. I. 2

3, 2. 2 2. 3. 2

well

3

I. I. I

nsions and the

Dime nic� in Three

10_

=

0

•

EXA M P L E 7 . 1

'

e

cubic 3D infinite well �f I nm ( I . 2, 1 ) state of a the (n.r' n�_, n�)on's energy n. (b) Where is I e P

k\cb

f. ,

I�',

E:l

. .

,

E,

1-: 1

,

f-'l

r·z

,,

cl

-- 1 .�1 eV " \ - n,6 e\l) "

1-.' 1 -

-- 1.51 1.'\1 - .\.4 cV

t-- .\.� cV)

\ ,\)eV

� - I Hq::\I)

" lU.J. eV

�-

\2,1 eV

E"huhll'l

::J hdA.. l1r A -' hdf.' The photon wilvclength\ folk)w Irom pru'\Qn' Rel.:lllling tholt the prOllu..:t lit' In the pre.. cnt unil.. i!'. 1 240 eV • nm. we ha\'c

1240 eV

.

nm

1 2 . 1 cV

A:\-_l

:=

;

1 2�O eV ' om -- -- = 1 .9 V

656 nm

1240 cV ' nm 10.2 eV

(Notr:

- 10-' nm

-=

122 nm

Some values may not appear to work out exactly. hut this is due only to

round-off error. ) The first and third wavelength" are. respectively. the ..ccond and

first me mber\ of the ultraviolet Lyman series. The second wavelength i.. the first in

the Balmer series.

Spectral observations are one of the primary "handles" by which we test the claims of quantum mechanics. but they are also an indispensable tool in chemistry. biology, astronomy, and many other discipline!'.. We discuss e\e� ments other than hydrogen in Chapter 8. For now, we !'.\mp\y note that each element's electrons have different quantized energie!O. and thus emit different wavelengths, a unique spectrum that serves

as

the element's spectroscopic

"fingerprint." Sending an electric discharge through a tube fined with a dilute gas is a common way to reveal such spectra, Another way is simply to heat a gas, although this requires very high temperatures (see Ex.ercise 3 1 ). Cold atoms emit no radiation-all electrons are in their ground states and stay there. If the sample is heated, interparticle collisions may become energetic enough to cause the electrons in some atoms to j ump to the first state higher

1M Ga;""

0

I. ....

n . n.. [lo " [ _ _ "' Ik H� Atom

..... ". "" __ .. '

J

I;ger a�a.

price we pay for simplicity in the poten,.,,1 This hefty expression is the factors between two derivatives in the r · . . y 0f haVlOg energy term. Its pecuhant but it merely gives a more compact fo and terms may be disconcerting, act .on everything t� its right-i ncluding f Each derivative IS understood to . . . mind that this exp . IS applied Bear course, 1/1. to whIch the whole thmg in Cartesian als parti nd seco simpler sion does the same lhing as the t arrive at the time-independent SchrOdt· nger nates. With everything in place, we . equation for a central force.

9

10

�, 1

[

_ __

2m ,.1

a ar

_

( )

a + ar

,.1-

esc

a 8a8

(

. Sin

a 8a8

)

+

]

� COO��

a esc' 8- ojJ(r 8 4» . a4>"

+ U(r)ojJ(r. 8. 4»

=

E ojJ(r. 8. 4»

(7-'51

To understand what this teU\ us. we break. it into pi�ce!> thal we ..:an ,.:rutinize �eparately. A." in the 3D inrlOi.te well. we sian by writing the wa\'c

function as n product of three functions, each of a different independent \:triable:

Here we IOtroduce some 'Very helpful bookkeeping. The \'ariable is in lower case. while thc correspondingfonction of that 'Variable is in upptrcase When

we insert this product function into {7-IS}. we obtain a �parale differential equation for each 'Variable. The mathematical steps by which the equation'!. come about are outlined in A Closer Look: Separating Variable'!. for a Central Force. In the following sections, we concentrate on what the differential equa, tions tell us,

It. Closer look

Separating Variables for a Central Force

variables in SchrOdinger equation (7.15). we first rearrange it slightly,

To aid in separating

'

(

0

)

0'

1{1(r, 8,q,) cse 8 a sin 8 a V1(r. 8,q,) + csc2 8 8 8 aq,2

AU dependence on 8ngle� 8 and q, \!i on me left and aU dependence on r is \In me right. The u...ual deduction appbea: Because the variables can take on values. indepc:ndcnUy. both sides must equal a consl.D.nt. for which we choos.t the: symbol C. Setting the right !Oide equal to C. festoring to R its argu. ment, and rean'anging a bit yields the ndlal equation. d

��

dr

(7-16) Now we replace V1(r, 8. q,) by the product of three functions, R(r}0(8) 0) or exponen k . this could imply sin tlal . function. As we now out by the particular nature of th. ed ru are r latte the i\ 0) solutions. but (D angle goes from a given val ue lhrou h '7l' J tha mu azi the Ao:. . spatial coordinate and it must do so smo th function 4>(4)) about the .,:-axis. the utIOn can do this. But the reqUi.re y.. that no exponenual sol Exercise 35 shows . n see soo we as ' ns, , usot'daI functto sm menl restricts even the solution should be sine, cosine, Or rne me is whether so A remaining question mistry. sine and cosine are on the application. In che nds depe It used . tion bina com mical bonding, where one atom reaches OUlto che to ed suit l wel are often. all they '" ' ' ' ' s. The functton cos'Y has Its largest magn'"ude tIon d uec rent we d'� ong others aI ' ''' ' Iargest along the y-axis ( wh'Ile sm 'Y IS , 0 or 1T. aIong the x-axiS. IS when , ifically a camp usually choose a combination spec Figure 7,8), In physics. we IS smusotdal !) We discuss the rea = cos Z + f sm Z . exponential. (Never forget n. Noting that . a ctio h fun suc r side con first us but let if son for the choice shortly, ution of (7-22) is sol a e, itiv pos be st D mu n cl>(4)) sinusoidal, the

,

'

��

:

ell

is

(7·23)

� mteger.

shows that if this functi n is to meet itself smoothly whe n cjJ must be an The smoothness condition changes by 21T, then _' this angular situation. a periodicity condition-has therefore given us a qu tum number associated with the 4> coordinate/dimension. We choose the s Exercise

bol

36

Vi5

m[ for this integer. Attaching a subscript to distinguish one

function from another, we thus have

For each

� uantum

� ber

nu

me

�

tha

=

0, ::t l ,

::t2, ::t3, . . .

I�

an allo�:�

(7·24)

arises-and two dimensions are yet to

come-a physical property IS quanuzed. The property quantized according to

Figure 7.9 St.mding wll\'es on me ..�·llXis...

I

,, I, , , ( , ,

,,

,,

,

,'\, -

'----"

"

"'t -

, -�....

,

,,

mj

-

%. 0

,

,,

�

r

\

,

,,

,. ,,

,

,, I

,

) � ",f' ,I I, , I

,, ,

�"

-

- � ..

,

no,

,

,�

,

, -�

IIIl == 3

,-

,, ,

,,,

0

2

,,\ ,

�, ,

,\

; -Y

'

the value of tnt is the z-component of the electron's angular momentum. Lz: This may not strike the reader as even remotely intuitive at first, but consider Figure 7.9. which crudely pictures the real part of einl/4>. The dashed circle repre sents a "cJ>-axis." For mf = 0, eiQ4l is simply 1 and is thus the same distance "above" the cJ>-axis around the entire circle. For mf = 1 , eil4> is cos 4> + i sin 4-. Its real part passes through one whole cycle, varying from positive to negative and back. as goes from 0 to 27T. Similarly, the me = 2 wave. ei 24> = cos24- + j sin 2cJ>. passes through two cycles in an interval of 27T. In general, the circle's circumference would be an integral number of "wavelengths": 27T1" = meA. If we make the independent argument that a "wavelength around the 4--axis" should be inversely related to a tangential momentum mVt in the usual way (A =

hlp), we would have A = hlmvt' where m here is the electron mass. Combining these two observations gives us 27Tr = mth/mvl' which if rearranged becomes mit = mv{. and mv{ is simply the classical ex.pression for angular momentum (r X p) in circular orbit. The orbital plane for the azimuthal angle is the : mJ

111

, .)1,

L

•

�,

' 00 I t

I

I .

'"

I

l' m

, ' c., I( '0 t

l

.

{

yOdIIIt .

lumm, JuruhtnJ our r-er

•

h••� anguw �

�

1\

diS "

"""" "

..,...., tfCIII

'

.

I

."unu'

� eJ,...

Directional PrC' hav Bc ..... . .... b I� h cd anv ,.1Iue from (� U> ' ''' " nnt \i f t . \ Il1l l

".we

. ties

1 ul lhlDl . we clll\llOl ", r.siuered Ihe radial � pe< 00. bul we can II \cui dI. rlccu.m would be 100 c As ell has a um qu e wave function mc , l (II. set h eac (JII �"" "'t ' Thus. because arc degenerate. I n fact , ShOWn the ground ..rale pt I'f! ce ex els ' lev y T Ie 7.5 de mo all energ nstr.'e' Increases as n-, lab acy ner th ge de Exerci"c 46. the Increases. . . b er 0f states as 11 ' qUlcidy grow mg num Gi ve n its spherical sy m mct We do is rather special . Actually. hydrogen entation-a componen t ori al ati Sp an , gular m( pend on not expect E to de ears i n eq uation app C? of at wh t Bu (7,31 ), 't be a factor. momentum-shouldn The fact that they do n ot e it? on nd depe es d energi pend Shouldn't the allowe In general, the allowed energles y. rac ene deg l nta ide acc fOr on f is known as nger equation before the h dro 6di Scnr the of t par ial gen. equation (7-30). the rad " is only in Ihe special case of e sim e. on nd depe do . rted ple specific U(r) is inse entally" does not A II)' de ial energy that E "accid ent pot via m ato . en rog IIr hyd simple case would lead to qu m this en m fro rgy ergy ene ial tion of the potent ctrons orbiting e nU In particular. additional ele Cleus levels that depend on e. s destroying the accidental de�eneracy thu , rgy ene ial ent pot . would alter the ental to chemlslry of energy on C are fundam e enc end an dep and d y rac Degene . apler 8. are dIscussed further 'In Ch

V��I �

I

;;,sf

e

07'

It

�

,:

�

:

Normalization

complex square of the wave funcl'Ian. The tern· The probabil ity density is the ' . . S drops oul In the usual way. As we noted m ee· poral part of the wave functIOn . . r the pola part IS and out, s real drop also . Now . 7 .5, the aZimuthal part tlon arrive at we , real also is h whic al. radi incl udin g the

1'I'(r. 8. (6 +

0'

.n ... fh e . ,,>,lf

-2 )(O-2rI3oo = 0

30 0

0 and r = -. but these are obviously minima. wht� P. It also overlooks the independent behavior of hole energies wilen they too are confined. but II is still a good eSllmate. A forbidde� 3\ elet.:trlc dipole tran.�. lion' rna} occur b} other proct:....t:... Charge may. lor rn'tance. O\cllIate

ma�nt'ti(" dipole. where a current

bI

loop penodlcally ..\l.ap.. t.hreetlon... But \lII;h an "anlenna" i\ much le\\ efficient than an electric dipole. \0 If'oln\ilrons ou;umng by lhl\ pnxe�.. � much .. lower, typically b} a ractor of 10'. O�llIiJlion.. e"'en more complrcated are po..\ible (e.g electric quadrapolel. but Ihesf" tend to be yet more InelTicient and ..10\\ Tran\ltion.. that can <J..:l,:ut only by !.IO\\< procc::...e::. cau\e cenarn atomiC \tate" to be unu\ually long h'ed .•

These are known as metastable \talel> and are Imponam to the operation of I la�r, di«:ussed r n Chapter 9.

PROGResS A N D A P P L I CATIONS

Figure 7.20 AI hIgh II, '0 many Matt:)

High,n States: Rydberg Atoms AlOm\ In which an t'1ci,:lron 1\ t'�l,:lted 10 \ery hIgh quantum Slale�. typicall),

lIon could 'Ipell

40 nnd hIgher. art: common lopic, or di)cussion

n

1 I I

art: av:uloble thol lhe proper combina_

oul a word !

nowada}.. Known a\ R)dbc'l: aloms. the), arise in many Ltetl\e d�n, 01 phY'K\ re\earch. While Ihe t'lectrons In gTrWII(J-IfIltt' mulilelectron Jlom� inter3CI ::.trongly wilh

eat:h other. ir e"Clled to a very high le\"el abo\e the ground stall'. an ekctron

In tim' alom

orblh a� Ihou!!h in a hydro

genlrle ;ltom. wrlh the nudeus and relatl..ely compact

�mdlnder ol lhe electron, comlllutlng an ilppro..imately poinlille po\llne co�. Wllh we3k.ly

bound electron..

orbitIng \0 far rrom the nudeu\. Rydberg mom!. are:

t:leCln.I'tallCdlly malleable dnd thu, well suited to �Iudie!> In many Mea'l or sludy. Rydberg Slate::.

or diamagnell\ffi

or panlCular Inte�!>1 dre Iho\e in w.hi(h t and nit

the mnxirnum allowed (both n

art:

I). for in these !>Iates.

the elet:tron\ orbit rnO\1 re,emble\ a clJ�\ieal circle Figu� 7.16).

(cf

Owing 10 the \Iow me3n� by which they can

..hed ener8y. the�e circular \tales of hIgh " tend 10 have

Once the subject or theoretical IInlihydrogen atOnl� moving slow enough 10

\cry long

Antimatter in the Lab

energy ph)'''lc\. Funhel"nlOre. le\'eh al

debate alone.

electron In occupy

con�iderable quantitie..

Irfetl me... with imponant 3pplications in high high " are closely �paced. and ellch level ha.. a large degenertlcy. allowing the :r

wide rilnge or very Intere::.!ing

combillllt/Oll.f (wpcrpo\ition\) or \tfy ,lIflerenlilil equation, (7-S) a� well as the �uired

th� energie)' correlopond"

An e\(:ctron i, trapped in .. quantum dot, in which it i& (onfined to a very .:,mull regIOn in all three di mensions,

If the lowe�l-energy lmmition is to produce a photon of 450 nm wavelcngth, what should be the width of the well (assumed cubic)? 22. Con�ider a cubic 3D i nfi nite wei\. (3) How many differ ent wave functions hove the well" ,

'llitJlmu .. hlf ft..,) aml .llt") In'ldC' th(' wC'II'� Ar(' Ci llnu C, pi,)"ti\i(', negll\tve, l,r lero') hllJ'f\"lIIl!. ul"rmpmllC' condillon,. lioli Ihe \l\low«i \ialue .. �If C, and C,. (0) H\lW Illan), lIIuC�ndenl quantum numbers JUt: lherd d , l hnd the all�lwC'd eMrJ,el> f I') Arc Ihl'1'(' cnerlpe� r�n which Ihere " not a unique Cl)ft'1:'pondmg wavc lun..:llon')

,h(lW thilt the I.:omcl nomlah/iltlon con..lnnt i,

gte� po\,>lble? (b) To how many differe", statcs do these

•.

What lun!';li,'n, w�luld be I:tc!,;('pll1� le ,tundmg,wBve

.".� \, \', .:)

boundary condillons, An eleclron " confined 10 a cubic 3D mfinite well 1 nlll on a 'ide, (a) What al'(' the Ihree lowest difft'rr>nr ener

, Iln\l ('

II\linlle- \$'\1.

\\'\\I. 1\1\'("1 I\'(,U 'I\ll,,'lhh ",h\'n 1\

\"ml'he, i\ 1$11\1,1

lin I

\.I!ti.:'

"I

m;h', .11\\1 Ih\,' ""'I.I�': \,11 1

" I\,,\\o C" 1\\\1)

1 ,hl'uIJ tIC" Ihe

Ih('

,um

�

I}

�� II"

Nt·\ � I)t.!.\'.� I )/('>, ..h,,,,..

..hl'ul,1 h\1'1... til\' 1\\cnt�(' hut the

" 1\\

I Wh� Ih" I' 1\\1\

\1t'lI·cll'/im·a \,I\lll' ni l � IcaIU\I\" Ilw l'i�llnlu, "PPf\,\\\'h ,d.:rrcl1

41,

hI III S"dill,, 7,' )

In 'NII.· nJ". to. Ihl' "\....r,llIlr hn the "lUM'\! \11 Ihc ;'1\Ij!ulur 11I11lllCIIIIUll ,� ,hllWfl l\l tlI.:-

What

'L " (

:",''lInPlnlCnL� 01 .U\gulllr momentum ,... 1\ J'O',ible tl> nenl of angular momentum, ttk' alUm I' equally likely to be found WIth any llll(lwed \·alu(,' of L,. Show that il the probabilIty dcn�i� lies for lhe'ol' dillerent p!)\\ibJc ,tale\ are added (with equJ.1 weightin!:), Ifle re,ult i, rndependent of botl! ¢ MId (J. 5J, A wave tundlOn wilfl J. nonrnfinnc w;l.\elength-howe\cr appl'IJ)(imale it might be-ha\ nOllLero momentum and Ihu\ non/em kineti.: enc'l!). E,·en :l 'ing/c "bump" h,t\ "ml'ti\." enc:f!!l- In eltiler Col.\e. we nn \3y !lUI thc function hOI, kincli.:: Clk.''l1)" hec:lu\c it ha\ cUl"\alUre-\l ,econd demalne. Ind(,'Cd. 1he !.anetu; energy operator in any WOrnJnolle \}�tem imohc_\ a ,econd deri\ati\e. The onl)' functi�ln Without kinetic energy would be a ,tr.ught line. A\ a \p:t;ial ca\e, ,hl\ Include, a con�U!nL whi�'h OJ:ly be thllu�ht ot a' a fUnctIOn with an infinite wa\elength By I�l('king at the cun·atun: In 'he uppmpnau' dim,mioll(.{). an\wer the following: For a given n, I� the kinetic energy '>Old) (a) '.JdJa1 JO the \tatc of I()\\.e,t t-tha( I�, ( "'" 0; and (b) rotational in the \Iate of high('.\( (-milt j�, ( '" II - I ? 52. We h,l\·e noted thai for a given energy. a,> I increase� Ihe mallOn " more like a circle oIt a comtant r.Jdius. with ,

the rotational cnelllY im:rea\ing 3S the radinl energy cor· n:\pondingly dcaea,\{'\ Bul l\ the mdial k.inelic energy o Illr the large\1 � valuc, l Calculate the ralio of c)(.pccta· '

tinn value\. rJdial energy to rotational energy. for the

(n.

f.

111/ )

(2. I , "" I ) \tale. U�e Ihe operator, KE"" KE""

'"";Jilr a photon and Jurnp� from the ground "ale to ii' n = 2 level. What wa!> the wavelength 01 Ihe photon? 68. Roughly. how doe\ the "tIfthonormoJl .. Tht m.un P'l'nl ll WI I( we ,,"alullr " pmhthihl) mltgral " H::r all .,..." 01., '., ur u( "'J.�J' lAC gel I .urt,urpns, In,I),). bul" we- n81ua1t Ilkh an mlC,rill Ill! til. -"'1 tIt �J-.1 � p. 0 Thill fg�n" III hr lrue ttlr.:.lll Ihe Y'1fl'IlI wJwI'C' WI' h.1I\C IlINI.IIIC1J "r "Iwlll akmnl ... o( w8Vf I'utl('lon!l. ll' B , Ihe partlde In a �'\' lhe hlrmonK: I)KIII.,m InJ ltat h�Jr\lg-en ..111m) fh . mlC,,.IIft, U\o�.al1I 'pace, l'k,,,,, Ihal t'\prL'��um (?,oW) 111m" nl.lnnalll(d unless a 'J('lilf 1.1 I' induJed "" IIh cht p,uba�I"I�

j

Comprehensive Exercises 1'9, lllt 1/1. /.II �Iak'-t��, 'I..I�' III "' hll'h m, - ll--h'l\ 1ll1" ! 01',111 pnlbu�IIiI�' ,/cn,l!) ...hll1.!1 lhc ,NI\I', .)11.1 '0 11 I' ollen f'('t('rrnl lll a.\ " 21'. 'Illlt' Ttl .dlllil ih prllO-lhlhl)

di:nltily 10 "lid; tlUI In tl.h...r '" J�'. .Intl tllU, till'l!Il"'e I'-In'

1M,j.� ki'kh \11 nkll«'ulat I'Illlkhng \1 Ith (Ilha ,lWnh, Ifl ,

""Imil' c/('I.:lfUII 111.1.\' /1"Uflk.' i\ "illt' IUl1dltl11 lhal I' UJl

.!Illlt'hnu� 1....lmhm,lllIm LII nUlltlpk' ".lll· fundllm, 11pt.'n 10

+1 + "':',1

it On!.' su\"h hytmJ ,1.1It''' I, lilt' 'UlIl t/I�! I· IM'/r'. 81.'\.'lIu!< Ilk' .sdmiJln�t'r l"lu.llion I' a 'inc,if Jif'....n::-ntmJ l"\juIIllLln, .. ,um llf �tllulI{ln' II Ilh Ihl' ",Ime ('nctR\ j, ... §lllulilln wllh Ihat ('nt'�� AI'II. nllmu/i/alilln l'tlfhWJb ma� � IgnllmJ in lhe /t11/1l11 II\� 4Ut" lllllb I "

Iii)

Wnle Ihls """H' llIn,'!llln anJ I" pR1hahihl) Jen-

sit) in tl'nn, til r.

Ihl

0. imJ I/J. Il\.! Ihi' Fult'r fi1mlUia

'" slmphJ., )\lur I\',ull,)

In whll."h tlf Iht' fllllt'" IIlg \Ia), dflC' Ihi, 'Iall"

tiiller fnml Ih pOln, 11 1."" "'!, 1.+ I .\OJ tP2_ 1 I' and fRlm Ih� 21'" �Iull': EnL'!}!) , RaJra.l lkpendl'nce of

t" hi}

... � deaslIY" A.,w.dtfMA .... ot .. �1iI)' decw,y" nu. "* I unto rderred 10 IU the- 1p.. \\'by" How ml&hl IIIo e prodlk� .IIi Jp. "'.lik'''

_ c.1.lf\Nder l'a11 puuc:b thai c'penent.'C .IIi mUluat lOI\1t

loren Tbr ..-I.uk.1 C\lUoIhon 01 rnot-. "201. ' "', . ancJ 1(11' p.rude 2 .. f, . JI'"1' wkno the Jt-c mean� a lime deln'� '

hid DO nltmal

(or putid(' "

I

•• ,

Show � Ihni: are "Iun "knl 10

"

.�

- I:llnlolJlnl

onO

'...1 - ."--a;1Io

....� ".. . lm,' 1

,', .. ,. ..oJ

m lm�

mr

+ fIIJ

In lllht'r \\I'flh. l� mulllln I:..n tlc: an.. lyu:J In '''''0

Pll'�-(:�: Ih.: �L'nl.:r lit rna" mollon .1I I.'on,I,umeu inllnlte. '" In the rhapIl.'r, Ilul " of m..,.. ttl I , I:.-hile 111-; I' the mass orthe (.rhlung negallie charge, (a) Wh;.)1 pen:entage t'rror is

lntnxlun:d 10 Ihe hyt.!rogL'n grounJ-..t::"e energy by

a"ul\llOg thn •

crl'I'" i Ii.ul,k' J'l"'\tl\'("

cbIu'1:" , 1l>4 III h",tfllit!:". but 1'Ift('

\Jll bo.� � \� IhI: '-1.1\\\ ' , ... th.11 1,1 the \'I("ll�'" by Ihc ('mlk'mh ttlr..:e I' �'\l·n b) l \'f'�i·'h1't·llmr {d (1" ...l1 lhlll r _ /11 uti. i, \h,� \tI • In!!u\;u rl'\.�\ll'n..:)' e'II!(II IH lhe lllill1t1l�Ull'il,lnlp pholtl\l Irn{u...nr) al l.'Hhl:r �LlI.I 1,1 hy\ll'\lgco', allowc\! encfg'�'·)

s.c. Ik e'pcl'\Ullun ,,,lUI! I.llh�· c\ed"'"" ).,I11C\1..: �ncrg) in Iht' hydftl��1\ };I"unti 'lilll: 1'1111111, Ihe ""'�IIIIIIIIt' ,'I the 1I1t,11 en�fgy (,ee I',el\: ...e flO). What mu', be Ihe \\o,l1th 1.1 II ,,:l,Ib,� 1I1tinll'" well. ill IeI'm, Ill' "0' for II, �round ,IJlc h) hale Ih., ,amc cnt:'rgy'l

86. �f'C''':lrill llllc" urt' IUlly due In lw,1 ellcd�. Doppler bnlJlknillg >Inti Ihe unr.:�n'\lnly prull;illle, The rdJII\e e e ",,,e!cngth l1ue to the flr'l ellel:t ( .. ) " gl\en by E,cl\:"(' Z,57

\Mt;IIIOn

HI

.lA

A

\i )'/"flTfm

c

",here T I' the lempcrmurc ot Ihe ,>ample lmd III Ihe light. The Vllriation due 10 Ihe wcond cHel'l (�cc F.xen.:bc 4.72) t" given by

m.l" of Ihe parllc1c� e1ll11lmg lhe

\

_

1(111) �a. w1lh I" nunu,,\lm urtN.t 1Wbu. � .

rmhtol"___ �II" lOl l m aNS ,,� IMlUmum. or IPheboG.

1\10 unlt'i .. fal. Wtwn ... tbole minimum '" mul· mum radII. II� I'MIIU' ", lit ,,"1W"Ioe, noli chaftll,q. MI '� rilJla' �U�lu; t'MlV) "O, .no.!. \I ""�,tc eneIJ)' I ('Ilurch "".II,,"�I hunl "" ."'11:,,1 mn:hanl\:' rob.lioaa' ,,'Il('Il) " ¥I\CIl h\ " �/. ""here I I", lhc mumcnl uf U'ICI" llll, ",hl\,t! hI( . "'''Int .. """,," Iii. !IoImpl) ",,.2 ,.) 1bc �" '''\l'n, '1'Ot'C'\J ,1.1 f'('lIh1:"hu" I� f'I !�� 104 mil. ('111";\1 \3\\' Ih II.I\v;uhu fl\l\n\('l\Ium 1M \lenh that the ,um of Ih," filil\ 1\;lllIllIa\ ,""I('l\lla\ ('nt'lJ) lind n.t.tlunal t'l'IC'raY 111'1.'

(,,\11;1\ .II \"C'flhC'ht'lI 1111,1 "phdl"n

,Hl'ftIf'mltf'r'

\1I�tlh" tl\t>mC',,\um I' n'n"f'''N \ (d ('1I1I;ul.lt Iht '1,11\1 ,,( Ihe l/-,a\U,l\l\llllll ,,,'tentl,l.\ C1\('I)I' 01"" nltll.lI\lml.l \'II':: \�� ",h,'n Ihe ,llt'Imple ca�e.

�

�

�

� �

lrtnctb

�

Applying the Physin

A magnetic field i.. applied to a material In which an electron is able to nIp liS 'PIn orientation and in which internal fields are negligible. (a) If an electron IS ' Inll1 all) m the low-energy orientation. which way I'> it!> �PJn? (b) ElectromagnetiC d ra lalion . , ' Of frequency f IS now applied. Whal magnctlc field sirength would lead t O this field for (c) Evaluate f 9.5 GHz. a typical absorption? ESR frequ(x1) 2m .p"(" 2) il9: h1

iI·ti

+

U(\" t21 - E

(S· I S)

when: U is the infinite well p.ltentiai enefJ)'. By the usual argumenb, each tenn in �nlhelt' must be Ii constant, which we have designated el and Cl' Setting each \enD to its constant yields a differential equation identicai to the one particle Infinite well SchrOdinger equation. except that E i. repllK:ed b)' C Thcrerorc. we can limply write dooNn the same ..ulution,. given In equations (S·I6). willi the C� ,ponding replacemenb.

As In the one-particle caq:, !.he pou:nllal energ)' dictate\ (8-15) a\ It stand� does nOt ha\c ,t, and '1 ..epatate. becau..e U{X1 (2), In general. need not be a

the �olullon Equation

• .

\Um 01 'oCpar.ue fum:tlOns of x I and Xl' The potential energy

u\OCiated With an inlemal force between two particle!. In a

\)�Iem i� u�ual1y a functIOn of their separatIOn. �ll - :cll. and

does not

ond

�parale mlO a sum For example. the electrostauc

potential energy ..hared by two elel.:lron!o i$

The quantum number. need nol be the "lme, \0 we u� an n and an n', and hecau . with thl' reMnl."tlon It 1 ndcf\ predu

on(' thin;: lu aher a .:1I0n\ that can be apc:f1nl(nLlJ ly \'cnhcd h'(lr) I t,;, . ui It I '" q U ite another to alter It merely }:Ie�au� It rC'n�n praJ.ll.:l101'l1 \'l 1/ h(' ,'rr/ ed expenmcntally (be..:au\e \\ c L-:.m't lnn.. V.hl\;h �tde h t f/lflll( · rc ... the tnctlOn e r , I ... nec.:e.....a[). wev . The tar-rC3(hmg nnd over I . Ho ,' .'\ e t"OU ld) "e diSCUS we ... oon ... cannot ... be ence explained uthervd\e. equ tiab1c cOn .... I 0 h t t lt e . h" r we a t P(x x ) "' a (; 'WI panu.:lc lahth \ ana.] 2. lea.\oe!o W might \' 2 o

fi'/"I.,

' " "Ju ...e � l �'"

li

H

II

I I re

_

_

tog

ngcd') We might try adding the �ame funcllOn .. Ilh the I nod 2

uncha

4

sm2

1:= P(x" ', ) L' .

.

4 1TX 1 L

�

, 31TX2

sm " - + .

L

4 L'

,m_

.,

\\ Ib.:ho.J

41T\} )1fX 1 01 I. L '

. symmetric-the same fu nc tion-under IOh::rchaogc of the rani. 111l s IS I deed B ut it is unacceptable because. quite \I mply, It I\n't the klUa.re of a ele labe ' S f the Schrodinger equation. To produce a \)'mmelnc prnbahlht)', den solution r t with the wave function. There are two ....a)" of modltying 0 _, 'Ia . . we mu'" � slry, rod · olutio i n equation (8-17) to meet the requiremenl.

�

(he p

ue ,

.1.

n

( ' '2) o//...(x I )"'..'(x,) "" .(xI )"',(,,) .1. (x I ) .1. "" (x,) (x , ) '" '" (XI ).,. (X2) "',' "'s

'fA

s

+

'.

1'-2

n

"' "

-

Symm2) 1 ' t "',,(2) would mean '" I . 0, o( r2, 82, metry to refer to he ch aracter af a state ,

lilt, Ill,

=

s

sym

ange ' when Iab els We use, the term exchexch n't change SIgn anaed, If it does . are s label when partic le etric', if it changes sign ' its exchange y mm etry is symm s hange exc sWItched, Its . ' , . anllsymmetrlc symmetry IS 0

"

are

8.3

The Exclusion Principle

ready for one of the central ideas of the chapter: Why i ... ..pin "0 behavior of multipart;c1e systems? 11 is fundamental to nature the crucial multiple indbtinguishable particles will he in a muhipartide of m that a �)\te symmetry, and II'I/ether it is symmetric or allf;.\vm exchange ,[�IC of definite of the system's parricles. All fundamental particles !ipill the 011 nlt'lric' depends is either integral or half-integral, and this divide, them into hilH� �pln s that predicts and experiment verifies that: Theory ories. categ \WO

We are now to

Sosons Particles for which s == 0, 1 . 2, . . . manifest a symmetric nlultipartic1e Mate " I 3 5 " 1es for Wh"Ieh !i' = 2' 2:' 2' . . . mum"rest un Feruuons Partlc anlisylllmcl'ric multiparticle Siale

Table 8.1 in Section 8. I gives several examples of bosons and fermion:;. It also reveals lhat all the familiar building blocks of nature---electrons. proton�, and neutrons-are spin.! . Being thus fermions, they always assume antisymmetric l\1ultip,rrticJe states. This fact may seem innocuous, but the consequences are re,ounding. Let us investigate the most far reaching.

Fermions: The Exclusion Principle Consider a system of two fermions occupying individual·particle Mates n and n'. If n and 11' are equal. the anlisymmetric two·particle state, equation (8·22). i� identically O. "',,( 1 )"',,(2) - ",,.( 1)"',,(2) � 0

We conclude that two fermions cannot have the same quantum numbers their spatial stale or their spin state or both must differ. This is known as the exclusion principle: indistinguishable fermions may occupy the same individual· particle state.

No two

To specify a particle's full sel of quantum numbers is to specify its state, so an equivalent way of expressing the exclusion principle is that no two indistin fermions may have the same set of quantum numbers. Although we ouishable • have considered only the simple case of two fermions, the exclusion principle holds for any number. 5 It was discovered by WOlfgang Pauli in 1924. and the achievement won him the 1945 Nobel Prize. The exclusion principle applies only to indistinguishable fermions. As noted earlier. this means that they must ( 1 ) be of the same kind and (2) share !he same space. Of course. all electrons are identical and all protons are identi cal. but we can tell electrons from protons. Their charge and mass serve as labels. We dOll't demand exchange symmetry for particles that can be distin guished, so the exclusion principle does not apply to a system of different

Exc\u!'oion prinCiple

�Se--.'c.'ral c.'nd-of-chaplcr e.'erci� dl""':Uo;$ � geIKr. aluauun of c.'quation {S·121 app1i�abk to

any nt>m

tier of p;l111Ch:� and leading to the \>arne C(ln..;ilNOQ. .... n as the Slatc.'t detcrmi.rwnl i L It � no

IV ' .. 04

�

"' ,., AJl1Iff"

. ' and. e le tron The .<eCOnd COnd ' ' l real SIIuat,o . lunctlOn... n, du tal l on c,ponentoall). "e

c .n n oto pr n'" Wge I )JIn. ..much a a\:e . . .. h . h n'. ", , , Ied fu are eenl r IOpic e Thes conden»les, . ' . (elect rons .1 0 the case of' form boson , Bose.Einslein '0 rtng pat . " s . Accord",g y, Ihey 1 . u' 11 ' in the other case - ) mo;; ato ' n IO . ferm . lIY. whole In lac' b050ns, ever the o,uperconducllV lUSIon principle. exc an by d e lraln eon, are no I anger

l

· artIcle s but the ns l

_

l

l

I

•

,�lunterpolnt to fermion". actually "prefer" to be in the ..arne "Iale. and Ih,.. i\ I,\hat S"e.. nt3tic attr t ac IOn. CompJcling the comparison. Z == I a neon differs by just one eleclr n ,odium and fluorine. but it'\ behavior is vaslly different from bOlh lIS � 2 I

•

8.9,

,

_

8.1-1.

ahrupt

' I

9.

�

�� e � ox�. �

f N

N

•

from

e

. fI sh l! I' full. It has no valence electrons to offer. as does sodium, and no hOles t i f 11, as

. (k'IC, Iluonne. It thus joins helium as a zero�valence noble gas. only gru�glng ly partil'lp.llmg in chemical reactions with other elements. A\ Z increases beyond .

10.

Ihe

v.�ence (now excludmg both fulJ "

==

3s and Ihen the 3p slales fill, .

and

�

1 and II = 2 shells) Increases exaCI IY as n.

"

'

.'

"

1,. "

---

,

' ,

-

' ,

'" "

--- '

,

"

•

�

,

+

*

•

1\

1\,

2

-

-

+'

..'

>.

*

L, 1

Bo

B

4

5

N

c

7

6

o

F

No

N,

M.

8

9

'0

II

/2

does during the filling of the /I = 2 levels. And just as the chemical behaviors of Z = 1 1 sodium and Z = 3 lithium are similar--each having one electron beyond a lined shell-so too arc the behaviors of Z = 1 2 magnesium and Z = 4 beryl�

limn, each with two valence electrons. This is the periodicity of the periodic table! Shown in Figure 8.15, its first column is valence 1 , containing Z = 1 . containing Z = 4 and Z = 3. and Z = 1 1 : the �econd column is valence Z = 12. The eighth and laM column does lIot correspond to a filled /I = 3 shell.

2,

The 3p Siales are rull, but the 3d states remain. Furthcnnore. as Table 8.2 indi cates, the 3d states fill only after the 4s. However, the energy jump from the 3p to

tbe4s level is large. Thus although Z

=:

1 8 argon does not have a full II

\1 is relatively tightly bound, so in common with Z

=

=

3 shell,

to and Z = 2, it behaves as

a noble gas of valence O. By these same arguments, Z = 17 chlorine, in the sev enth column, is chemically similar 10 Z = 9 nuorine. Both arc valence - I .

Si

13

"

.

•

§

•

'"

J!

I ,

.

,

•

•

•

•

I

• - -

•

, r . -

• I •

•

,

J

•

,

• •

i ·� I : w

N

,

,

: •

�

•

, •

•

'

. �

•

· ';

�

• • · -

r;

�

· -

,I

$

- -

•

•

·

• • •

· • .! -:

! � 1 • � -

,

•

�

tates.

II == fluorine art in in n S "" V " e (;U I e 4(0.053 0.050 nm =

valence electrOn Sodium'!> lone

Zeff = 4. 2

nm)/Z,"

has 11

::=

3.

3 nm)/Z,"

9(0.05 0.180 nm =

theless about the same size as Z =:. flourlne is never . 9 (Note that _ of fluonne s " = 2 valence eleCllons each . l e ma l l s e 0 ou, Acco,dmg 1 ned . We would expect the interveni ng I hydrogen.) so 4.8 are scree 4.2. y nl o of e t the �even II = 2 electrons should would see a charg hly twO charges. b � roug en scre fI as SIX others orbiting in the same range I electrons to e extent. Each som o t other al�o screen each about 3 IS reasonable. screening of tive effec an letely scree ned .by all lower·1I elecirons. II. of radii. so comp were ons == 3 electr ' elliptical orbit s·state, however, IS a highly If sodium's II only I . An ' of Z . tive . should see an effec nce InsIde the lOner clouds. increasing liS prese le iderab trOn cons gl\lng this elec 2 2"1 etfccti\e Z. approximately 3 / = 2.25 times an 11 == 2 is s radIU rbit II � 3 ? of fluonne because sodium's In hydrogen.. an than three times that rodlus I� more Um sodi effectively .,maller charge. The an rodius. higher n. also orbits es beJng In a beSid tron. valence elec . Z Fluanne has

_ _

.

:::;

�

,

8.5

Characte ristic

X-Ra ys

lowest·energy states u P IO completely fiJI the . alOms. electrons In ground.state . d vaI ence e Iectfons can Jump around am ng xClte . um energy. E n some maxim olve no more Ihan le s of ecwnward jumps inv Do . els lev ed fill un higher ns in the ult ra violel nge i ndicales, producing photo 6 8.1 ure Fig as S, tronvoll glh" In Section · much shorter wavelen made to emit be can ms ato r, Howeve

� r: 3.3:

Figure 8.17 Characteristic X-rays are produced when an inner-shell hole is made.

All Jl;i!"eter.ucd

°

o

... ..

". °

b

o

.,/'

•

•

n=1

..... eje ..:tmg an

..... orb lt , illg electron

and U¥king a hole

° 0 . whl�h i� filled b} eled tpn orbiting 0,�....!. an In a higtj"er shell " o ,/ creaullg an )V'ray photon.

we learned that electrons smashed into a target produce a

continuous spectrum

of X-rays via bremsstrahlung-but something else also happens. As Figure 8.17 illustrates schematically, if an accelerated electron knocks an orbiting

electron out of an atom's

illller shell, a "hole" remains. An electron in a higher

energy state can then jump down into the hole. producing a phmon. Of course.

this leaves anmher hole, so the process can repeat. Inner-shell energies are

often thousands of electronvolts. and photons produced in transitions between

them are therefore in the energy range of X-rays. 103_1OS eV, with corre

sponding wavelengths of

Figure 8.18 An X-ray specmtm. Characteristic X-rays

�

1 0 nm to lQ-2 nm.

Because atomic energy levels are quantized, only certain X-rays can be emit

ted by a given element, and each element is different, so these

characteristic

X-rays serve as tingerprints by which we can distinguish the elements.

Figure 8 . 1 8 depicts the X-ray spectrum for a particular target elemenl. Super

Wavelength

imposed on the continuous bremsstrahlung spectrum are some of the target's

characteristic X-rays. We use distinctive notation to refer to the X-ray photons

generated in inner-shell transitions. The II = I shell we refer to as the K-shell.

the n = 2 as the L-shell, the II = 3 as (he M-shell. and so on. These letters are A subscript also used (0 indicate the shell at which a LTansition

tenninates.

Figure 8.19

Ka

X-rays.

advancing from cr on through the Greek alphabet designates how many shells higher the transition began. An cr LTansition begins one shell higher. a {3 two sheJls higher, and so fonh. Thus. a transition beginning at the L-sheLi

(n

•

= 2)

•

and ending at the K-shell (II = I ) produces a Ka line. An LfJ line is from the N

= 4)

within

a given shell are not down to the L-shell. (Transitions shell (n energetic enough to be part of an element's X-ray specrrum.) Inner-shell elec tronic structure is largely IOdependent of the behavior of the vaJence electrons,

with their temperamental periodicities. so the variation of characteristic X-ray

wavelengths with Z j.:, relatively smooth. Fig ure 8.19 illustrate!l. the smooth variation of Ka lines.

•

•

.'

50

, tiO

z

EXA MPL E e.'

. otbInng a cha1geofZarr given in Section 7 8 as neodymmm. 1bt efdPS tor ll1 � b Of . K X·ray Z = 60 gt

_

zt,,;t fSDIIJ* lht W � the

�1I�J

r ON

�

�Jl)

The pboCon t

1\

b;fTOII

otMf clouds, and thai bef re o 5. � ilboul halfoftM L-�hell clouds add 10 the scree 15 o

[55'

he - -IJ6. eV 22 A _

y minus us fimal.

A

_

-

-

59'J

'

.,

/-

3.7 1 X 1a' ,V

=

:z

3 .7

1X

Iran\j.

/ ()4 eV

O.033 nm

between 0.033 om and 0.034 om Wave1e 1 eod)'mJum KK X·-ys n_ 8,, for many elemenlS using simil 1. engths wavel ar e ) e�lJm:lI apprO)(IJ:n 59 Excn:ise h a· f F' dilla 0 e I 8. fils Igure model 9 1 Ihe t w well Actus n

,

" ,

Iron) and �ho"s Jus

h0

ore

,I.

IJ

.

ways. Those produced by a "'-J . . X-mys are used many ,_ CharaclensllC IO \V 1l X-ray source for crystallography. Con common • are ·aJ ren rna verse rarger ly. ,amples somewhat unknown are also very usefu tho� coaxed from l. B y X-mys With tabulated values and measUring matchmg chamCteostic re,ali.V , e s elemental compositio . learn much about the. sample n and \4-e , leS, mtenslt con. . . electron IS not the only way to exci te chara . centralJon. An accelerated cIensh. . ' l c ic art p ) es are (alpha also comm X.l'3ys. Prolons and helium nuclei only use . . d ' X-ray emiSS . as partie "" e-mduced ion (PIX and Ihe leehnique is then known '" . E) . . techmque IS liS sensitiVity, this due to a low O N J advantage of The pnm_ e r back . contl The X-rays. uous ung brem bremsstrahl sstrahl ung of ground level spec . of the charged particles and has [rum is caused by decelel'3l1on nothing 10 · . do · · with the specific larger. Given comparab'e kmellc energies, a proton ' w· . Ith a much larger mass than an electron, has a correspondmgly s maller a ecel era· " . " lion. so il emirs less "braking rad latlon. _

�

lJ D V A N C E D

8.6

The Spin-Orbit Interaction

In this secrion and several that follow, we will study Some of the Way" . In which the atom's various angular momenta interact We start with the sim . P �t case, returmng to the one-electron hydrogen atom. In an atom, the electron, with its intrinsic magnetic dIpole marnenl . . " essentiaJJy an orbiting magnet, so it should interact with any magnetic fi . ,� . le present Our simple solutIOn to the hydrogen atom assumed thal lhe only POlen· U·aJ energy In · Ihe alom IS · due to Ihe Cou'am b attracuon · belween nucle us and electron. This is certainly oversimplified if there is any magnetic field pr ese . nt. ' We need look no further than the alom uself. MOVing charge!ii produc fr e ma g. nerie fields. Because il orbits, the electron produces a magnetic field . , and om lts

�

fel�

-

-

1240 eV nm

t

Illog

the tkdron'lj Inllial energ

I l:,mi 1240eV ' nm forhcglve�

I

Assume- 1;�tV� ---.... ;:

In

orb.lS a charge of 59, due 10 screening by .

X-:II�COCIrnbution� from

K-shell eJcdrUl'l and

tJOA. llortMb "

.

�

t\,() The Spin.()fbit IntcncbOl'l

rehiCh ....

Figure '.2.0

pictured in Figure 8.20. the lIucleus is in orbit. producing a tield in !">pe'tive. the electron's intrinsic magnetic dipole moment has an orientatIOn energy.

The t"lectron feels a

netic field. dut" to the ··ortnting" in the ...arne dirt("-tion a!o l..

A detailed study of this interaction is beyond the scope of the text. But

319

mal·

proton.

'th fairly simple considerations. we gain a good qualitative unden;tanding

.... ld a reasonable idea of how much the interaction might alter the atomic 'Y levels. Strictly speaking, this "new" interaction invalidates our naive an "

e�'b . . . h drOgen atom solution of Chapter 7 . However, we WIll 'iee that the mterac· ergy is small, so the effect can be considered a minor perturbation on an li n en ption. eS'ienlially correct descri

�

The orientation energy

of a dipole

IJ.. in a magnetic fIeld 8 is

U=

-

• - - - -- -

'" What the prolon '>tt!o

f.L ' B .

Here the dipole moment i s the electron's intrinsic ""s' related to i t... spin. and

c"4 '"'

due to orbital motion-hence the name spin�orbit interaction. Thus the fIeld is we write

u

=

-""S ' Bdue lo L

8 (due 10 prolon "orbit"')

(8-23)

e

To obtain a rough idea of B. we assume that the electron "sees" the prOlan

I'

orbiting il in a circle. as in Figure 8.20. The field at the center of a current loop is given by 8 = J.LoIl2r, and the current J we relate to as follows:

L

Here we have

assumed that

whether the motion

is

viewed

as proton

orbiting

electron or vice versa. the orbit radius and speed are the same. Therefore, by multiplying and dividing by the electron mass, we relate I to the electron's

orbital angular momentum. A s Figure 8.20 shows, the B felt by the electron is in the same direction as L, by the usual right-hand rules, so we have

(8-24)

Finally.

inserting this and equation (8-8) into (8-23), the orientation energy

U= -

-p.. S · Bdue toL

(

e

- --S me

)( .

I'oe

41TTI'I..r3

)

L -

-

is

(8-25)

The energy is high/positive when the two angular momenta are aligned and low/negative when antialigned. This makes sense, for if S were aligned with L. Jls would be opposite L and thus B . and we know that a dipole's highest energy is when it is opposite the field. As we might expect from Section 8 . 1 . the spin has only two possible components along the internal field. However, the values,

�S -�': t:J - - -

_ _

�

_

_

What the electron I L h' PL' l'\,ldly pJrillll'l. II I' lef! a'c. Finally. Lr and 5r couple via a �pin-orhjt 1I1tCfilCIioll angular momentum J l' wherejr may take on any vuluc betwecn hl fllflll II towl

....

.\rl and «( T + (r) in integral ,tep�. For !ir = D. this i" the single vulue f " .. 1 . while for .� I' - I . it Tllay be O. I , or 2. Thus, we ..ee that the excited

hH.:h '

II

-

dt\"lron hns open to II many different 2p states. The conventional nOiation i.. �L\c:n ill the margin, where

( r fol lows

the same lettered scheme

11"

ror ..ingle

c1c,,:trun,. except uppcrca..e. Thus. the «('T ' sr.jr) = ( 1 . 0, I ) state i, thc ..inglC:I 21P I qate, the ( I . t . I ) state ,.. the triplet 2JPt state. and sO on In this IItllBtion. the grrllllllJ ,tate i .. the ..inglet 1 '50, There is no triplet 1 '51 �tatc . for

the \pin ,tate mu..t be anti ..ymmetric when two electrons are in the same 'PII-

na! !ltatc, Table 8,3 ..ummari/es how LS coupling governs stalCS in heliUm

.... ilh one electron excited. In general. different LS coupled states are of dilTerent energy, Of the many factON competing 10 determine lowest energy. the major ones are the follow LIl,: ( I ) The preference for an antisymmetric spatial state favors a symmetric

'pill �lale-3 larger $7' (2) The reduction in electrostatic repulsion afforded by baving multiple ( '* 0 electrons orbit in the same direction, "pas\lng olle

farger Cr (3) The spin· (A complete analysis can be found in any

another" less often than if coltnlerrevolving. favors a

orbit interaction favors a .H!wllerj.,.. dedicated atomic physics tex!.)

With so many states split in so many ways. it should not be surpri�ing that

\pectrn can be quite rich. Figure 8.30, though not

10 scale, illustrate.. the

\\ealth of spectral lines in excited helium just through " =

are forbidden by selection rules:

IljT = 0, ± I

3. Some transitions

UT = 0 -f+jr = 0 )

As usual. these are based on angular momentum conservation-the emitted

pholon has unit spin-and the ability of the atom in transition to oscillate as an electric dipole (sec Section 7. to). Because ilsT mUSI be 0, the lines naturnlly divide into those among the ST = 0 singlet states and those among the Sf = I triplet state... Interestingly, from the triplet 2351 Slate, a jump to the

f...h II,ele tron

C. (05C)PIC

'Iotal

2

31po

0

h�P

3'P

o

,

2

3'P ,

2

3'D ,

2

3'D ,

3

3'D ,

2

example of a metastable slale. is a good allowed. This not her than electric is ground stale r only by means o slate can occu und gro s. which ar invariably Transitions to the atOmic olh'-Ion . such as inter ration gene n dipole pholO ratton of lasers, and this particular al to the ope slateS are centr ble Metasta slow. helium-neon Jaser. in the common ited explO is m one in heliu nt is subjec ted to a weak external mag_ that If an eleme II is walth noung d. spill accordmg to m)T by the are further ennche es lin al clr �pe its netic field. to (8- 35 ) ma be used directly. with t. equallOnS (8-33) . leadlflg nomenclature fro Zeeman effect In fac mls (, s. and j. Somewhat ng laci rep Jr and (1' s1" "normal" Zeeman effect, Ie physics persists. I n the the early days of atomic ular mo me ntum alone, e �>componenl of orbital ang ell, are o;plit according 10 th Before spin levels, as noted in Section g"mg an odd number of e was ineXPlic mpl even number, as in Exa understood, sphtung into an Zeeman effect. We now know that the "nonnal" and known as the anomalous . nt aloms whose tOlal spin is O. behavIors merely IOvoh·e muillvale

�

�

�

8.1 8.6,

�

:

;'��

Figure 8.30

) 'So

2 ISO

Helium'� rich �pectrum.

3 1P

l

2 1P,

31

02

) lSI

3'P

3'D

\

-2 eV 2lS1 -2 1 eV

We won't delve deeper into multielectron-atom excitation spectra. eJlcept

10 identify an important alternative to LS coupling. It involves a spin-orbit coupling between eacll electron's S and L, yielding a quantized J for each, followed by a coupling of all these individual J vectors into a quantized grand 10lal angular momentum JT" As the final step is coupling of individual J vec10�. rather than 5T and Lp this kind of interaction is known

as jj-coupling.

it

predominates in high-Z atoms, where the intense magnetic field of the highly

charged "orbiting"' nucleus favors an "immediate" spin-orbit interaction for each electron. Left until last is a J coupling that reduces the relatively weak electron repulsion.

PROGRESS A N O A P P LI C AT I O N S Controlling Quantum Coherence in Spin

It is

impossible to men.tate the role of spin in modem techno

logies. and the developing field of quantum computing is an excellent example of its utility. Most schemes for building the

basic unilS of a quantum computer rely in some way on �pin-of electrons or of whole atoms or of nuclei. Many exploit both spin and the convenience of the quantum dot formed in a semiconducting �olid. In quantum computing,

can perturb the confmed electron�. A recent ad..-ance by a team at Harvard essemiall)' neutralized these effeorne semiconductor

electrons are begun in one dOl in the � ':>patial ...tate.

quantum dots, the nuclear ,pin.. of the surrounding atoms

requiring, of cour..e. oppo.,ite ..pin... Via external c¢ntro\s-

the r'llh:nll�1 "IICJ} J\ .lIlctn/. ;and oot' d«trolI " nullied Mlin w,,,,,... , , !.l.lt k�� b\ ;t NIner through "hilh "';ill' fUIKII'1f!1 un f"'S§/lunnd .h JC'pl([eJ In Figli/e 8 .ll flIto �mlu,ndud"f nudC:1Il .IlliXl the \.ep;!f3Il-d

'r'"�

t'le\:lnm\ I.h"��rlll.t. aJJmll .l roll\J

to sand

wich a semiconductor between ferromagnch who!te magne

tilations arc aligned. as tllustrated m Figurt It 32. A potemtal

differtn,e would coax electron.. from the �(lun:e ferromag ,

net y,.hose \ptns are aligned WIth lb m.l�netualtfln. W,thout

1 I,

Summary

337

Figure 8.32 A proposed spintronic device. where !tpin preces..ion in an external mag· netic field controls electron flow from one ferromagnet to another.

Source ferromagnet > •

-

1

Externally applied magnetic field

1

1

:� �� Semconductor i

would arrive aligned with external influences. these electrons accepted. be However. an applied and ferromagnet the drain

ing its polarization, particularly at interfaces between

materials, are major hurdles,

Chapter 8 Summary

For an electron in an atom. the quantum numbers n. (, and m(

are said to specify the spatial state. while m, 'ipecifi� the "Pin slate. up or down.

Fundamental particles possess intrinsic angular momentum. spin. that is inherently quantized, The measure of this property is the value of s. a characteristic of a given kind of par

called

c particle's intrinsic angular momentum S is

ti le. The relationship between this numerical value and the

I)h

For a multi particle system. the probability density must be

unchanged if any two particle labels are e�changed. Solutions of the SchrOdinger equation not violating this requirement are symmetric or antisymmetric combination'i of products of individual-particle states. For two particles. these are Symmeuic

(8-6)

quantized according 10 the value of a new quantum number. ms'

"" m/l

. = -so -s + I. . . . . s - l . s

"'

The electron is �aid to be a �pin-{ particle. because s and it follows that S i"

\111. 111

(8-21 )

Antj�ymmetri..: (8-121

The :-eomponent of a particle'� intrinsic angular momentum is

S.

-

spin current are being punued, and there is no shortage of

ye:t to be made workable, Other methods of controlling

Vs(s +

-

innovalive proposals. Practical problems loom, however. Efficiently injecting a spin-polarized curren! into a semicon ductor (the dominant fabric of modem circuits) and preserv

magnetic field in the semiconducting region would impose a controllable precessional frequency and would thus control the degree of acceptance at the drain. Alas, such a device has

S=

1

Drain

ferromagnet

Although both give symmetric probabilit)" den
lame

lull

Iod

01 quan.um numi:rc'n ,. l..kntlcaJI}, O. 00 '\100

Ind/l"nIUI�hl(" lC'nruvoi mI) (I!:cupy !he' Orne of its nod. dangling ou l al high energy.

10 aJIl!n \oIllh Ihe field (mough il would oscillale unles!> It could \h�d ene�y). One \oIuh angular momenlum

Con�ider to"� 2 and 5 in the periodic table. Why

can.not \'-by?

should fluorinC'. in row 2. be more reactll-e than iodine. m row 5, while lithium, rn row 2, j� les.. reaclile" than

Dot-\ cln;uhlling charge reqUite btJlh angular momen tum and magnelU: moment? Consilkr po�jlj\"e and neg afl�(" chali�� Imultaneoud) circulatrng and

rubidium, 10 row 5? 15. Discuss wh:1I is nght or wrong about the follOWing �Ialement Noble ga ...es corre..pond to full shells.

cQuOlen.:u-culalrng_ J, Summanlt the" connC"f;lion be,ween angular momen.um

4_

SolvI08 (or aUcmpllDg 10 �ohe!) a 4-electron problem is not twice a� hurd II� �olving a 2-electron problem

ondlcales advanced questions

2.

�son or a

rermlon IS milependent of Z, Instead depending emirel),

A� individual atom\. would the�e behave: as bosons or as femlion�? Mlghl a ga� of either behave as a gas of bosons? Explnin.

'h allnlcll\t IllfCC' by IIlle""("nIn8 10l'.er.energy eleclron�. The III

ilnlJ\ymmetric, or neither? Explain. Whe"me"r a neutral whole a�om behales as a

10. In nature. lithium e.\i�(s i n (\010 i�OIopes: hlhium_6, With three neulron\ in liS nucku�, and lithium-7, with rOur

Ihe}' OCCUP). ConsC"quendy,

dec'ron\ may I"Il' I�r rom f t/lt' nucleus and SCrel'ned from

availllble 10 plll1l(1pJle

the simple"sl tenns.

on the number of neutrons In Its nudeu,. Why? What,li It lIbout th,\ number that detenmnes whether the alom IS a bown or a femlion'.'

'Ion pnn.:lplC' Onl}' t\o\o t1("\;lIOn� rna) occupy the 1()\I.�t· eMIJ)" \pJllal \1011("

III

("�p«l their mullrpanJ(le" ;l,p..1tI1l1 Slale 10 be symme1nc,

Tht' \d\rOOln�t'r '"'Iua"on c.lIInO( be whC'J

probk'mall�

pie?" E'plain

attraction All other things ocing equal, would you

Comple:'C' ch..,.;.C'nlahun 01 lhC' multld«lIOn alom I�

•

other f rarot

�lUd)". but aJ�o lIs\ume Ihat neither is del'iated too from the [("nter of the ,hannel_ Your tri�nil lI\h " Wh), i!o there an e:l;clusion princi.

16. Lithium is chemically re:Ktile. Whal if e1ectron� were

�

�

quantl/JltOn ilnd me Stem·Gerlach e�penmenl.

spin- IOSlead of spln- . What \'alue oJ Z would result

Compare and con.rasI UlC' angular mornen'um and mag

in an element reacli�e in roughly the ,ame way as lithium? What ir eJectron� were in�tead spin- / ?

netic moment R'lated 10 orbtlaf mOl ion wilh those Ihal are In/rltlS/e

17.

5. The neutron comprises mulhple chal"8ed quarks. Can a partlclC' thai IS electrically neutral bul really composed momenl? Explatn your answer hydrogC'n ((

- 0) Stem-Gerlach appdl1ltu� of Figure

is aligne"d with

!he fi,-.,t bul totaled 90c about Ihe X-lUis,

in\tead of the";:. Whal would }'ou see emerging al me

rn Figure 8.16, to remove one of helium's 2�.6 eV of energy. h it\ energy \oIhen

electrons require�

/9.

orbiting -24.6 eV? Why or why no(' Early on, the lanthanide.. were found 10 be quile uncooperative when auempl\ were mJ.de to chemically

8 3, you place a \ccond �uch apparalus whose channel so that II!> B-field line� point roughly in the )'-direction

would be Z ror the first nobl; gas?

18. As indicated

of charged con�tituenl� hal'e a magnelic dipole 6, SupPlhe thai at the channeJ'� outgoing end in the

What i f electrons were ,prn-� in,tead o f \pin-t What •

separate them from one another_ One (ea�on can be

20.

seen

10

Figure 8.16. Explain.

Figure 8 16 C' an mtinite

36. \cn/}' Ih.Jl lhC' normalilatlon eon�tant gnen in E\ample � Z j, nlrrCCI for both ,}'mmelric and anri\)'mmetric \loJle, and .\ mdl."pendenr of II and 1/' �17. The gener.. 1 fonn for �ymmcrric and anli,ymmetric

11

I

(b)

overlap--and functions A and B approach equal energy. as do function� C and O. Wave functions A and B in diagram (b) describe essentially identical shapes in the right well. while being opposite in the left well. Because they are of equal energy. sums or differences of A and B are nOW a valid alternative. An electron ina sum or difference would have the same energy as in either alone. so it would be just as " happy" in A, B. A + B, or A - B. Argue that in this spread-out situation. electrons can be put in one atom without violati ng the exclusion principle. no matter what Slates electrons occupy in the orner atom. 41. What is Ihe minimum possible energy for five (noninleracting) spino! particles of mass m in a one dimensional box of length L? What if the particles were spin- I ? What if the particles were spin- ? 41:. Slater Determinant: A convenient and compact way of expressing multipanicle sillIes of lIntisymmetric character for many fermions is the Slater determinant:

44.

45. Exercise 44 gives an anti�ymmetrk multipanic\e state for two particle� In a box wilh oppoIate with spins opposite and the --.arne quantum numbers i�

!

0/1" (xl)m,1

iJI",(x!)m,\ iJi",(xJ)m,\ 1I exercise. we look at swapping only parts of the state-spatial or spin. (a) What is the exchange symmetry-!>ymmetric (unchanged). antisymmetric (switching !>ign). or neither--of multi particle states t and It with re!>pect to swapping spalial slates alone? (bl Answ'er the same question. but with respect to swapping spin state<Jarrow,:> alone. (c) Show that the algebraic sum of slates I and U rna) be written

(X)nlJj.

columns).

What property of determinants ensures that the multiparticle state is 0 if any two individual particle Slales are identical? (b) What property of detenninants ensures that switch ing the label� on any two particles switches the sign of the mu!tiparticle state? (a)

43. The Slater determinant i!o introduced in Exercise 42.

thOt if state, /I and n' of the infinite well are occupied and both �pin\ are up, the Slater determinant yields the amisymmetrio.: multipanicle state:

Show

where the left arrow in any couple reprC!.enb the spin of particle I and the right arrow that of panicle '2 (d) Answer the same questions as in pans tal and (b). but for thi!> algebraic sum. (e) Is the sum of ..talh I and II �till anli�ymmelric if we swap the panide..total-spatial plus spin-states".' (f) If the two parti cles repel each other. would an)' of the three multipar tide states-I. 11. and the sum-be preferred" Explain. 46. Exercise 4S refer" to state... I and II and puts their algebraic sum in a !>imple fonn. (a) Put the algebra!.: difference between these state:> in a "imilarl) §imp!e form. (b) Repeat part.. \dl and lel of Exerci-..e 45 but for the algebraic difference. "Upy ... 47. A lithium atom h3.!o three electrons. � roo.: indi\idual-panicle "tate" corre...ponding to th
= ( 1 . 0, 0,

+�). ( t . O. O. -�). and , ,:!.O O. +it l stn1

'" 0.0 (r," , "1.1.1 0 Ir}'l . and "2 O. 0 I�)' to represenr lM Ind.VlduaJ·pattKle u.teJ when occupied by

�d�J, apply the 51 lhe equation numerically and verifleen it in a fonn obtained later in Exercise 7.92 that further adap15 the differential equation to a numerical solution.

.,, { f(l;

j(x + �x) = 2f{x) - f(x

I)

_

�x) +

2Z;X)

_

},)

E

We study lithium's lone n = 2 valence electron. To a good approximation, the effective Z produced by i15 three nuclear protons and the roughly fixed cloud from ib two = I electrons is given by /I

Z(x) =

7.5exp(- 126x) - 5.5 exp(-1.1 1x) .... I

(a) Plot Z(x) from.r = 0 to 10. Does it make sen;;e at the limits':' (b) Lithium's valence electron is not allo�ed in the full Is level. whose twO electrons have energies of roughly - 1 00 eV, or about -7 in the units orlhi!> exercitate'!> have equal energy? Can you c' �otume and lhe numbers of different Lim!> of parudlOSl

chem,cal �p«ie.. h migh' be in'� b} hc.llm8 at co..,....n ",tum. (.1£ > o. .1\. = 01. b� fltt

upiUlS'oo r.1£ = O • .1V > 0). 0< in a chrmical ndtllt, tilt 'oJu","" DOl Iht

reacUOfI that chanse.

Internal «nelllY

- � -

T

(9-4)

A glance at equations (9-2) and (9-4) raises an obvious question: How on Earth do we take a derivative wilh respect to E of a number of ways? We

'In hr..1 Inn,fer hom ')� 1 10 sy>lell\ 2. tht

d;",nkr of..

rf.

O_11� 0 1'" N ·- 1 0

001�

AI "" 50

O,O_�O O,O!5

I----�--=::::oe ::: ;o...

o

...... .. ...... . .. . � . .......... ..-I"__

�

"

40

30

50

An a..erage Q\cillalOr occupies its fifth energy level. The probability that parti cle

i ha, Ihe

avecage energy-that 1// =

(

quanlum numbeJ'\ add 10 45-would be

� -s

= =

45 + 9 - 1

�

5

and Ihe remaining nine panicles'

)/(

8.86 X 10'/1.26

X

50 + 1 0 - I

10 "

50 =

0.Q705

)

The probabilities given by (9-9) for all values of 1/, are shown in Figure 9.6. The

cune drops ..harpl)' as 1/, increases, which leads to an important conclusion: Varying the energy ofJU�I one particle causes a sharp change in the number of ways of distributing the remaining energy among the

other particles. The greatest freedom to distribute the remaining energy occurs when that one panicle has the least energy. Therefore, Ihe more probable state for a given particle, the state i n which the number of ways of distributing the energy among all particles i s greatest. i s one of lower energy. Equation (9·9) is cumbersome and limited to the special case of harmonic oscillators. In a system of infinite wells. for instance,

squared, and the expression

En

is propoilional to n

would be entirely different. However, it may be

_�hown that in the limit of large systems of distinguishable particles. converge to the Boltzmann probability.6

n

p

all cases

b�

"'I'btdtn.m;;,n and lboJc,;,f lilt forth.:,>!mng

"qlllltum l diSU'lN..l"" Ii delennt 10 App:I).l,\

H. � II U. be,1 Itl _ tao. Lilt �h'hty 1\ flSty/ til'll :and t>eeause KICID, the den, al,OO,

IOfdIIU .Iantics dIt,tllmdll'iuc, 1Uld dJtlcrrOl.�'.

This is the probability that in a large system at temperature particle will be in stale n of energy

T.

an individual

En' where 11 stands for the set of quantum

numbers necessary to specify the individual·particle state (e.g._ n, e. Probability drops exponentially with energy.

me_ fils)'

II i� important to note that the Bolt7mann probability and a1l the di!>tribu n�)'et to come can be applied to ,,� II

changes by I , E changes by

hwo'

are

there in an energ) range dE.�'

,,"0 the ratio of the number " I" ,tale,

�i)\� per change an energy is just l/hwo' More formally, differential number of state, dE D(E)

1

hwo

\9-27)

�

[

This density of states calculntton mny seem too cute and easy to be true, oot it is COrTect. Because others later will be quite a bit more involved, it I� helpful to consider briefly a case only slightly different. What if our �y�tem were, say, a collection of particles in a box? In this case,

£

would

J �ain be proportional to a single quantum number rI. but squal'('{1 (cf. Sec

u�n 5.5). Contrary

to the oscillator'S equally spaced energies, the levels

g�1 farther apart as " increases. Shouldn't a " density of states" decrease as tnergy increases? It is left as an exercise to show. by the same procedure OL'

above, that if E

Increases.

0::

,,2,

then D(E)

0::

£ - In.. It does decrease as energy

We close the section with a basic question: What would be the average

fntrgy in a system of oscillators if the temperature were high enough that the quantum levels could be considered " closely spaced" ? This is the limit for \\hich we have tailored (9-26). Inserting (9- t 8) and (9-27),

Several things cancel top and bottom: the multiplicative constant

A,

the total

number of particles N, and. perhaps surprisingly, the density of states. While D(E) always cancels this way when it doesn't actually depend on E, we are

careful to put it where it belongs, for in several cases to come, it will depend on £. Now integration (see Exercise 38) over all energies from 0 to inlinity gives a very simple resuh: (9-28)

I> our switch to integration valid? We obtained equation (9-15) via summation:

It is left as an exercise to verify that in the limit of closely spaced energy levels,

""0 «

k.T, equation (9-15) becomes (9-28). It fits'

Ity 01

• ,".

.�

'1,1'

161

9.4

Classical Averages

applicable only when Ind· The Bo ltzmann distriburion. is l'ioll ngu ' I \ha b There . are quant u m Ignored be i/j, may s particle thefTn Od identical or Y n arr Je ul�hab dISllRg . are bUI llc}' the Bol t m , lems in which the particles z a . n '1\. n d l')l . ribu t' dl'ilnbulJon. because i l i� the is often said to be a. cJasSICaJ IOn c OrrecI . one cal . quantum mecham where beh l i hml the avi n systems in ors c Oov u a erge . 1 0 ICaJ the class . . Classical averages naturally Involve not summation Over IIo Wed quantu inuum of "Ckl%ical ')tar eS slates bur integralioD over the cont III th at . · n coordm ates. d In eartesHl . dYdZdv-'dv IS, Pan;. x cle positions and "cIOCl. tles. .\dl'. y En become.... E(x )' IS 0 l' infinitesimal elemenl. and Ihe energ 1 ..t .• 1;:), kinetic energy depends on velocity and pOlenilai ener� ie's e P nd on PO . 5itio .. I ·' raIher than energies We d n Because we Inlegrale over Sla 0 ' nOI . need a me uely assu mfin den. , man y stale sil} of slaies. bUl lhe fact that we lses a n i nt nly principle lells U � er. eSling and subde Question. The uncerta s t ;� P QSlilo n e known simullan be! ously a cannOI with nd velocily/momenlum abmI u� . pre I. . of a contmuu m of C si . speak to o sense no makes such n 5 so " reaJI)" tates. Tt, . . . e e an r I S between pOSIllOn and velo ' mescapable granulanI)". a spacing Ctty Val ues · I eSIi. e h fhat e\-·en can be Shown th H N · Istmcr. d· can be declared lndy at the re ' , IS a o ne. mechanical states an to-one correspondence between quantum d Ih ese . . . "dI.SlJ .. o. guishable pos!llon-velcx:uy !!otates. so that aCcount ing for the g ra nUlar i "h, would just introduce a proportionality constant Ihat in calc ulat'1 09 an ave r ag e would cancel top and bottom to common quile calcula I!I it te averages 0 In c1a�sical systems. f qUa nt llies for speed, instanc or n e, auo Fonu nalel other than energy-loc We nee d no a model for all averages. To a! new appararu!!o. Equ31ion (9·26) erage an b . . ar I. · I wllh · [rary quantity the expOnenti pur It In the Integra Ju_st . al Boll llllan o probabilily factor. (Note the analogy to expectation value i n qua I u m mecha n. )/2 ic,. where knowing the probabIlity density, /",(x , We find t�e a . verage . of · th e Inlegral with the prob ' . any funellon 0f x simply by pUlling H 10 ab Iltty') . . . ..I .. OC followmg example Jllustrates the pomt.

.

.�

� �v\

becalJ�

�

e\

�

,

i\

Q.

.

EXAMPLE 9 . 3

I J ,

-

By appro�imaling lhe 3lmo.)phere 3!. a colu mn of da\sica/ pan icles of ass a( caitu/ale the avera e � IcmperalUre T in a uniform gravitational field g he ght oIn f an . air molecule above Earth \ �urface

g.

St)lU"I�N

Probability depend� on energy, and in this example. we have

£

=

KE +

, U.,.,..,. .'-

=

I

(

111 ' -2 � '* , .

". + � ', .

+

yo, )

+

mgv. •

I

'

2 nn·-

+

mg),

We may use (9·26) willi two simple change�. We average ), rather than E • nOled. there is no need for a density ofSlaies.

and, �

9.4 Clusical Averages

v=

"nit constant

NA

367

h' NA exp[ -(�ml,2 + mgY)/kBT] dtdydzdl',dl')dl': JNA exp[ -(j 1/1\.2 + mgY)/kBT] dxdydzdl',dl\dl':

canceld\'�

dl'..dl' dl' J :

Sincefty) depends only on speed. rather than on velocity, we further simplify things by using nor rectangular velocity components bUI spherical polar coor dinates:

I'x' l'y.

1'_ --) V. e. c/>. By analogy with dV = ,2 sin 8 dr d8 dcjJ in spmia/

�ngular integrals

coordinates (Se�tion 7.4), the "volume" element in I'elocirv coordinates is tN, dIll dl': --)

v2 sin 8 dl' dH dcjJ. With this replacement. the cancel. and integration o....er speed l' alone remains:

"PoJ)al"""� ,deal JUC' 1\;I,'e ,n!emal degree. of f-oom_ rotalJON-t and vibnlJJ>lUJ. wh>eh dqIend ... "'.-�r,>m< in tb,

·

\

\

'

•

!

! : /!

(

j

and disuibullOO 11 r - 0 The ftnni·Dirac ,." ,.... }-emll rft'l!(i}' t.. \lIUe ..... ltmpcnturer.. the � kJr 1oW r 0 mil( ,ink from ,u ..r d l EfO

E '!!

,

,,\

r � I,t

II

9,6

",I lOw T ,.�� Ptila

I).fa e....

o ,� L

0

'.

_

--

,

hNU,h H AI wne " urn T: t" c••

" AI

f. F1

� f

-

f

•

Tne Ouantum

Gas

n\ to a \ituation more reaI'I�h. c ''-A_ um dl.,tnbUlio UI4l\ . '. the quant annl .... no n we do, we enter an area 0f ph)"" \\e mmon ongln. A" ilco III c J c f"":llIatl\f' J,ltrol..:! of Ihe mo" aeli'e In r'''ar h ' mvohmg . h., b<en un' 1'I9!h Ihe Ih". ,,"" , .. wQUId fiInd completely baffimg. I ph)".IC\"t a ·\:la.....ac3. are f t-o.:hilWlf" that . h N masSive particles �.I" one In ",hie ider Clln, \t 'IIo e "The " ". tern ... (The massless h .10 a cI�...\Cal g a u\e" mo\et.: are a:. \J \ " In thf'.'( dllfl(n'Il an: nol eon...denng ordinary gase \ Ho."er, " 9.1. un Se QuantumOu

371

F9" 9.14 Cl;Mlcal di�ting\li�hable particles at high T. and t�o different lo\\,·T tdA1L\'f'. (Adapted from Nobel lecture by WOlfgang Ketlerle. Rei'. M(l III

r iliffcrenual numbe _ _ _ _

-

tlf 87TV - -- ::::: --I'

range

c-'

the u�ual number per unit

1.'11' ,tate' in range

--;J£!h .

D(E)

�

tiE

=

energy

87TV 7(E/h )'

811'\' - £' -

h' c'

important way. Whereas f u differ.. in another applicati on ' ' " ,. �Qn fi 'The: of ma,,'" obJecls 'S con ned 0 some • fi"d number ,ch , Id wh ,\>OUI " l n '1sumjng equilibrium between atoms and photons' the rate at Whlch · . . atoms make upward transHlons from level I 10 level 2 must equal the d ward transition rale his is known as the "principle of detailed balance." .. ply put, the two emISSIon rates must add to the absorption rate.

�

�:�

9.8 � l.&ter

A\pon N2

+ B�um N2 Y(a£) = Bab\ NI Y(,1E)

Solving for Y(.l.£), Y(IiE)

�

N,

N2

Babs

'[he ga5 atoms. being diffuse. obey the Boltzmann distribution. �o the ratio NI ' \'2 is e-EII*BTje-EJkBT = etlE/ksT. Thus. Y( liE)

�

s j --'Bs '-:::--= A po Bab

,'Elk.T

stim

- --

B,I»

Here is where the "intriguing conclusion" arises. We defined the quantity of photons of energy 6E, but we know what this must be! uE) H as the number energy density (9-47), with d£ (i.e., the photon spectral is the photon gas the atomic energies £1 and £2) replacing hi in that fonnula. energy; not one of The t1Ei*oT is in place where it needs to be, and the only way the re�t of the denominator can be correct is if Bs1im = Babs' This means that if the numbers the two levels happened to be equal, a stray photon of the of atoms occupying as likely to induce absorption as stimulated emission! just is ncy proper freque don't want these rates equal. We want stimulated we laser, a in Of course. emi��ion 10 predominate, but how? As the reader might have guessed, simply making N2 » N1 would do it-Rwm would be much larger than Rab�-bul such a situation is certainly not !he way things are in equilibrium, where number drops with energy. (Note: We a.>sumed equilibrium to prove that B\lim = Bab�' but they are indeed constanle,. so they are equal even when the situation is far from equilibrium.) In fact, such an overpopulation of higher levels is crucial 10 laser operation and is known as a population inversion. Being unnatural, it must be established by some e�temal means. Bul so long as it exists. one phOlon of the proper frequency. perhaps the result of spontaneous emission, may become two, which become four, and so on, very quickly resulting in a large number of coherent photons. Before investigating how a population inversion might be established. let us take a look at the basic elements of a laser, shown in Figure 9.23. Because spontaneous emissions could iniliate separate "outbreaks" of coherence in dif ferent directions, the medium ie, tuned to amplify the coherent light injust one, Parallel mirrors are placed at the medium's two ends. Photons not parallel to !he axis have relatively few opponunilies 10 induce stimulated emission. while those parallel renect back and fonh. providing many opponunities. The length along the axis is tuned 10 a standing-wave condition L nAn, so that waves moving in opposite directiom constructively interfere. In the reJO,inal l.ucr. Wllh tptr.1 n... l llube \urrouDd'ml R$CIIWJ' Qlur, of ruby

h(' n ..i� so., B)' m.tkmg lIle mm"Or 81 one end only partially refiecling. a frac _

III.n vf Ctlhc:-renl lighl i� allowed 10 exil lhe cavity.

There are \('\"Cral "a.p of addmg [he exlemaJ energy needed [0 prom olr d('dflln� (0 the higher le\·els. One j" optical pumping. used mosl often I n �."d·\I�Ie' Ja...el'. ",·here Ihe Ja�r medIUm j). a rransparent solid. The medi u

fra:

1\ \uhJeer. who�e optical pump h a nashlUbe wound around a ruby crystal. the "las Ing" m;uerial. Figure 9.25 shows a one-pass laser amplifier, i n which a bank of high-inten'otty lump" " charge\ up" neodymium glass plales just prior 10 a lase r beam" enrry. When Ihe beam enlers. atoms in the glass deexcile and a much

more p<merful beam emerge).. (The beam i\ polarized. and light is passed

Wllht1Ul renection by orienting Ihe plates at Brewster's angle.)

AnOlher means of energ} inpur is elecrric discharge pumping. used in ga, lu\cr,. Arom, are rai\ed 10 higher energy levels as lhe discharge ionizes 'oml� alOm\; and produces generally violenl mol ion. Because excess hear dissi� piuion /\

Ie, ... of a problem in a gas,

U\ oppo,ed to pulsed, operation.

il is u.suaJJy possible [0 sustain continuous

•

.-\ VC\lflg problem remnins: Optical pumping can increase the population

of a higher energy level onl} when the number of electrons ar the lower level

Figure 9.25 A one-pas, IUlIer amplifier. An lRU is a replaceable unil of neodymium

gl.MplllfC'

Flashlamp cassatte

Slab cassette lRUs (2)

Q � l'h< L.a,

�

!

I'

• • • • • • • •

Population inver�ion

389

FIg"'" 9.27

FOU1_le�'el laser.

•

--E, --

1 1 JJ

epopuJale�.

\

og a large _, do _ ...,.Jlcide .. C'" attract new applicahon\ Ibtu ch.tra..'IC'rUtic."s come under clo..er

17. Cla" I�all). what woulJ he the a\er,.. energy of. par.

\\hether the den�lt)' or ..tale, ,h(lUld be Independent or

tide

f, llfl ln(rca! isn·t a

thennl)dynamic �)'stem, but the �eneral idea Stl\l

applie\o, and the number of combinatIOns is tractable.) 21, Con..ider a room divided by imaginary line.. imo three equal parts, Sketl.:h a Iwo-a,.i� pill! of the number of way, of arrn.ngin� particle, vel'ou, N.,I and N .... for the cao;;e � n.. �� Note that Nm,Jd1t: " not llldcpendent, being of N � I(}"

Nlch

Nn&ht

and - Nlcfr Your axe.!- 'hauld be COUM N Nn¥M' and the number of way' �hould be repre.,cnted by den\lty of Shading. (A form for number. of wa), applu,: able to a three-Sided room 1\ given

Appendix J. but

the queslton can be an,wered Without II.}

22. The Stirling approximation. JI

•

III

Virr J't tt2 e-J , I�

very handy when dealing with number.. larger than nbout

100, Con�ider the following ratio: the number 01

ways N particles can be e,'enly divided between two halvc\ of a room to the number 01' wayj5)\' for large N.

(b) E�plain how thi\ fih with the claim that a\'er;tge behaviors become more predictable in large sy"'\cm,

Section 9.2

23, The entropy of an ideal monnt(lmir.: ga, i, 13I2Wks In E +

NkS In V - N.ls In N. to within an additive \:omtant

Show that lhi\ implil!l> the com:r.:t reiation.ship bet\I.'ccn

internal energy £ and tcm�rature.

... ChIp.."

SiIbec"CII MIll_ ."

K. The dIOpOm show, .... .y"""" lhaI mlY ..cIwIg< _ tbmnoJ and .. e d lU llcaI lt,

....onductm' .. puII.Oft. Btau.. boIh £ and I'mol chlnp. we ,OnSJdtr 1M entropy of eacb sy«m to tit a (Ull(1lon of bach' S (E. VI. Considenn, the e..\change of lhmnal entfJ)' unl y. we lIIuN In SectIon 9.2 lhat II ..... J"h.5OOlble to define 111 1.1 d..SIa£. ln the mOlt �n· craJ �ar.e. PIT ,s abo dcfinN as �mcthlOg. tal Why huuJd pr6\Urf \:"URlC" IOltl ria),. and II.' \Iohal mlghl PIT ht equated.' IN()I�: Ch«k In 5tt whether the unil\ make �n.'it.) Ihl (Jlven thl\ rcl,UlllO'Jup, oJIow that dS " dQIT tRelTlC'mt'C1 the fl",1 h&w llf Ihennod)'nOtmll" .)

I_ I

28. In a large! ..ystem of di\tinguishable harmonic oscilla.

ttlf', how high doe.. me temperature have to be for !he probabiht)" or occupying the ground slate to be le\\ than 29. In a large ,,},tem of di�[inguishable harmonic oscillator how high doe� the temperature have 10 be for the pmba tile number of particles occupying the ground state [0 hi; Ie" thlfl j '] .30. ObtaIn equatIon (9- 15) fro m (9-14). Ma�e use or the followmg sums, correct when �tl < I :

�?

I

�

2:X' � I -x 11,,0 � x-:: :"-" 2:1Lt" � -'( I x) ' " 0

31. Show that "'Iualian (9- 16) foUows iTom (9-15) and (9- 1 01 ' temperature and M and .v given m (9-16) and that between E and n in (9-6), obtain equation (9-17) from (9- 12). The first sum gi\en In Exercise 30 will be useful. 33. Show that in the Iim.it of large numbers. the exaCI prob abdity of equation (9-9) becomes the Boltzmann proba bility of (9-17). Use the fact that KU(K - k)! .= }(A, which holds when k « K. 34. The euct probabilities of equation (9-9) rest on the claim that the number of ways of adding N distinct non negative :ntege� to give a total of M is (M + N - 1 ) [/ IM!(N - I )!J. One way to prove it involves the follow ing trid. It represents two ways thaI N distinct integers can add 10 M-9 and 5. respectively. in this special case. 32. Usi ng the relationship between

\(10 K. I� bndl)' put in conlBd · wllh a ·'hut' ubjC1.·I. I"l - .IUO K. amJ 60 J of heal flow, Inlm :he hU1 111'IJ�l ln :he \:"old (mc The objecb are then K'polI3ICd. Ihen temperature, ha\in8 l."han�ed negligibly duc 10 1MI' large 'IIC\ la) What are the \:"hange� in cntrup)" t'll ('a�'h ol'ljt'd and the ,y\(cm a\ a "hQle'� tt'll Knuwlng \ml) Ihal lhc,c (lbjel."1\ Mt' in l'ontact Jnd lit the 81\en temperature" \\ hat i\ the ral io or the proba hlhtlC�' (Ir thclr being Jilund In the ,ec{lnd (final) state hI thell �lf Ih�lr 1'I
E\

Note that hydrogen atom energies are E " - 1 3.6 eV/,,2. (b) What is the limit of this ratio as /I becomes vcry large? Can it exceed I ? If so. under what condition(s)? (e) In Example 9.2, we found that even .11 the tempera ture of the Su o s surface (-6000 K). the ratio for It = 2 is only 10-8. For wh at value of 11 would the r'Jtio be Om? (d) I� it re3listic that the number of 3toms with high It could be gre3ter th3n the number with low /I? =

'

31. Consider a system of one-dimensional spinless panicles in a box (see Section 5_5) somehow exchanging energy. Through steps similar to those giving equ3tion (9-27), show that

38, By c3rrying out the integration suggested just before

eqUillion (9-28), show th31 the 3verage energy of a one dimensional oscillator in the limit kaT » fi.wo is kaT. 39. Show that in the li mit /two « kaT, equation (9-15)

the number 01 wuy� illr YI,ur di\tribution,

(d) Culculute the outllher 01" ways il" thcre wel'e' (I pacti-

cle� III tl

I , and none hig.her_ NOle

saml.' IOtal energy.

(e) Find ,ll le:\\1 one other t1i�tributi\ln In whi..:h the

I I osciHlltors �hare the ,atlle energy. and calculate Iht' number 01" W.lYS. (I) What do your finding' �ugge �1 ! "

Section 9.4 41. Show that the nn' �peed of a ga, molecule_ defined a" \'rm. -

_. 3kV9 \ '., i� given by "\.17: BT'm.

42. (a) C3\cu131e the avcf'.Ige ,peed of a ga� molecule LO a classical ideal gas. (b) What is the ,erage velocity of ;l gas molecule') 43. (a) Using the Maxwell speed di:(: 52), the doeflnition

�l21 +

Section '.6

I ) - l\ and

N-

II1II

SL ShO'\\' tI\at, u"inl equation (9-)6), density rJI ....

J .'·(E)D{E)d"� �

(9-381 follows from (9-)7), Dcn,ilY of Mates (9·)9) don not dcpead. 01'1 N,_ total number of particles in the sy'tem; neUher docs the .... ,il), of ,late, in equation (9·21). Why bOt7 60_ For ;l panide in l one-dimensional (to) box. Ell is pro ponil1nal tll a ,inglc qUlntum number n. Let .. UmpIify thing.. by ignoring the proportionality facwr: Ell ,.1. For I :\D 00:\, f... " , " = , the 20 and + " +

59.

•

",he f\lr B In di,tribution\ (9-32) and (9-33\---<ardul u..e III :!: ",ill ,ut )'our ",ori.. b)' about half. Then plug Nt,:i.. In and ,hoI'; that for a ,)'..t�m 01 �Imple harmonic

"".1101101"1. th� dl\tributlon\ become

"!

. ,

"�

_

:

00\ i, I-alrly ob\·ioulo. (II l'he table �s . start on

.__,ElaE •

� .I,'

and

,-..,.I.T

S(E)m -

YlJU will need the lallowing integml:

I

' 1"[ 1 ' 1/8).

J��{ Br:t. 1 r

'

d:

SJ. E,erci�e 52 gives the Bohzmann distribUlion for the

'pecial cll�e of �imple harmOniC O�CillIl10rs, �xpre.o.sed

NhwrJ(2s + I ), Exercise Interm�of the eomtant ti. 53 g'�'e� the Bo\e-Ein. (c) Plot Ef(T) vmu, koTff. from 0 to koTIf> '" 1.5. (d) By what

percent doe, the F�rml energy drop from ilS maximum

T'" 0 value when toT ri'C\ to 25% of fi.?

57. E'(en;:i�e 54 calculat�' the three oscillator distributions'

£ = 0 \'alue, m the \pecial ca,c where "sT is

l&. Using

a very common appro'(lmatlon techniquc. show that in

';;',4,� and I . for

the more general low·temperature limit, k T « 6, the occupation numhl:l' occome f.lksT.

tht di..llnguishable. bo�tln. and fermion cases. �spec

ti\ely_ Comment on th�!'C re,ulls. (Not�: Although we a.\sume that "sT
\'hw/( If +

1), we aha sllll

assulm Ihat le\eh IlR cJo�ly 7 ";Jkulalt'� Ih�' minimum total energy in a !(omliom ;tIlIj i\ applicable 10 conduc li un l'/cl'tnm\ In lI IlWI,./, The (II't'IlI,i:e panicle energy i� the Mal encrg)' Ji\'iJl"1.l "y the numner of panides N.

68. E\cl\:i,,�

-j

N=

o

In�ert the quantum gas densilY of Sillies and an expres_ sion for the distribution. usi ng j: to distinguish the

Bose-Ein�lein from the Fenni-Dirnc. Then change vari J IIble�: £ ::: yl, and factor B,,+\ /(sT OUI of the denomina, lor, In the integrand will be 3 faclor

C)- I

== I :!:: c, a .sum of two inlcgrals results each of Gaussian form. The integral thus becomes two terms in powers of liB. Repeat the process, but i nstead lind nn expression for UlOldl III tenn� of liB, usi ng

U.�ing (I

+=

�Y" em ,II' \PIII

.�hl)n' th.." hc: ..\cruge panidc energy E ofII conduclion

cl«" nm al 1,1\4' tl'l1lpt'rJIUre IT iit 0) is (J/S)£r' This (vIm I' ,'llm'...niC:nI. br:-in,l! rulher �impJe, and il Clln eas ily t'IC' put ," Icrms Ilf N. I', and m l'ta equation (9-42). 69. Thi\ pmfJlem imc�ti�ltle\ what frJclion of the anlilable chtUJc mu�1 he Iran,terreJ from one wnductor to unulbr:-r ItI pn,Ju,� a IYPlcal c(lntao.:t potential. (a) AS:l rough "rpn):{imafion, freal lhe I.:OOOUl.:tors as 10 em x 10,'m .;qUiin:' plale, 2 I.'m 3pat1--3 parallel-plale capae ;1II(-!j(1 thai q 0= n'. where C ::: £0 (0.01 m�IO.02 ml. How mUl'h dwgt' musl be translerred from one plate

hi lht other 10 pmdu,'t' II polential dift'erentt or 2 v·, (bl Approx;mllitly ",'hal fTUl.'lion would Ibis be or the lotal number of �'ondul'lion electrons in a 100 g piece of copper. whi"h hi&!> one conduclion e1cclron per alOm?

/ Jf(£)O(£)d£ wilh a re\ult derived In Example 9.6 1hould go to 7ero and obtain a rough ..nlue. (a) Starting with ..N'(E)FD eltpres\ed as in equation (9-34). !:Iho..... that the slope d.N"(E)ro'd£ at £ = £F is - 1 I(4kaT). (b) Based

on part (a). the accompanying figure is a good approxi

normal ga!>, all molecule!>. on

""O when T is small In a mation to N(E) ..

such a.. air. when T is raised a lillie.

average. gain a little energy. proportional to kaT. Thus. the mternal energy U increases linearly with T. and the heat capacity. aUlaT_ is roughly constant. Argue on the basis of the figure that in this fermion gas. :b the tem perature increase.. from 0 to a small value T. while some particles gain energy of roughly kaT. not all do. and the numtxr doing so is ale;o roughly prop!..lttionaJ

Section 9.8

80. The fact that a la!;.er\ resonant cavity so effectively

to T What effect does this ha\'e on the heat capa�it)·"

(e) Viewing the total energy increao::.e as simply

.lU

::;:::

sharpens the wavelength can lead to the output of

(number of particle, .....ho'e energy increases) X

�veral closely spaced la�r wavelengths. called

(energy change per particle) and a'>Summg the density

longitudinal modes Here we see how. Suppose the

of states ie; ,imply

'ipontaneous emission �rving as the seed for stimulated emission b of wavelength 633 nm. but

particle energies. e;how that the heat capacity unJer

somewhat

tional to (ksRIEF)T (TI)'ing to be more preci,e i not

fUllY, WIth a line width of roughly 0.001 nm

:1

con..tant D o,'er the entire range of

the� lowe,t-temperature conditions should be: propor

410 � ' _ IUIly _while, rar Ill< �') .,.,..,.", 'I IUbjcd 10 .everal corrediocu from etfC(li we

,port.)

di,tributeJ unifonnly throughout a sphere of radiu\

7 x 10 ' I � m?

(bl How Joe' this result fit with Exercise 87'>

89. When a !>tar ha� nearly burned u p its internal fuel, it

may become a "hile dwarf. It is crushed under Its O\Iin

1

..

enonnou� gravitillional forces to the point at which the

e,clu�ion principle for the electrons becomes a factor. A smaller �ize would decrease the gravitational poten_

0

lial energy. bul as'mming the electrons to be packed into the lowest energy ible Quantum states of a single electron in the pres ence of two "atoms," represented by one-dimensional finite wells of width L

and separation a. We choose the finite well as a model because it is capable of holding an electron bound, as is an atom, and it allows wave functions to wan

der through the classically forbidden region, as the unrealistic intinite well would not. We must allow for mingling of wave functions among atoms.

413

stales. which molecular . pairs of ve states. ha . lS" d atOm" P " atOIl _ of "olate ' 10c O' wO " T ' differe .. ,0. � rns and p." e: 10 su rg nve lion co

at I"ge 5Op",.

(e) (b)

L

(ol

energies for the fOUf 10West nctions and fu e . ' II. They somewhat resem· OWS wav tion IS sma 0.1(a) sh ' ose separa wh / Figure 1 s ' g e fi'ntte well. (The .op light a sm s/atom 0 f en Ons w . func.t. states in .hree, the ligh. red below" gy wave energy below " has . est �ner red low ur dark distorted antmode. ble the fo odes, the ) only one badly ur antln one has f o has red a larger separalton, A. dark gy. red one ener m ow O'to . a' h'tgh and . and the b m palfS has .wo, for are essen.ially equal, and 'hei, palf eS . of .he lower these s.at , r eve How rg,eS nt in the right atom and 0ppo. y coincide. ), the ene uall I(b virt 10. ilar. also become closer. At Figure very sim . er pUlr have tions are of the upp func eS gt e r ene wav converges 10 a single energy. atom. The each pair left (c), the . . l 1O site in ' equal energy. The low·energy paIr on. Figure . approach rati epa oer s . pairs ,.ill iaro algebraically added and sub· ce that eS tha" ' f cOinetden stat no .wo . is 11 to It ds, the two In o.her wo separauon ted '·alOms." at large 0'0 isola .he of converges 1 atomic teS s of .he two " � I s.a combination r ' are the ea lin .ed trac combinattons of the different states that are stateS are nverges to co r molecular pai wells, there would be three upper there .hree larly. the es. Were states. Simi . stat erge to three m ato ration would conv 2 lSola.ed. large sepa a. ich tWO " � · ncome together, theU stateS, wh n N atoms olecular general. whe n � I m In related m es. N olecula of sta. r I band, � or mn a se', isolated.ato ne 'o form and b' b t , and molecular com so te . her N,slll m sta.es form anot ISola.ed.a.o iC stateS m ato 2 energy of the isolated-atom state n � around .he . ster stateS; their clu rgies Figure 1 0.2 Illustrates band's ene n deereases. n ratio give sepa A on. atomic wells!atoms to produce apart as .he states of N and spread of atomic ning e atoms nbi the cOt and do belong to separat ' . electrOns can schematically e WhIl ntrate in thIS chapter cular bands. e 26), we conce N.state mole (see E"rcis apart far e. atoms are by the whol when those are shared that s state iatom on trUe mul.

I

.

:

'

I

1 0. 2

Mol ecul es

energy than the se(>'state is of lower . the molecular hen w s molecule eleetrons IS crucial. As they Atoms form atoms' valence the of The behavior lower e nergy may resuIt. But araled atoms. . atom orbits, a + ated· separ molec le, two ve to their u possible case: the H rearrange, relao st imple the s g iderin � conS e rgy may be en 1 by t01ll begut e Th . ? e re 1 0.3 how . W depicted in Figu tron, elec I � P us the neg· 0 .y : eUC ener "ts I.�n protons and one on's energyelectr the ( I ) ( " ) he positive twO parts: ' 'ded mto ' dIVI wt·th the protons-and " . potential energy il shares . ... t , onS1°der f!.Pil the electron s ns attve, attractive proto two . ' energy shared by the tial repulSive poten "

C

10.2 Molecuta.

FI9"r. 10.2 At small separations, each atomic state becomes N-�tate band.

FlguAi 10.3 A simple molecule.":t+ an electron shared by two protons a dis tance a apan.

an

At large atomic separation. elec::trons occupy 11 '" I and

�;:��r

11

'" 2

molecular

Slates

�;:�'�r

��... '-}em Probabl'I IIY/ch" Ib r 0 . Ihc:' rna)!)nl) with the inlervenmg eIectron cloud is the protons ,hare both n roo l 'lIo : r: n Thl>' JH energy. bond''\ lower of Ihc!' molecular H,- mole ''ould , of course, yield the neutral ron , . d elee r _ a ..ef,;on cUJe . , Adding drasllcally complicate . s thi� . rhe a n POl ectro I enll"aJ tiel but Ju..t as fOf I�U ground state has both electro ... however. the gues ns . . . . hl in o ' m Ihe encl1!J. A. By �ublracting the molecule's final energy from ill'. initial for both case!> .1f := -=- \ (�ee Exercise 40), we obtain from equation (10-4) thc pos l eV) than those involving molecular vibrational transitions (>0.1 eV), which are in

energetic than molecular rotational energy differences « 0.1 eV). not to proper scale, Figure 10.21 illustrates the relationships. The curves represent two different electron states-perhaps the ground state and the firsl excited state. When an electron is in an excited state, the atoms are still bound to each other but are farther apart on average, and the interatomic potential energy well i" wider and shallower. Even so, for each electronic stale, there are closely spaced vibrational levels and even more closely spaced rotational levels. Thus, while we might expect a molecule to produce a simple speclr3.1 line when an electron jumps from one of its allowed energies 10 another, vibrational and rotational energy changes break each line into many, yielding a much richer structure. Note that symmetric molecules. such as H2 and N1, lack an electric dipole moment, so they do not produce spectra due to pure vibration-rotation transitions. However, the distinctive structure of Ihe':.e levels is superimposed on l)pectral lines arising from transitions between elec trOn states. (The oscillating electron cloud provides the necessary dipole moment in these case�.) tum more

Although

Fig'" 10.21 �lol«ular \lbratlOnal and rotational le\-els ror

1....0 . different dettrun k\d\.

HI�hcr

ela:lf\ln If:o\el \'Ibr.ltllllUl

le,d' II

I I I

l :; ' = ) " :� ��' ,

Lower ek.;lrun

:2

, 10 � tol 11;' ,

D_

ll

ROlallonal e�e \ J

Db....

1 0.4 Crystalline Solids lei u..

now

lum

from mol�ule... the bondmg of a relati\'ely 'imall number of

atom.. into \OmewhJI larger unil", to caCd matter Indude.. ,tudies of 'iohd!l, hqulds. and other syslems in which the beha\'ioT'i of man)' alom' together must be understood (such as Base

Ein,tein conden..ate". di-.r.:u."c,ed in SectIon 9_6). In the following sections, we restritt our attention 10 cl)'�talhne wILd... All elements and compound:. form :.olids at sufficiently low temperature .. a cr)stal lattice, in which certain alom� and/or high pre\,ure mml often a

1-a,:t-(o!1lltmi lubK II�d

.

are found at 'per.:ific location, in a mlcro"copic unit thai is repealed identically countle.... time, in all dimen,ion"_ (Some materials form amorphous solids. in whICh anglc� and bond length.. are \0 Irregular thai the location of one atom i� cs..enlially unrelated to that of another only a few atoms away. Familiar exam

Side �itw HC'O\Itln;d dOMCSl l'3'l.etl 4hi:p>

0 •

----

'

.

.

n

0

-;;

c.n

Simple ,.;ubic .. ,)

ple� are gl:c>, and rubber,I TypIcal atomic "pacinI! ID a cT}lotal is 0.25 nm to

0.5 nm_ or 5 to 10 Bohr radiI Geometrkal con'ldemtlon, ,how that In three dlmen�lOn.. there are only 14 pcMible lattice 1)pe5. or wa),,, in which atom'l can be arranged in a regular geomelric pattern. (Exc:rCl..e �5 gi\'elt a 'Imple example of"geomelrical con�ld c:ratil,m" in two dimen,]Iln�.) Figure 10.22 "how .. �\'eral of the rnO:.t common. The bod)'-ctnltred cubic I' a repetItIOn of cube'i wllh atom... at the comers and (enler. It ilt found in the ,olid form, of the periodic table-s fir-Holumn ele_ .. in a number of tntn..ltion metal", iuch a." ITOn. chromium. and menl'. a., well 3. tung,ten The face-centered cubic i.. aha a repet itive cubic sUllcture. but ....11lI alom, at the center. of the t:ub.t:/II("tl lD,tead of al the center of the cube. The:

10.4 Cryltalline Solidi ept helium) solidify in this structure, as do many transit.ion ele DObie gases (exc g copper, silver. and gOld. The hexagonal close:st.packecl includin . ments ,tr\lc(Ure has a six-sided symmetry and shares with the face-centered cubic the of resulting in the smallest volume per atom for a lanice of idenbcal distinction found throughout the periodic table �pheres. Eleme ts wi Ihi st cture

�

�

� �

ar�

helium. magnesIUm, ZinC, tttanlum, osmIUm, and many rare earths. Compounds same crystaUine structures as elements. For instance, sodium chlo form in the ride is a face-centered cubic structure in which a sodium-chlorine pair replaces

the individual atom. In facl, the sodium and chlorine ions each i.ndependently ntered cubic arrangement. because the atoms of one are a fixed form a face-ce those of the other. Cesium chloride. though resembling the from displacement

bOdy·centered cubic, is an example of a

simple

cubic structure, fonned by

each ionic type independently. Besides their lattice geometry. crystalline solids are often categorized according to how the valence electrons are bound in the solid. Four categories are generally recognized. Covalent Solid

ill a covalent solid. such as diamond, each atom shares covalent bonds with those surrounding it. resulting in an unbroken network of strong honds. Such

solids are relatively hard. due to the inherent strength of the covalent bond. and have high melting points. They are

poor

electrical conductors because an

\a1ence elecLrons are locked into bonds between adjacent atoms. The crystal lat uce assumes a geometry determined by the directionality of the covalent bonds. In the case of diamond, it is face-centered cubic. Ionic Solid When atoms with nearly filled shells meet atoms with weakly bound valence electrons. the fanner may seize electrons from the latter, producing an

ionic

solid. The solid is held together by the strong electrostatic attraction between the

ions. Thus, ionic solids are relatively hard. with high melting points. Because the transfer of electrons leaves both positive and negative ions with noble gas electronic structure. electrons are not free to respond to electric fields, so ionic

solids are poor electrical conductors. The ionic bond lacks directionality. and

the lattice geometry is therefore determined by whatever arrangement leads to

the lowest electrostatic energy. which depends on the relative sizes of the ions. in any case, lowest energy results when signs alternate-that is, the closest neighbor of an ion of one sign is an ion of the other. Because ionic solids depend do not fonn ionic solids. on asymmetry between atoms.

Metallic Solid

elements

Except for noble gases. al l element!; have valence electrons. But in most cases. no arrangement of

covalent bond!; can join all of a given atom's valence elec

trons to those of the atoms surrounding it. An element or compound with "leftover" valence electron ... forms a

metallic solid.

When bound to their i,oiated atoms. valence electrons have relatively high

energy. In a metallic sohll. however. the atomic valence states mix. much as in a diatomic covalent bond but I'nl.:ompassing all the atoms in the crystal.

This

.a9

to • . ne. ... beoor1 bY potassium .I"ne. b) ,hl"nne alo and b) �.. put . 7 IOIid (anned -.w ttl. I;iod _ _ 11CA1I 'LE

.

__

.... ,,1

__

qedL _

.

--':�. - I. When -valence .7'1 " dIIorine

+. "'II '711 "-

logeth". chlorine will . . ........ _tng complete ....hell po"'llwe and

l('IM. 1be: kIIU �houlJ tht-n f\1fffi an K)ftl,;' ",>W1IunH.'hloridc whd. In..thid � � IC'MkI d\k1rmO not potassi um .;u fonn an hlflll; dtd. I.' ilJrrul('al ar('l4n ..III .,. dIIf(' tk(trOOi �)mmttrk.lIl. I). i\"�,iWQ h.n. illJ� 'J t�1ron In. phm

.#. I)� ,..tilt. "'-' 11 �11I n..-c ItnJ 1,1 � o:�·\akntl) Wllh mullirk .1then

� Il In a 1.11tK'('. \\ Ilh lC'fI" \'ef C'lt,,:trvn.s, I"lIlill1�m �ll the 'MUlti' t )I'IC '

.;I.nJ l '

N.'l m" dn\,:(', Iharnlnm". Il>e II"""""... . cI'"_'en_.,"II.IJ) morl"n" ,..,e.

'" bre�" "t"" '"'0 . " '""I" 00;' polOnti ; od" pori "" ..1.

In ' "y, om "," ,;o,k."

'0.23

"""'..-- p" � �.

n

r 11 = 3

.

,

f- .

·

'\\

••

..

•

\

\

,

dX

,

I

,

,

,

,t

I ;

. ,

\

, ,

•

,

,

Stutes

j

well in one IMge ionil" the UI .,.;/hlJ d'

\

pOlenti�L

.

�--.-- '

f----- ' .

Large well

432

, , ,

./

,

,

,

,

,

,

,

,

, , , , ,, ,

.

,,

,

,

,

,

,

,

,

,

II

,

2

. .,' ...

�

Energies i a band

cluster around the

corresponding

,mgle-alom , '" "ergy. .

,

•

...

'.

'"

n

,, , , ., , . ., ,, ·

·

,

=

band

,

,

. ,

,,

,

, ,

,

,

/

• ,

,

. h aminO
potential energy is high ha e relmively lv high IO al energy.

; ,• ,

,,

\

where the ionic

"

,

". . " '. ".

"

,

,

-------

band

,

,

='.�' '.

band

.. ., .

'

.

4-atom crystal

Single atom

kmt'IiC

... ...e\en (Mlme1olo'hal mi...-.hapenl an\t 1btkft. � Re'(l-Iower wa\'e tunctlon ha �, 'Otl' &I\'C\ It a energy. 'I�k'!n.m. "iomewhat lower. but 1\ alMl

.

.»-t'" it the wrong wa\clength to be lCo) between all Ion... Ito< ptltt'J1tial (fItlt) " higher. Thu.... tt"i lolal energy .... nOI much lower than the top-of-the-

tIJI'd ,tatc. The nc,l-lower \Iate . ..i, anlinode.... \\0 lower 'till In kinetic 00\ biJbCf again to potenttal. and 0;,0 again 1\ only ...lightly lower In total energ)' 'tbC wttom-of-Ihe-band \tate. with ft\e anti node!".. 1.. nearly at 11, mal/mum at

all lotCratomic pointo;" Thu.... whIle of !\Itill lower kinetic energ)" It I' 01 \'cry h potential. again re,ultmg m only a \light decrea..e In the tOlal The ..itua

DiS I tWO ,hange"i dra of electrical conduction Ihat may be under�tQO(.I

from a classical point

01 view and 10 identify those that require u quulltllm�

mechanical approach. Figure 10.26 Beh;\\i\lr of allowed electron SillIes as atomic spacing varie\. MInimum N alomlc

�hlle)

B�d width

AtomiC 'P' ing

•

, "1 ... .. � . •_ .. Ck;tl l' •

BOlIld, and �ar' 10 a one ,... 10.27

di�1 crystal

. .

..

. .....

.

I

) .....

-'1,-----,1f�r¥) 1!4p , II c

2 bud

!

" -.

:�

are free 10 move aboUi i n ondu o piclure. electron, Cl ,. In It\( da....IL:al ions_ This i s a th nec ti s ary , , Ingrt_ io", " Ith the po" colh, "flee 001 th,) ma) electrical current is proportional to e applied anding why the ndcN 10 dlent Were the'e no call'ISlo u , ns, asing function of time, being an IOcre . l'idd ralher than < constantly would ncce erate, field c electri proant to a con�l electron.. ..ubJed represent a retard'Ing force ons Colhsl nt. " Increa>;lOg curre - In dUl,;\Og B ..h:adily ously attmn lermmal speed in whIch a bal· nc tanta m .. �t >; almo . e..-.en�e, electron and the decelerating effect 0[ acceleratlOg field the n bet\\ee anee I " reached , ' clllh",on... Speed ; moves randomly with an avera typical electron ;: CI�...ica\ly. 11 with a positive i n , !I IS ) "t ", After a co\lisio the tempcralu chaneten,lIe o[ - n as another. With no externa e ectr.ic _ onc d-lrecllO monng 10 But when an e,ectric a.. IiLel) to be the electrons is zero. \'elocit)' of all . . . a compOnem field, the B\'erage collisions between the tLme . electrons gam, 10 the nt, pre!\C I' time field average belween colus define 1" as Ihe . lle the field. Let given electron of \'elOClt) oppo� a may have time. in inSiant n time AI any but an average eleClrOn Ii,ions. or collisio lime, [ . span 0 - g [or an arb-Itraty 1eraun eee a ' 1 ree 'i been 1". It WIII thus have acquired an adde •

I

,

1 I

_

_

'

found Il!; folloVo-'!i:

j•

charge

charge

time . area

-=

distance · area

distance time

:::

e X number distance volume

time

field.

the

It" I'J '" the number of free charge carrier:. per unit \'olume. We "ee that ..tic 3 current den,it)'. is proportional to the cause, an electric which I' (/ftc!. . .. , . y l constant IS . own c i l . 1m d " y 0\. the as t e con pr oportlontu lt h t v u \, law The reciprocal of ils resistivity p--and is given the symbol u.

::rial-the

j = uE

( 10-7)

vity decreases as the collision time 't decreases. One way ecrease the collision time is to increase the temperature, for faslcr-mo\'ing

(I � �ical1y. conducti

��

c1es collide more frequently with obstructions around them. Thu\, the conductors decreases as temperature increases. ductivil)' of o l Cl Much of the classical view of electrical conductivity is valid quantum 'banically. Collisions are still viewed as the origin of electrical resistance.

:e � �

er. we find that in metals. though Ohm's law still holds. the collision larger than can be explained by a theory in which electron\ may m ' e is much llide with all positive ions. The quantum-mechanical explanation i" that near edges of bands) the states allowed an electron in a periodic crys( Ctpt 1. are essentially those of a free particle. The electron is a wave. upon which the \'

posith'e ions have little effect. It is not the regular array of positive ions that up�elS the otherwise free electrons. Rather. deviarions from regularity perturb

the electron wave and thus determine resistance. The most important devia

U(lnS at ordinary temperatures are vibrations of the positive ions. These increase with temperature. contributing to an increase in resistance. (The facl thai conductivity in metals tends to vary as T to the first power is further evi

dence that ionic motioll. not mere presence, is key. See Comprehensive Exer cise 77.) At temperatures below about 10 K, vibrational motion is so

diminished that the predominant sources of collisions are microscopic lattice

m i perfections. These may be either point defects, due to atomic vacancies or lmpurities. or more widespread disruptions of crystal regularity. In any case. the quantum wells at certain locations are altered, and the electron wave is perturbed. Unlike ionic vibrations. the abundance of lattice imperfections is largely independent of temperature, and so is the resistance lhey cause. In all solids. electrons fiU bands from low to high energy in accord with \hI! exclusion principle. In a conductor, the topmost band is only partially filled: this band is where lhe conduction electrons reside. (Lower full bands do DOl participate in conduction, as we see in Section 10.6.) To a good approxi

mation. the conduction electrons behave as though moving freely inside a macroscopic well (the entire crystal) with no ionic potential. Their energies are those of an electron

gas (Section 9.6), and the energy of the highest·occupied

siale. measured from the bottom of the band, is the Fermi energy, EF" As

depicted in Figure 10.28, with no external electric field, as many electrons

" I.... .,, �

..... .. Os- to

'l\idl

ickf pllCftYl (k�(� 1nOl'C" da.1J,l(tl Ji ...,nf':Itn •

;S �

\

IIra:.J Jt.Ift'.

""'"

,

,

.... .t

.

.

'

,{.., > 0

0

hlled

....-- Eleclric field

tJt,..-!n( ',tId · 0

/

direclion as in the opposile direction· Theke{ffe in" � in onc Ct of � ukl � mel\ �''1 enta toward Ihose in wbich 10 shi fl clectron mom I' field mal e\le =: P/h) ,;tn the mo menta of all the conduction elecl fl�iI� [he tield. Although I� ()pn ' r tOns only among e higheSI�energ W't' ,h ttt , � Ii« mal IC I_� a net Shill .

e

�

1'.

Y stales. . £f' where the momentum IS 10 the direct'IOn belo" y l h l , t g i ,["Ie' ' of '" fnlm . l.Ie � where I" I IS O�posl le. Classically or qu ' above l:.F' tates just to the top of a band that IS separated from the band above-the empty conduction band-by a large gap. The ncxt bigher allowed energy is �o high that ordinary electric fields simply cann(.1t bump electrons up there. The electrons cannot occupy any state.. they were not

alread) occupying. so there is essentially no response to the field. Even at

fJtber high temperatures, the number of electrons energetic enough to occupy 3. \tute In the conduction band is negligible. The term semiconductor is applied to materials that are insulators at zero temperature but in which the gap between the valence and conduction bands ii'. ,mall. By somewhat arbitrary definition, an insulator with a band gap smaller than about 2 eV is considered a semiconductor. With such a small gap, the tem

�ra1Ure does not have to be very high to produce significant occupation of the conduction band. Semiconducting clements commonly used in electronic devices arc sili

con and, to a lesser extent, germanium. Both are "covalent" solids with the

�ame lattice geometry as diamond (Figure 10.13). Nevertheless. they are prop

erly treated via band theory, as though all atoms share all valence electrons. The relevant states are the four tetrahedral states. Four spatial states and two ,pins could form a band of 8N state!:i, which four valence electrons per atom would fill halfway. BUl this band splits in two, just as the bonding and anti bonding atomic stales split in the carbon-carbon bond of Figure 10.14. The lower band of 4N state!:i is the full valence band. and the higher is the empty conduction band. The bands are separated by an energy gap of 1 . 1 eV in sili

con and 0.7 eV in germanium. Diamond shares the same band structure in it!:i n :::

2 states, but its 5.4 eV gap qualifies it as an insulator.

The Conductivity Gap

By applying some statistical physics from Chapter 9, let us investigate more closely the link between temperature and conductivity in insulators and

nIptJ)'.

10

1ID.lt'\:trm, 1'lI.':Up)

.

"""'

£ oIIoy. .. ""'-

:finJ

tht nUlllhcr ('\ClteJ Ill lh(' contiucl1(\n b..md at T higher than 1-", bolwm �lf thlll t'land. " hll'h I'

iF,.lr

.,.

0, \\« il\h,'�r"t'"' lr\\1l\

{k(lIlI,e S(£\n fnlb 00" quickly, (''(tcnding the upper limit tn

infllHty Hltru The mtcgration i( III I U, tIofh h"le� .md t'1«Jrlln, wn mbule III �'urrenl In the J,rt...III'n lli' an aPJIItc:'\t t'lft:lri.: lidJ

.

,.

Al.t�(1m

111•

.\1 tIouum, "I tlalld�.

" J:rdk " larger. ..,.., - 'I' ''' _ _ m

, ,

'

"

�." AI wp, of tlanlh, ..,�

., .__ .� �

J�f)dl..l ilnd men' are

negaI1H'_

,

�lIh Figure 10.27, In most region.., it is a plot of a free particle in a macro �:()pit.: well, wilh £ = II�A.2n.m and lhus an effective mass of ffl. The positive ion, ha\-t little efleel on eleclrOns at lhese energies. Only near the band gaps is

the curvaturr ditferent, At lhe bottom.. of the bands, the effective mass is less

than m (the cunature d1.£JdA.2 is larger). At the tops of the bands, it is negative.S Here the force» mtemai to the crystal are such that !:he electron'S acceleration is

-

-

(lppo�ire the eXlernal force. (See Comprehensive Exercise 78 for an instructive

mechanical analog of negative effective mass,) NO\I, lei us put the ingredienLI; together. In an ordinary conductor like cop per, electron' fill )Ialts to around the middle of a band. where they are essen tially free panicles of ma.s� III. In

n semiconductor,

however, electrons fill states

1\) the lOp of the valence band, where the effective mass is negative. Because hole energy vane.. in the opposite sense, we might imagine Figure 10.37 flipped tlrc!rl': flC'IJ anJ nln�nl

-�

up..ide down, with opposite curvature. Therefore, at !:he lOp of a band, and only there, a hole has positive effective mass, and thus it moves i n the direction of the e'm'mal field. Figure 10.38 illustrates how conduction electrons and valence hole, respond to a field.

Doping The semiconductors we have considered thus far are referred to as intrinsic semKooduCIOr5. In these pure materials. the number of electrons thermally

'....,. dYt t".... 10.)1 .. mt'lr\._ pt ... ..... 1CI.*' IJ, Thf dftcfne .... 0/.. eko.·, .-0. 111 --" .. oCIIt II 1IdbdIr' Thf t\pI.aMbon gf ,., �-.:"-ioo IS tI\M Jbt p.niculIrty . u. " dill ...

elements-we produce f¥lrinsic semiconductors. These have a large majority

tttoq inttmIJ 'QrttS lw,_ tilt dkl and tilt tloo.-aoa lit MIdI. staltcon"oe) lilt nlrnW r"n.� l{l

greater concentration of charge carriers than the pure intrinsic semiconductor.

dIr 1Otid • • ",hilID (""!urI). thl: t"io:I.uvn "ooid .... -.er)' 1nIIW'�.

e,ciled to the conduction band necessarily equals the number of holes left behmd in the valence band. By doping-addition of small amounts of impurity of eilher conduction electrons or valence holes and, i n either case, a much In the following discussion, we assume the intrinsic semiconductor to be the

1 , 1 !,.,,\){\\I1m :1'".'"

r1l \ ;\ mum�

tn:>e: a covalent lattice of telrovaJent atom'. ,ueb u, ..ih,.:on or

.\0 o·type e:tlrinO

T=O

acceptor state.) Thermal excitation readily bumps electrons from the valence

.

band to acceptor states, leaving freely moving holes.

Again there is always some thermal excitation across the entire valence_ to-conduction gap, creating electron-hole pa irs in both n-type and p-rype semiconductors. Both thus harbor charge carriers of the "wrong" sign (i.e.. valence holes in n-type and conduction electrons in p-type). known as

minority carriers. But far more abundant are the majority carriers: valence holes in p-rype and conduction electrons in n-type. What makes doped semi·

a

conductors unique is the ease with which we c n vary the sign and abundance

.

of the carriers via impurity type and concentration. As we see in the next sec·

tion this is central to modem electronics.

A Closer look

Effective Mass

a particle moving Ihrough a medium. and

Section 6.4 shows Ihal the velocilY of a panicle. called group

\Ie/DeilY. i� related to its mailer wave properties UJand k by

I'

.

paruclc

: 1'

8"'"P

dw dk

=-

As demonstrated in Example 6.3. Ihe dispersion relation rnat gives (J) as a function of

k

accounts for rne effects of a

medium on a particle moving rnrough it. Thus. we may write the acceleration of a particle in any medium. including a cry.�ralljne solid,

d dl

as

0 = - 1'

.

parodc

( )

d dW dl dk

=- -

J2w dk dk2 dl

= --

Effective mass is defined as the ralio of rne exlemsl force to this acceleralion. Ignoring losses due to frictional forces. it is the external force alone that does rne net work on

the force times

the panicle's velocity is (he work per unit time (power). �

dE dr

-

Using E = ItW, we then have F" ,' _ " " " � -.

=

dw dw dk � h -- = dr dk dt

11 -

or, canceling vpMI,cle on both sides,

FUI

=

dk h -;;;

dk

Ill'parllC . ,c dt

1 0 . 8 S emiconductor Devices

how doped semiconductors may be exploited to produce twO \..et U� nOW study important circuit elements. most e of th

The Diode

applications, we wish to allow current to flow only one way. The ill many

most basic is a rectifier. which converts the sinusoidal\y varying (alternating) current supplied to homes into the direct current needed by most eleclronic circuitry. Actually building a device that does this automatically, known as a requires some ingenuity. The primitive solution used evacuated glass tubes housing bulky electrodes and heaters. The advent of doped semiconducbrought a revolution. Asymmetric current flow can be achieved through a

diode, tors

simple physica l asymmetry : charge carriers of different sign in different regions. We produce a diode by joining an n-type semiconductor to a p-type �emiconductor. The area of contact is known as a p-n junction. As shown in figure lOA 1 , if we apply a potential difference with the p-type at higher potential. known as a forward-biased condition, free holes in the p-type region and free electrons in the n-type flow toward the junction. When they meet there, as we discuss below. they annihilate, or recombine. Current flows

continuously as electrons are added at the low-potential side and holes are added (valence electrons removed) at the high potential side. On the other

hand. if the n-type is at the higher potential-a reverse-biased condition

both holes and electrons move away from the junction. A region devoid of free charge carriers quickly fonns, and current stops almost instantly. Energy-level diagrams help clarify the behaviors . Figure IOA2(a) shows an

unbiased condition. If we simply join a p-type and an n-type, without an exter nal potential difference, electrons "high" in the n-type conduction band cross to the p-type side, raising all electron energies there by repulsion. while corre

spondingly lowering them in the n-type. Few electrons need actually cross

before an equilibrium is reached in which the highest occupied levels-the

figure 10.41 The motion of the charge carriers in forward and reverse biased p-n junctions.

Hole" added

Electron-hule r&ombin"ll\)O

If e e ® e ,'� 0- 0 e '-0

...

..

'

.

'

..

EI':"lron� JJ!.kd

.....

...

n-type

p-type

Electron noYo

«.tl

"---'lN1fIr-I 1 \�- +

Forward bia)'

Free !.:harge depleted

Holes removed

.

'

+

e e e

,.......,

p-type

•.

. •

th� mu�h lar!!er f.. nUll in� uur the �'\lllt'dN

Figure 10.46 A MQSFET i.. well suited to i ntegrated circuit .., The g;lIe,

The bipolar transistor discllssed above is just one of many semiconductor amplification devices. Another is the field effect transistor (FET), one type of which is the MOSFET, depicted in Figure 10.46 and discus!.ed in Exercio,;e 71

,eparated from the scmi..:onduclOr by an rn�ul;:urng layer, (;ontrob �Iectron now from the ,ource 10 the doolin,

ib electrical properties and ease of fabrication make it one of the mOSt common ,witching devices in integrated circuits-the heart of today's ubiquitou"

Dr.!in

microchip electronics. A silicon chip can be modified so that doping is arbitrar

(-wJ[c�lUnI

"

ily n-type or p-type in successive minuscule regions, creating transistors smaller

than I f.lm2 in area. By appropriate doping level in single n-type or p-type

regions. either resistors or highly conducting ;'wires" can be incorporated

hence

,

Gate (-ba,e)

I p

S()urc�' I emitter)

"

- l lJ.m

Ihe name integrated circuil. By the early 2000s, circuits of a billion

rransislOrs could be "written" into a single chip. Figure

10.47 exhibits a typical

IJ'Jnsistor density. Densities continue to multiply every few years.

1 0. 9 Superconductivity

Some materials at low temperature lose all electrical resistance. Figure 10.48 shows resistivity versus temperature for tin and copper. While copper retaino,;

resistance down 10 the lowest measurable temperatures. tin's resistivity plum meb to zero at ils

critical temperature Te' a characteristic of the material. In superconductor. About 40% of the

such a stale. the material is known as a

natural elemental metals are known 10 become superconductors at low temper ature. In this section. we discuss features common to superconducting materi

als and the mechanism by which superconductivity occurs. With no resistance. an electric current established in a superconducting material should persist indefinitely. without an applied voltage. Indeed, many e�periments have been done testing this very prediction. one indicating a min imum time for significant current decay of 100,000 years!

Figure 1 0.47 A modern computer processor. The 'mall chip in the center. only 143 mm2 in area. coma;ns about 300 million transislOr�.

Fig,". to."9 When II drops �Io� Ib cnllcal lemper.uure. a �upen:(lndUdOr c'l:peh magnelic field line!>.

,.,..,. 10.... Copprr .. h u� tw rh', (nf:.1 tnl tm«, .hrlc lin be('omcs 01 uptn:ondtKlor

i . 2 .. -

f

:

o

In

,.

8"" . O. T
. ,ourc'. or ,,,,h 01 ,ueh ""h an 'I"m'" ' n,,"n.1 ,rr:ty' dln" . 1".,. pn" ",m cf

, . D'

10.60 fh� I.!c'lgn l\t I , t)rl\·,J.1

\"opI('lllbt-r :!�. :2607-26(9). If the untenna/CNT length I" not th(' l't.'n'-'Ct

flgUfW

":11.l1uple of the light \\u\eienglh, or if it... oncnlallan i, pcrpcmhculilf tfl thl..' !Iptt pvloU1z:l!ion. lhe reception become... meFricient. Rounding out their

l Onln { -.:\'lk�·I\'f)

n',lIm\'. Ci' , ,n,, ",uperconduct, though ugrun 3t fairly 10'\ tempernture...

\1.10\ of the upphcations nDled above employ �o-called I\lultl\\ulled

�

('T�. d p,\�ted In Figure

10.6 t . of which combination" are emile"'''. Ol,co\cr

ICC. I\ml the I�,ol prope-rtieo;. and u�c" of CNTs are multipIY'ing :1I a dizzying p..

1\'Jlkr j.. cncour.\gcJ to "carch the Internet as the knowledge unfold .. Fmally. It I'" worth mentIOning a few variations on the theme. J\I"t ll'" ..lIi .,)n ca.n be replaced by quadruvalent germanium in some semiconductor lIpph

�'alloo� tlnd by trh'alentlpentavalent ( I l l .V) mixtures in others.

llunOlubc ... hllve

betn fabricated from silicon and from boron-nitride (I11.V). The latter �holiitl ,(.lnd up betlcr to high temperatures and exhibit superior resistance 10 oxidu \I(\Il lhan CNT�. but it is too early to say whO! role either of these newcomers \\ill e\cntually play. Figure

10.61 A mul!iwallcd carbon nanotube.

eN l !l' r It'll'll.! ("" 1It'd tt....''''''I\'r).

I\", CNT l' -

--'--

SI..'lln;e

\-cIlIIUt'rl

-----'-

Gale (-�)

II brra1� fhousunds 01 Cooper pair;, The liberaled elec[ron�

UgItt o.t.ction and fMf'l)' G.pt PhoIOn dc't«tlon i\ "Ital In IMn), arul of K1C1JCl' A\UOOOm)' I� �ps the tint cump,," Whllt fbi: IbIllun of Idcsropn 10 rotM'f

I

art pn'mul�d "UP"ard" acrOl.\ the gap. Ihen [unnel through

h,ht (and .ltb£t part) of Ibt dectroaugnmc Ip«truml ha\e

the m�ulah.lr 10 the Olher �upen:onduclor. resu ltmg in II

1TI(',..urJNe charge tramfer (Tunnelmg of e lC(lrons Mill m

Cooprr p.:II.... I� ,upprr�..ed b)' an e1ltema/ magnelJc field,) by the STJ is that b.:\:au..c thc= number of elecrrons promoted is proponlona/ to UlPlhlr.;fIVUY "",)' he 11t'0'� IQ the JlClt m, JUDlp. The thO empry nmdu!;l1on band The,.:' re�pond to lin e.\l('m.a1 li('ld JU\I a, do dedltms In II condu�·lOr. The resulting \'iIIlt""e band \lk:.IInu('\ aho !;onlnbule 10 det:lrical condw,;li\, Ily Tbr)' art lnllwn '" hole\ and re\pond 10 an external field a.)

wh)" both energic\ in the lower band �hould be roU!!nt)" C'tIual to (hat of the n = I atomic state and why both enc:rgie\ in the upper should roughJy equal thai of the n :! atomic �Iale. =r

2- L'pon what definitions do we base Ihe claim Ihat the 1/1, and �1r \131eS of equalions (10-1) are related I. r

�P,

'

3fld Y JU\t as 412/', I!' to ;:.1

J. Seclion iO.:? discu�ses u-bonds and ,"-bonds for p-�tate� and u-bonds for S- Sla le� , bUI nOl 7T-bonds for f-sUltt\. Why not? 4. or N�. °2, and F2. none has an electric dipole moment but onc docs have a magnetic di pole moment. Which ' one. and \\hy'.' (Refer 10 Figure 10.10.) S. It takes le�, energy to dissocillte a diatomic fluorine mole_

cule than a diatomic oxygen molecule (in fact. less than one-third � much). Why is it easier to dissociate nuorine?. 6. Why is covalent bonding directional. while ionic bonding is not? 7. For the four kinds of crystal binding--covalenl. ionic'

moleculat-how would the density of Lhroughout the solid ? Would it be constant, centered on the atoms. or largest bet\\-een the alom!>? Or woul d it allem3le . with a nCI charge denslly positive at one atom and negative at thc next? 8. Why should magnesium form a melallic solid? PII'toIIl\(' I;h4l',:e �'anie"" The energy necessary 10 break: the ionic bond between a 9. The numbcr"\ of C()nductll,m banu electrons and "alence sod i um ion and a fluorine ion is 4.99 eV. The energy band h(lJe� c.:an he: ,ancJ I�I) by doping. Of Introducing into the to separate the sodium and fluorine ions that necessary /4111\'(' It �m.tlll frJ!;tllln \11' impuril) a,om�. Tri"alent lmpurit) form the ionic NaF crystal is 9.30 eV per ion pair. al�lm� )Idd a r-')pC' '\('mkllndu�lor, in \\hich hdenct holes far E1tplllin the difference qualitatively. fllJlnumhc:r l'ondu!;tllln cI�(n)O�. whik an n-type \emiconductor, a me m l consisting principally of copper alloyed Brass is 10, where �"Undu'"tHln ela:ltlln� prNtlminatt. resulls rrom pentava· wilh a smaller amount of zinc, whose atoms do not len! Impunry alllm\ ImportUllt de\'icts are rrude by assembling the crystal lattice but are i n a regular pattern in allemate \1LnOUS �'Oll1blnatlOn' of n·type and p-Iype �C'mjconductors. Join somewhat randomly scattered about. The resistivity of inB ant or e&;h type in II p-n junction yielcb a diode, which higher than thnt of either copper or zinc at room bra�s is aJl()w� e/cclrkld current III pil'>\ (lnly one way. Three together temperature, and It drops much slower as the tempera rtlrm a D"lUIsl�t(1r. II baMC clemen! of mplitication. ture is lowered. Whal do these behaviors tell us about Many melah and II growing number of novel materials electrical conductivity in general? /().\e al l te,i\l.lnce belo",,' a malerial-dependent cntical temperaII. Explain the dependence of conductivity on temperature turt. bc�·t'ming �u �n'onducto� CUrrtnl is carried by electrons for conduclOrs and for semiconduclors. tw,'und In C\II,lper pall1t.. the rt,uh of an attraction mediated by 12. In the boron atom, the single 2p electron does not com tht cry,tal Jauice. IUld all patrs mO\'e as one. The effeci is bro pletely fill any 2p spatial stale, yet .raNd boron is not a ken bJ high ltmpcnuure and/or it \trong: magnetic field. conduclOr. What might explain this? (It may be helpfUl to consider again why beryllium is not an insulator.) 13. The bonding of silicon in molecules and solids is quali tatively Ihe same as that of carbon. Silicon atomic stales become molecular states analogous 10 those in Figure Conceptual Questions 10.14, and in a solid, these effectively form the valence I. Thr left diagram in Figure 10.1 might represem a tWO and conduction band.f. Which of silicon's atomic stales alom cry�raJ with two ban�. 8a�ing your argument on are Ihe relevan! oncs, and which molecular state corre we Linelic energy insick eilher individual wel l, explain sponds to which band? metll1!ic, and

valence electron s vary

decrease the. conductivity of a conductor I" What factor> 1I'i \emper'J.\ure increases? Are these factors also present In II semiconductor. and if so. how can its conductivit), vat)' with temperature in the opposite sense" Il. Based only on me desire to limit m;nonlY carriers. why would silicon be preferable to gennanium as a fabric for doped semiconductors? 16. Why docs the small current flowing through a "versc biased diode depend much more strongly on tempera ture than on the applied (reverse) voltage? I7· In II concise yet fairly comprehensive way. explain why doped semiconductors are so pervasive in modem technology. 18. It h often said thaI Ihe transistor is the basic element of amplification, yet it supplies no energy of its own. Exactly what is its role in amplification? 19. The "floating magnet trick" is shown in Figure 10.50. If the disk on the bottom were a permanent magnet, rather than a superconductor, the trick wouldn't work. The superconductor does produce lin external field very similar to that of a permanent magnet. What other char acteristic is necessary to explain the effect? (Him: What happens when you hold two ordinary magnets so that !hey repel, and then you release one of them?) ZO. The isotope effect says that the critical temperature for superconductivity decreases as the moss of the positive ions increases. Can you argue why it should decrease? 21. What is a Cooper pair, and what role does it play in superconductivity? 22. Describe the similarities and differences between Type-! and Type-II superconductors. 23. In a buckyball, three of the bonds around each hexagon are so-called double bonds. They result from adjacent atoms sharing a state that does nOf panicipate in the Sp2 bonding. Which state is it, and is this extra bond a u-bond or a 1T-bond? Explain. 24. What are some of the properties of fullerenes that make them potentially so useful?

Exercises Section 10.1 25. Fonnulate an argument explaining why the even wave functions in Figure 10.1 should be lower in energy than their odd partners. 26. The accompanying diagrams represent the three lowest energy wave functions for three "atoms." As in all truly molecular states we consider. these states are shared among the atoms. At such large atomic separation. however, the energies are practically equal, so an

electron would be Just al< happ), occupying any combi nation. ta) Idenllt)· alg, but Il! can be :! I , then the allowed photon energie!: obey equation ( 1 0-6).

.n. Vibration-rotation speClra are richI For the CO mole

cule (data are given in E",erci�e 42), roughly how many \'013tional levels would there be between the ground vibrational state and the first excited vibrational state?

�2. The carbon monoltide molecule CO has an effective spring con'itant of 1860 N/m and a bond length of

0.1 13

" one-dimcnsionlll ealcu� \.).\ion is in�tru..:t;\'e, C'(lnSlo.ler an infinite line of point charges alternating betwt'C'n +" and

... the lc) For thc'>C energies, by what per\:entage dot atomiC ..eparalion fluctuate" classical vibrational frequem:y (d) Cl1kulate the_ p- and lhe rotational frequency W w"t- =

10\'01\'«1 h'r a 3D lani�e. Nt

per

be helptul: In(1

Section

,

" I

2 band and the oo\t('m of the II

:III

3 band fur a

W8\'e number have \"" tl)' d,lkrcnt energie).

48. A, we ,ee in Figure, 10 :B. in a one·dimen�ional !.:ry�tal 01 finite well�. top-ol·the·t'land ....Ialc\ do\e!y re\Cmbk inliOltC well state'i, In la!.:t. the I'amou\ partldc-in.a.bOll. energy formula give� u fair \'alue for the energle, of

these ,tates and thus the energies (If the band, to '"hlch they belong,

(8) II for rt to thaI formula you u,e the

number of anllnodc\ to the ",hole W3\e

functlllO. '"hat would you U';C for the box length L? (b) If. in'tead. the n in the fannula ....ere . laken to refer to bdnd n. could you �lIl1 u�e the fomlula? It \0. what would )'1..11,1 u\c foc L?

in the lower-energy electronic ...tate? Explain. What

vibrational energy levels.?

-L{- \)"'11

wells. Explain \\-h� the� 1\\0 \Iah.'\ 01 rou�hl)' equal

(c) Explain why the energ;e� 10 a band do or do not

about the rotational levels?

Ie�'el!." (c) And what effect doe� it have on the

-

one-dimensIOnal "cry�tar' con",lIng 01 !-e\en finite

the same. farther apart, or c\m.er together than lho�

(b) What effect does it has'e on the rotational energy

"{)

10.5

the II

tional levels 10 the excited electronic state to be spaced

on the bond length and force con...tant" Explain.

+

41. Make rough ....kelehe' 01 thl' '"a\e IUnctlon, at the 1\IP uf

�3. From the qualitative shapes. of the interatomic potential energies in Figure 10.2 1 , would you expect the vlbra

atom (a) What effect. if any. does the replacement have

ion. For \lmr\u':lty. ",,,,urn... that 3 po�ltive charge 1..

at the origin Th... fulloy,.m� puwl'r �nes e'pansion W il l

CO might absorb in vibration-rotation transitions.

from H,! 10 that a deuterium atom replaces a hydrogen

with a uniform

Wh)? (bl Calculate the ele..:tm\tatk potenltal ene'll)'

nm. Determine four wavelengths of light that

"-'. A noted in Example 10.2, the HD molecule differs

�(',

....paung bet....«n ad.la..:ent loppoo.ite) charges of Q, la) The cle..:tn\\tatir.: f"'tenltu\ tnefl), pt'r ion i.. the same illr a g!\cn Pl"ltl\'e I(ln a:o; lilt u " \'en Ilegati\'e ion.

49.

depend on the sile of the cry�tal 3., a y,.hole In Figure 1O.24.lhe n

I bilnd end\ at k

'"

4rrlL,

while in Figure 10,27 It end\ at rrla. Are t.he�e ..:ompill1· ble? If �o, how?

50, Assuming an interatomIC 'paCIng of 0.15 nrn. ot'luto a rough \'alue for the width (in eV) 01 the n one-dimensional u)\tal.

=

:! hand to 3

51. The density of copper 1\ 8.9 X 10-' kg/ml. it.. Fermi

energy i� 7,0 eV, and It h..� one condur.:tion el...ctroo per

Section

10.4

atom. At liquid nitrogen temperature

45. 1\Yo-dimensional lattices with three- or four-sided sym metries are possible. but there iCaller plot (c) Repem pan (b). but chOOSing

B In lhe " Impuri ly" alom. pUlling irs bollom 31 -2 unih. Cd) Discu!.s how Ihe impurilies added in pan�

- 0, I for

(h) and fc) cotre\pond to atom, I,I,ho\e ...alence differs from thaI of lhe intrin�ic atom!., (e) If each inlrinsic

110m comes with two eleelron\. and Ihe impurilies

come wilh one and three, R:specII\'ely. which slat�s would be filled in part:. (b) and (c)? Remember Ihal

there are IwO �pin Slate,. (n Discu,.. the overal l result

of adding Ihe impuritie�,

N u c l e a r Phys i cs

11

Chapter Outline 1 1. 1

@

I \ .2 1 1 .3

I tA

Basic Structure Binding Nuclear Models Nuclear Magnetic Resonance and MRI

1 1 .5 Radioactivity 1 1 .6 The Radioactive Decay Law 1 1 .7 Nuclear Reuctions

. n the preceding chapters, we tended to study ever-larger objects with photons and electrons. one-electron moved to We began � tron atoms, then to molecules and solids. In some sense, n the multielec

I

n a[o l.n" ',nue the trend would lead to other disciplines-chemi�try. engineering, to co and �o on. We now head the olh�r way. .

contam nuclei, studying the "atom" usually means Although all atoms of its orbiting electrons. The nuclei are assumed to behavior on the fIX'using nuclei . But are not always inert. For one thing. they can s n tiaU y inert

b< e se

' and such nuclear reactions release spontaneous1Y or otherwtse, ra f menl, grrnous amounts of energy compared to chemical reactions. In nuclear 0 ttmg e1ectrons t hat are 0f I'title concern. It IS true that they ,"h �ics, it is the orb" ,

P } bound to the nuclei, but compared with the energies within the nucleus,

�ctron �

bindi ng energies are usually negligible. The nucleus is the focus. hard problem t? solve. While much has been learned, it is nfortunalely, cs contmues to be an active area of research. auClear physi , ' cannot be overstated. h ex.plalOs the tance 0f nucIear phYSlcs impor The

�

and nuclear weapons; nuclear magnetic resonance perotion of nuclear reactors produce images of the body's interior; and radioactivily routinely used to

�I �

ergelic particles emanating from nuclei-is both a useful diagnostic tool, via n

dioactive tracing and dating. and a mounting disposal problem. as we con artificially produce more radioactive materia1s. Moreover, tinue to unearth and

the study of nuclear physics is an essential step in our quest to unravel the fun damental structure of the physical universe.

11,1

Basic Structure

All nuclei consist of protons and neutrons, known collectively as nucleons. Masses are given in Table I I . \ . Introducing important symbols, Figure 1 1 . 1

depicts a helium atom-two electrons orbiting a nucleus of two protons and twO neutrons. We use the symbol N for the number of neutrons and Z, the

475

-

-

-

-

�

-

-

-

-

-

-

-

-

- -

�-

- �-�-

.., , , , ... -

I 6126J17 )( '0 .17 ...,

"

•

••

I 6'49273 x 10 11 ..,

o

If •

•

-

9.1(9)( 10 u k,

•

....

hr.

.....

-

� '=>

J.(J07276 u

I 00tIM.'I u

� "'AA X /II � u

lor rbt number of prolons. The ..ymbol A. caJled the IRI5s

II &he lOCal number of nucleons.

At we found in C'bIpIer 8. an element's chemical behavior depend, onl}

a. die IIUIIIber of eJecuons miring il.\ nudeu.�. equal 10 Z. But nucle, of .he .... demenI rarely beha� alike jf they have different numbe" of neulmn,.

ftar ...-.xe. while tbt nucleus of ordinary hydrogen 1\ "" mply a prolon. about 0.01$" 01 hydroaen IIOmS in naluft' have nuclei con ..ie:nd on e orientation of the nucleon not �urpnsmgly, there IS no simple formula for this force, .. ins, Perhaps indeed has yet to be fully characterized. Nonetheless, we can explain quite a bit from what we do know about it. ore

�

::a,.,y :h.ich

Sta bility : A Theoretical

Model

engaging in an extremely strong attraction, we might lmag With all nucleons could bring together any combination and have a stable nucleus. vast majority would be unstable, disintegrating in a time span the owever. [allging from an instant to an eon. A few would be stable. The following discussion of nuclear stabiliry may seem rather s.peculative, forces in the nucleus have defied formulation of a compre but the complex htnsive theory. so we must resort [0 a model-well-informed but simplified guesswork. [n a later section, we discuss well-established models thai refine we one we begin here, bUI it is good to gain a quahtative understanding of the \arious factors fust. As we do. we won't speculate whether any particular combination of nucleons should actually be stable; we will simply argue ....hether a given factor should tend to make the combination more stable or le�5 stable. A nucleus is more stable when its constiruems are bound in a state of lower energy, requiring a greater expenditure of energy to extract a repre q:ntali\'e nucleon, Energy is all-imponant.

'ne that we

�

Two-Nucleon Nuclei

The simplest possible combinations of nucleons are two protons (p-p), two

oeulIOns (n-n), and a proton-neutron pair (p-n), Whkh should be most stable? . Let us look at the factors The short·range strong attraction shouJd create some. thing like a narrow but deep potential energy well. and if the force doesn't dcare" whether the particles are protons or neutrons. the weU should be the \.llIlC for all three combinations. This condition is no guarantee that any would hold together-wells can be too shallow to have any bound state.!. (see Exer cise 5.38)-but at least they should all be the same. On the other hand, the p-p combination ""ouJd mclude Coulomb repUlsion between the proton!>. which v.ould rai� the energ), and this argues that it should be somev.hat less stable than the other IWO. .. are ValId. but the ltuth is that onJy the p-n. known � a These argument dtUteron. form' a bound nudeu'i (that of the stable deuterium alom dio;cussed earlier). 1be n-n and p-p don't stick together at aJl. Why" The internucleon aruaclion i, .,pin-dependent. and II i... stronger ""hen spins are aligned. We will :e 1.5 Ie.. 'loigOlficant In larger nuclei. but here it call.. . • pendcn e find thaI milo d thai Ii crucial in all nuclei: the exclu�ion principle. factor r e h t o an play into

Fig.". 1 1 .4 The basic elements of the internuclean (strong force) poIe11tia1 energy-a strong. mort-range anractioD with a repulsive hard core. Sudeons Iolrongly atUaCI

lpo!cntia] energy drops). but only when close e/1Ough to "'toUch.�

ur,}

f -'7'!!'...-----" H-+ . rn -U

.

•

•••

:,0

.............

�p!

When roo dole, die.. han:! can::s repel.

,

,.... f1.l

The dnlrcroo'UtUlJ'Oll

Mtd profua buund In. well rnullIIJI

" -+�-r

from IIIftt w.:tt� poIftIuaJ cntlJ)"

Proton�

WId llC'utrons

art" I-pln

i f(mlion�, '0 they cannOI occupy

the same

�.)\I("m. The' lo�(�1 energ) po\�ible for Iwo nucleons in a "dl �Id haH' bolh in Ihe ground �piliioll �Iate with their ..pins aligned (the !OWe In lhc

�

mentation of slronger allm:l.on I, 001 ,ut:h a stolle is forbidden 10 the p-p and thC' eu-Iu�i(ln prinl,:iple. h Ihi' indeed why Ihe p-n forms:, but the P-p

I)-I) by

and n n do n,,(') E'p'-'nmental endence \lenfie.. thaI the deuteron i.� ani)' ... no exciled bound Slates, jusl the one �ly bc.-.und: 11\ potenllal well ha �mund slate, .\IO(f'O\er. the neulmn olnd proron ,pins art' aligned (its total SPin -

I� II. If til< p-n cannot �md any other way-with '>p,ns opposite or widJ one nudeon in a higher 'patial \tJte-lhen the other two should IIor bind at alL The enC'Iog) in the deuleron i� depir.:led in Figure 1 1 .5. (The repulsi\le nucleon

core, are i�nored.) Th'o nudeom occupy the lone bound slate in a well that ari\e\ from their mutual altrJclion.

fIsIu,. 11.6 RwumS tntfll} ret

Arbitrary Nucleon Number

nuclcvn JuC' II' Ihe ,'rpng mh.'mud("(ln

We no� com.ider nuclei compri�ing any number of nucleons. addressing in rum the efted� of the Ihree main faCIO,...,: the strong force, Coulomb repUlsion,

"lIr:atlion poly fill: �mJ"C'sl nudt.

t1a\e' lew Ix>nu. (Itt nudl'on, In 1.l11!C"

ilnd the e:(clu,ion prinCIple.

nucld. many lIurk.-ons J� !umlYnutd

Strong Intemucleon AUraction A two-nue/eon nucleus has only one bond. but a three·nucleon nudeu� ha.� three. Thus, the fonner has half a bond per

nlJ/1ron and the lalfer one bond per nucleon, A four·nucleon nucleus would ha\e m bond" or one·and.a-half bond� per nuc/eon. so a representative nudeon would be harder to extract than in a three·nucleon nucleus. This mcrea\ing [tend n i bond� per nucleon does not conlinue indefinitely. however. Nudeon� attract, bul they also have a hard core thai causes the nue/eus to

l .,

grow like a collection of hard spheres stuck logether. The intemucleon aUrae· lion i\ \0 short range that a given nucleon attraclS only Ihose immediately surrounding it. �o each surrounded nucleon will have Ihe same maximum number of bonds. (We ,ay that the force salurates.) Of course, nucleons al the surface are not completely surrounded. but as the nuclear sphere grows, the number .11 the surface is a diminishing fraction of the lotal. The ratio is essen

F/vu1"9 11.7 [pu/mnb rrpul\ion faj'e, prI" "n eneq!ll,·'.

lially area/volume =

41Tr 2/� 1I"r3 oc

IJr. Thus, the average number of bonds

per nuc/eon should inilially increase fairly rapidly as the nucleus grows, then gradually approach a con�tanl, as more nucleons become surrounded. Figure

Cllulomh

J ,

I

1.6 show� the trend.

Rather than bonds per nucleon. however. we now speak

of binding energy per nucleon, Binding energy is the energy rnal would be required lO pull all nucleons apart. and binding energy per nucleon is the frac·

E,

E,

-I---,;�·l

-t--�:

Scurrun

T

;--

lion required for a representative nucleon.

I:

E,

Coulomb Repulsion All pairs of protons in the nucleus repel. In effect, this positi\e potential energy shift� all proton energies closer 10 rhe top o f the finite well in 'II. hich all nucleons are bound. as depicted ID Figure I I .7 for the case of helium-4. In such small nuclei. all nucleons are

"

touchi ng:· \0 there is slill a

,(rong net attraction between all pail"!. But in large nuclei, paiJ"\ of protons can be too far apart to attract via the strong force, and these pairs add to the nucleu, an uncompensated nel repulsion. Thus, while under control in small

Figur. 11.8 Binding energy per nucleon due to both the �trong inlemudeon attraction and Coulomb repulsion. t.ur�e nudei hone proltln\ thai do nol

Small nude. ha\c:

allriltl.'� \

Ie..... NInth pc:r nudeun

remove lrom lin

2

A

Coulomb repulsion should be an increasingly destabilizing factor in The energy needed to extract an "average" nucleon becomes pro ones. larger Combining this with our previous arguments. we should smaller. gressively expect the binding energy per nucleon to vary as tn Figure 1 t .8. Accordingly. �omewhere between the extremes, there should be a mass number more !ltable than all others. for which the nucleons are the most lightly bound possible. nuclei,

The Exclusion Principle To this poim. it would seem that the beM way to produce a stable nucleus is to build it of essentially all neutrons, for only pro tons repel. This argument overlooks the fact that protons and neutrons tn the same nucleus must obey the exclusion principle, each independently. Putting Coulomb repulsion temporarily aside. the most important consequence of the e'(c\usion principle is that for a given A, the state of lowest energy would have equal numbers of protons and neutrons. Why? Suppose that Z does equal N. as illustrated on the left in Figure 1 1 .9. The lowest energy state consistent with the exclusion principle would have aU the Figur. 1 1 .9 I gnoring Coulomb repulsion. the exclusion princi ple argues that for a given number of nucleons. energy should have N = Z. Slanmg wIth equal

•

numben. of nCUlron� and

f: E,

' higher energy level_ _ _ _

\

-

•

E,

E,

_

./

E)

�7

Neutron

d conveni ng one to

an

the other force� a nucleon to

pruto"",

E,

__

the lowest

Ez

E,

Proton

.

--=- : - - - -

- .�--=..-..:. :=.=..;..- =� --

..... ft. to In .... nucki. .heo Coulomb�ltJOCI becomes l.I,,"ti·

e.I. dw JowaI tnftJY IAouJd

�N>Z

Anllmn cnerJ} Ic\-e!, filled. c&:h �ilh an 0ppo"He-�p,"s pair. up 10 some maxi_ mum cnergy_ Comlde-nng only !he inlemudeon altraction, protons �ould occup) 110 lde-nllf.:aJ ""cll and would fill le\eI� 10 the same maximum energy. NtM. keeping A that a dtcreas(' b)' J in Ihe prolan number would mean an 1f)(:Tea.\e b} 1 In the neutron number. Becau�e all lower energy levels are filled,

ncubun wQUld ha\e 10 occupy an energy levcl higher than the previ00\1) occupied prolOn leH!I. PJld me 10lal energy would increase. The argumenl applic\ )1.1\1 � well If we chan!!e a neutron 10 a proton. Thus. Ihe slate Z :::: N is

lhc

.. -+.....;.-j-

.. -+.....;.-j-

.. -+.....;.-j-

.. -+.....,-1-

�

of 10....('\1 energy (If elmer top level initially had only One neutron or prOlan, we might ha\'C' to changc a I.C':cond particle before the energy increased.) UIU_\ now faclor in Coulomb repulsion. In �mall nuclei, where essentially

all nucleons lauch, all prolon repulsions are overwhelmed by the strong attrac_ tion. so Z :!' N should be me more �table Slate. In lurge nuclei, uncompensated Coulomb repul�ion, may shif! me prolon energies upward enough to align wim

different neutron 1C'\'els. as depicted in Figure 1 1 .10. The lowest energy slate with me lap.. oflhe proton and neutron levels roughly equal-should then ha\'e

more neutrons than protons. A balance is struck between having eXira neutrons \toe as nonrepuhl\e "glue" and forcing them into higher-energy states by the

e).du�ion principle. O\craJl. our modcl �uggests mat a represenlative nucleon should be most ti!!htly bound-in a 10we�1 energy stale-in a nucleus of intermediate mass number, but il is nOI a function of A alone. The bindlOg energy per nucleon

should ha\-e a rtla/h'e maximum at Z a N for small A and al N > Z for large A.

Stability: The Experimenta l Truth The \alidilY of an)' theory rests on ils agreement with experimental evidence_ Obtaining evidence to support our model of binding might seem 10 inVOlve pulling a nucleus apart while measuring forces. bUI it is much simpler than that. All we need to know are nuclear masses. If we were to pull a stationary nucleus apart into separate stationary nucleons, we would have to expend our own energy. which would increase the system's inlernal energy and therefore It� ma�s. The mass of the parts must be greater than that of the whole bound nucleus. Figure 1 1 , 1 / bears this oul for our simple!!t nucleus.

deuteron mass = 2.013553 u proton mass

,.... t t.11 A dtUlC'ron ""C'IJlh) IC'" lhIn the sum of liS parh

+ neutron mass =

mas� of pans - mass of whole

=

1.007276 u + 1 .008665 u = 2.015941 u 2.015941 u - 2.013553 u = 0.002388 u

We o..ee lhal we must do work to pull the deuteron apart. For an arbitrary nucleus. lhe energy required to pull all the nucleons apart-Ihe total binding 2 OIJ�.H u

energy BE-is lhe difference between the final and Ihe initial energ ies which ,

Proton

Nrutroa

1.007276 1.1

1.008665 1.1

o

mc IIOnra

•

dIoKtiw: IMIIS'iah move about or how they slick to one another. radioactive IIOIDI lie

iDIrodIlCed, either as a loose mixture or by being bonded 10 specific

IIIDIcaIJ.r 1fOUPI. 1be subsequenl behavior of the materia] is then easily fol. lowed by the lelhale decay of the radioactive tracer. In one of many biologi.

cal

research

applications• • slerOid molecule is lagged (covalently bondtd) WIth tritium. • /J entitlei'. and then mixed with a protein. The .H�rojd·s abilil), to bind to the protein is clearly indicated by how much trilium ends up stuck to the proIein. Let us now look II another characteristic of radioactive decay that I allO much e..ploited in ICteDlific investigation.

1 1 .6 The Radioactive Decay Law

All forms of radiOKtive decay fundamentally change the parent nucleus, and the nuckus may decay in a particular way only once. A good analogy is a light. bulb. which can bum out only once, after which 11

IS fundamentally different

from a wading lighlbulb. Furthennore. decay!>. are governed by probabililles. It is irnpo5jible 10 know exactly when specific nudel In a sample will decay. One

nucleus may decay righl away and another after

very long time. In a large sample. however. there should be a predictable a\'eroge time. characteristic of n

the panicular decay: it may be long for the a decay of isotope X and shon for the /3. decay of isotope Y. The lightbulb analogy is again helpful. One neVer knows when

8

IighlbuJb will bum out. but in a huge office building with thou

sand", of IightbuJb X. thert would be a predictable number burning out per day. A rtliable average lifetime would be apparent. In a similar building with thou sands of lightbulb Y-a competitor's lower-quaJity unit-a predictable average lifetime would aJso be apparent. though it might be much shorter. No mailer whar value the lifetime might be. if we increase the size of the sample (building). the number decaying (burning OUI) per unit lime should increase proportionally. In OIher word�. the change per unit time i n the number

of nuclei present should be directly proponional to the number presen!.

dN ex N dl

-

We make this an equation by using the symbol A for the proponionality con stant.

(Note: In this seclion. N stands for Ihe number of nuclei. rather than the

number of neurrons in a given nucleus.) Becau�e the number of radioactive nuclei that are present is constantly decreasing �uires a minus sign,

dN dl

=

-AN

(dNldl

ually taken to be about IJ X IO-u. Carbon· 1 4 dating WO(�!i only for (formerly) Ii\ ing organi ..ms. \\'h.ile a planl or animal !i\'e\. It e .....:han1!c carbon "'Ith the eO\Lronment through Its food

,uppl) and the le,_' lhan J MeV. Thu.., In a reacllon In whrc:h about 2.w uramum nue/com �nd up near A = 1 20. binding energy All tilt rea:tion\ in ( 1 1-16) ha\'C free neutrons among (he products. As DOkd in Section 1 1 ..5 m C'OIUI«tion With "ponlaneous fission_ the reason IS thai

beIvy nuclei ba\'e • higher ntutron·to--prolon ratio than do hghler ones. The JRICIICe of IffIIItipk neulrons ImOng the prodUCI!O of Induced fission is faleful

for if we colIecI muy nuclei together, a

cbain reaction

....

�

IXlS\lble. Suppo

tbIt eKIl rmction liberlles " product neub'On.... If a neUlTOn I'> reqwred 10 initl. _ . racIioa IDd each reaction produces n neutron!). then each reaction may

� .. �"h.hect. Il.' t,here wlmld be too much area from which to lose neutron,. A ...phere !\iI" Ihe

flllnt01Urn �urf(lcc area to volume ratio, However. e\'en for a "phere. there " tI 111LnU1\I1I1I �i/c. occause the rutio incrca�e� :IS the silc of the sphere dccrca"e....

1l11puntlc" come into play becaU',c they may lIb�Ofb free neulron... without Ii..,

...umlng, 1i.) compen!>.ate, the "...!>.embly would have to be larger.

\\'hether an a....cmbly i .. critical " determined by the multiplication con·

\tanl 1. Ihe a\'crage number

of reaction, I;ct ofT by a given I\:l\l..:tlon. \I three

""utRln' were liberated In each reaction and each induced another reaction.

the number of r!.!uctiono; in each generation would be 3 time\ Ihe number in th!.! prt,;ou, generation. and k would be its maximum possible value 01 3. How

�\I!r. the reactiom, in ( 1 1 - 16) show that some fissions liberate fewer neutrons, Lo,�' tit the surfnce and absorptions by impurities would further reuuce the

multiplication con:-.tanL Thu,. to give the true energy rclea\cd for tI given gen

�r.ltion, we repl.lce " in equation ( 1 1 - 1 7 ) by k,

'\ �ustained chain reaction require... a multiplication constant of at Icast I . If l > I, the energy relea..ed mcrea\cs exponentially with time. If k

I , succes

\i\e generations would all release the same amount of energy. giving n steady

chain reaction. If k

or poltntlaJ \(lUrcC'o;; of power. fj,,etting It equal to q� \/ showl> that a deuterium nucleu� need onI)· be accelerated through about I MV to tu..e 1,I.Ith another. ev·en unaided by tunneling. Although not an obviously practlt:al route to fusion power, thl� alternative method of �tid..ing .. recently been demonstrated in a deuterons together ha surprisingly convenient way that may find numerous

. 11.38 The eicctric field around a pyroelectric cT)" [1l1 and a ' Fig .... �hdrp lip IIlmll!� deuterium lind dri'(" il lo....ard a deuler3led larget, Vtht'rt nudei,lt 6,,1\," (Xl'Uf'). 101

. in the early siages cause changes in ratios of certain trace elements. Thus, by comparing

Medical Imaging: Turning on Gammas with a Neutron Seam Medical imaging ha, seen the estabh,hment ofa

gamma intensities.

numbc:r of new technique, In receO! decades. among lhem

ccrn by a nonimasive scan long before present �canning

NSECT might

identify regions of con

thl." IT \Can, whn:h employ, X·rays. and MRI. which uses

techniques would notice anything unU dnd . few years are found in nature

Ifl

small abundunfo."eS Ih31

do not chan¥e at J.1l mer man)', many )can, How is Ihis po�\lble" (Him_ Natural uranium and Ihonu m have l't'/)' long half-Ji\e\.)

9, Why dlle' fi,,'lon of he:a,y nudci tcnd 10 produl:e li'ce neutron,')

10. I n bolh

( I I-IS)

and

fusion, which ekploit the vel')' tight bindin, of inrenncdiale

O-D rcac.:lj�'n" ln clju;Ulun ( 1 1·/8). 1....0

deuleron, fu\C 10 produce tw.I} particles, .. nuckus of

A = 3 and It free nu..:k"tlfl

M.lU ds nf!utrons. the tol31 binding

37,

n

t

a,

""

are

section 1 1 .4

All target nuclei used in MRI have an odd number of protons or neutrons or both. What does this suggest about nuclear spms? (Not�: 80th the proton and the neutron have gyromagnetic ratios.) 35. In electron spin resonance (Section 8. t). incoming elec· tromagnetic radiation of the proper (resonant) frequency causes the electron's magnetic moment to go from its lower-energy. or "relaxed," orientation, aligned wilh the ex.temal field. 10 Its hlgher-energy antialigned state. MRJ IS analogous. A quantity commonly discussed in MRJ IS lhe ratio of lhe frequency of lhe incoming radia· tion to the external magnetic field. Calculate this ratio for hydrogen. You Will need to adapt equation (8·7) to !.he proton. Note that the proton gyromagnetic rauo, gp' 155.6. 36. MRI relies on only a tmy majority of the nuclear mag· netic moments aligning with the eXlernal field. Can· sider lhe common target nucleus hydrogen. The difference between thl! aligned and antiahgned states of a dipole a magneuc field IS 2JA..B. Equation (8·7) can be used to find JA.. for the proton, provided that the correct mass and gyrOmagnetic rallo (gl? 5.6) mserted USing the Boltzmann dlstnbuuon, show that for a 1.0 T field and a re�onable temperature, the

)4.

10

-

are

Q

arc

=

... " �_� . .. 0.

CIIIIJY would Ktually dlCmur. bplatn wtaM .t wrong e.

�

with the naIVC' atpmcnt. il lht reooll lpced of tht dluJhtef nudeu� when ,.,Ho 0 dcaY''t CTrcal aU mobon Il!o nontd'II\1i�bc.}

_ " .6 51. Given IOJbally 100 a of plutoOJum.239. how much time mUlt pus for the amou.ntto drop 10 1 g? 51. 1l1e IMial decay rate of a wnple of a eertal.n radioac· tlve ilOlopc II 2.00 X 10" � I After haJf an hour. !he decay rate 11 6.42 x 1010,'1 Detennine the halr·life

of the itOlope.

51. Tbc half·hfe TII2 1$ not the a�ngc lifetime Tof 3

radloacll� nudeu!> We find the a"'erage lifetime by multlpJyma t by the probability per umt IInlC P(t} that the nucleu\ will " Iwe" thai long.lhen inlegrating over all IItnC. (a) Show that P(t) \hould � gi\'en by At-AI (Hint. What mu�t be the lotal probability?) (b) Show that T - Tlr../ln 2. 53. Eighty centurie� atter its dealh, what will be the: decay rate: of I g of carbon from the thigh bone of an animal" S4. A fOiU,il specimen ha.\ a carbon· 14 decay rate of 3.0 s' 1 . (a) How many carbon· 14 nuciel are pre�nt? (b) lrthi� number 1\ rn the num�r that must have been present when the animal died. how old i� the fouil') 55. A fossil \pecimen has a 1 4C decay rate of 5.0 s I Ca) How many carbon-14 nuclei are prt!otnt? (b) If the specimen I� 20.000 yean; old. how many carbon·14 nuclei were present when the animal died? (c) How much kinetit: energy (in MeV) is released in each /3 decay, and what is the tOlal amount released in all fJ decays 5in� the animal died') 56. A bone of an animal contains tli mol of carbon when it diei. (a) How many carbon-14 atoms would be left after 200,000 yr') (b) h carbon-14 dating useful to predici the age of such an old bone? Eltplain. 57. Pow.sium-40 has a half-life of 1,26 X 109 yr. decaying 10 calcium-40 and argon-40 in a ratio of 8.54 to I If a rock sample conllllnod no argon when it formed a solid but now contains one argon-40 atom for every pot8ssium-40 atom. how old is the rock'! 58. (a) Oetennine the total amount of energy released in the complete decay of I mg of lritium. (b) According to the law of radiOllctive decay. how much time would this release of energy span? (c) 10 a practical sense. how much time will it span? 59. Gi"'en initially 40 mg of radium-226 (one of the decay product!. of uranium-238), detennine (a) the amounl that will be left after 500 yr, (b) the number of a parti cles the radium will have emined during this time. and ecl me amount of kinetic energy thaI will have been

released (d) Find the decay nne of the radium at the end of the 500 yr. 60. Ten milligrams of pure polonium-210 is placed in 500 of water. If no heat is allowed to escape 10 the Sur_ g roundings, how much will the Icmperature rise in I hr? S.ction 1',7 6 1. Determine Q for the reaction

62. Calculate the net amount of energy released in the: deuterium-tritium reaction I 2,0 + J, T - "2He + on 63. In an assembly of fissionable material. the larger the SUr.

fnce area pcr fissioning nucleus (i.e., per unit volume)' more hkely is the escape of valuable neutrons. (a) What is the surface-to-volume ratio of a sphere of radius ro? (b) What is the surface-to-volume ratio of a cube of the same volume? (e) What is the surface_to mlume ratio of a sphere of twice the volume? 64. If all the nuclei in a pure sample of uranium-235 were to fission, yielding about 200 MeV eaCh, what is the kinetic energy yield in joules per kilogram of fuel? 65. (a) To release 100 MW of power. approximately how many uranium fiSsions must occur every second? (b) How many kilograms of U·235 would have to fis. sion in I yr? 66, 1\yo deuterons can fuse 10 form different products. Although nol the mosl probable oUicome. one possi. bility is hclium-4 plus a gamma particle (see Concep. tual Question 10). Calculme the net energy released in this process. Consider equal numbers of deuterium and tritium 67. nuclei fusing 10 fonn helium-4 nuciei, as given in equa. tion (1 1-18). (a) What is {he yield in joules of kinetic energy liberated per kilogram of fuel? (b) How does this compare with a typical yield of 106 Jlkg for chemical fuels? 68. For the carbon cycle 10 become eSlablished, helium-4 nuclei must fuse 10 fonn beryllium·a. Calculate Q for this reaction. 69. Ignoring annihilalion energies of the positrons, how much total kinetic energy is released in the silt·step carbon cycle? (There is a quick way to answer this. and a much slower way.) 70. (a) How much energy can be extracted by deuterium fusion from a gallon of sea water? Assume that an average D-O fusion yield is about 2 MeV per atom. the

I!un:"

(b) A modem supertanker can hold 9 X 107 gallons. How many "water tankers" would be needed to supply the energy need:. of greater Los Angeles, consuming electricity at a rate of about 20 GW, for 1 yr? Assume that only 20% of the available energy ac(ually becomes electrical energy. 71· A fusion reaClion used to produce neutron beams ('lee Progress and Applications) is 3 2 I iD + I D - 2 He + on

,

M\uming that Ihe kinetic energy before the fusion is negligible compared with the energy released, calculate the neutron kinetic energy after the fusion.

Com prehensive Exercises 72. The binding energy per nucleon in helium�3 is 2.57

MeV/nucleon. Assuming a nucleon separation of2.5 fm, detennine (a) the gravitational potential energy per nucleOli and (b) the electrostatic energy per proton between the protons. (c) What is the approximate value oflhe intemucleon potential energy per nucleon? (d) Do these results agree qualitatively with Table 1 1.2? 73. For the lightest of nuclei, binding energy per nucleon is not a very reliable gauge of stability. There is no nucleon binding at all for a single prOlon or neutron,

111

)·el one' is "table (so far as We' knOW) and the other is not. (a) Helium-_' and hydrogen-3 (tritium) differ only in the switch of a nucleon. Which has the higher bind ing energy ptr nucleon? (b) Helium�3 is stable, while tritium, in fael, da:ays into helium�3. Does this 5Ome how violate law,,? 74_ You occupy a (me-dimensional world in which beads of mas" 1110 when isolaled-attract each other if and only if in contact. Were the �ads to interact solely by thi� uttr"olclion, it would take energy H to break the con tac!. Con�equently, we could extract this much energy by stick.ing two together. However, they also share a repul�ivc forcc, no mattcr what their separation, for which the potcntial energy is U(r) = O.85Halr, where a is a bc"d'«;ay Mode, and Con'lervation Rules in the Standard Model 1 2 6 Panty. aw¥e ConjugatIOn. and Time R�eBa1 1 2 7 Unified ThC'une.. and Coo.mology

r"f"hc \oColt�h 10 lind the fundamtntal building bloch of nalUrt tw 1 been gwnS nn fur lung tunt_ Ancient ptu� po!olUlaled •

..- die: "'�lfld compn£

'"

h

parucle of energy ,lE could appear and disappear without "erifiably up�et so long as it existed for no longer than ,lr :::::: h/ �E. ,n \ g energy conservatton the temporary problem with momentum conservation.) By covers iS also a force whose mediating particle has mass must be a shortargument, ge force. If a massive particle must be created, there IS a lower limit on the

; �� :

ount by which the total energy could deviate: the mass/internal energy of This, in turn, implies that the maximum time the particle could lnal particle. Even at the speed of light, the particle could travel at ive is 6.1 �

hlme2 .

\Ilrv

most

ax � cat :::::

h

-

me

h 1 range ::::: - em

(12-1)

If lhe mediating particle has mass, the range of the force is lirruted. Conversely, a force whose mediating particle is massless could have infi nite range. With no mass, there would be no lower limit on the energy of the mediating particle. Its strictly kinetic energy could be arbitrarily small, so the

time and range could be arbitrarily large. Consider the electromagnetic force. Although it does fall off with distance, it nevertheless reaches infinitely far, so

it would have to be conveyed by massless particles. Indeed, electromagnetic

forces are conveyed by the exchange of massless photons. An electrostatic

repulsion between electrons. for instance, is conveyed when one electron emits a virtual photon that is absorbed by the second electron.

We will return to equation ( 1 2-1) when we discuss specific fundamental

forces in Section 12.3. Here. we merely note that it has been used in several

instances to predict fairly accurately the mediating particle mass from knowl

edge of the force's range

before the mediating particle was first detected. This

ISone of the most convincing arguments for treating forces as the exchange of particles.

1 2.2 Antiparticles

Before delving further into fundamental interactions between particles, we must understand antiparticles. For each kind of particle, there is an antiparticle

that shares

essentially all the properties of the particle except that it is of

III

_ 0." tI

" "n _ t .., � .. ... i_ _

ClfIPlCiIe dtIqe. Section 14 introduced the positron-me antiparticle of � eJoc:croa.. 1be poUlrOII bas the same mass and spin as an eleclTOn btu is ofpas_ ime cblrJe. An el«Cron and posib'Oll can be crealed in me process of pair production. When they meet. pair annihilation may follow. in which they diuppear and Iheit mass enerv is convened 10 pholon energy. Many OOltr

anap.tides have been found. such as the anliprolon (negalh'ely charged) and

antinturron funcharpd). II mighl seem mal an uncharged neutron has no prop erty to disliJlguisb il from irs anlipattide. However. the antineutron's distinct

idenbfy is coofumed by the fael that it does annihilale with the neutron

�

whereas rwo neuuons do nor annihilate. Acrually. as we will see in Sectio

12.3. the neutron is not fundamental-nor is the proton-and its internal struc

hft disbnguishes particle from antiparticle. (Some uncharged particles. such as the .0. lack such distin,uishing sU'Ucture and are their own antiparticles.)

The convenliona.! symbol for an antiparticle is (he same as for the particle but with an overbar. The antiproton is thus p and the antineutron ii. An alternn_ live convenbon is often used for charged particles. The positron is usually rep. rtsenled ef- rather than e -, and the antiproton is somelimes wrillen p-. Similarly. the JJ. f- and JJ.

-

are anlipartic/es

of one anOlher.

The existence of antiparticles is an experimental faci. bUI there is a theo

retical basis: relativistic quantum mechanics. Let us pursue jusl enough of this advanced theory to gain some idea of how combining relativity with quantum mechanics might suggest the existence of antiparticles. The SchrOdinger equatIOn for a free particle 1ft one dimension is

A2 a2 2m ax>

---- "'(X, I)

=

Expressed in lenns of operators, where becomes

- jJ' "'(X, I) 2m I

ih

a aI

-

"'(x I)

(12·2)

'

p = -ih(ajax)

and

E = ih(a/at). it

"

= E "'(x, I)

As we know, this equalion is based on energy. In the absence of external potential energies. the kinetic energy of a particle. enerxY E. However. because for kinetic energy.

p2/2m

p212m,

equals its total

p212m is not the relativistically correct expression

= E cannot serve as the basis of a relativistic

replacement for the Schrooinger equation. A logical basis is the relativistically correct expression

(2-28):

(12·3) To obtain a relativistic matter wave equation, we might try inserting the appro priate operators and then have each tenn operate on a wave function:

c'jJ' ''' (X, I)

+

m'c' ''' (x, I)

=

£' '''(X,I)

11.2

( 1 2-4)

Klein-Gordon equation, this has been shown to yield correct �nown a� the the behavior of spinless massive particles at all speeds. Par about tiolls redic the related Dirac equation. In nonrelativistic quantum obey spin de� with spatial states that are solutions of the SchrOdinger meII)

;\ 2 )( 10

Me-V

I�t Me-V 'II MeV

� I Me\' ' I MeV 50 MeV

�\II1I.lJ",nR qllJrli �)t.'Ild the �tlange

J,I/I (c d

,' 10l)

II

11. 0

H7 lc\'

("")

"-IN.l

II

n.o

50 lcV

\

Ib antiparticle KO. A d,...cU'.slOn of this miKing may be found m higher-level . le�l\ on particle phy...ic... Yet anolher mlrtTl... ic property given m Table 12.2 i.. strangeness. Sen... , bI). II is the \lrange quark that endow... a particle with slrangenes"" By arbl

tr.tI)' sign chOice. po......e ..-,Ion of one strange quark gives a strangenes\ of

.

I,

of twO, a "trangenes.. - 2; o f an antistrange. a st.rangene! energie� are comparable to the nucleon·s ..... 1 GeV. Be,ide� the ptl\ilron, co\mic ray\ ...cned a� the source for the discovery 0. 14 in the 193(h and 1940, of the muon (m :-:: 0. 1 1 GeV). the pion (m.". J4 GcV), ilnd Ihe kilon (m... = 0.49 GeV). After the kaon. or K-meson. a great =

man) (llher hadron� were idenlified. But co...mic rays "ere soon no longer the Ix,t energ) source. The panicle aCCeler310r had arrived. Although not realiled at the t,me. the discovery of the kaon had brought

10

three the number of quarks �ubJecl 10 study. The fir�t hadrons discovered

"ere merel), differem combJOu(iom of the same three quark�; the up. (he down, and the �trange. Indeed, after a decade or �o of finding recurring pat. terns, sirong suspicions aro�e that the known hadrons were composed of a

�maller number of more fundamental panicJe�. The three-quark theory that came to be accepted is due to Murray Gell-Mann. who was awarded the 1969 Nobel Pri/e for his work of 1964 (and by whom the name quark was chosen). The theory was widely embr3ced. but it was not until (he deep inelastic scat tenng e'{penmenls (Section 1 2.3) of Friedman, Kendall. and Taylor in 1 967 that the fir..t direct e'tperimenlai evidence of quarks was obtained. Still. other more massi\e Quarks lay undilud Arl'hl,.·,llh1n,l ,,\,,1 Q1Q(" t thaI C(lll'CC\illilln III lertolt I1\lmlx-r mOl) 11\11 l'oe a UI\I\\'"",I rule \#Ihd�". "Inhl ..t "II pml'e,"c'" ('on"'I:(\I,.' the Ihr\.'(" Il'rh1n "'tm�r' .adcr1.111 t.. (\t.lt ......"'. inl!I"! b.lI)·\,n number '\ I + 1, Ilnd mt" linal I' I sn'mnlClr) 01 the \\e.o.. (on:e

.appeal 10 d'k' Iwficit, ,., Ilk' fk"Ulnno Hdll,:II) rrkr\ 10 the onenratlon ot I . a tklinlle !kilo:. piIl'tJC'k., "flO tdarwC' 10 ,1\ m"�nlum [lcdmn.. do nOI hne 'iOO ,11 lCln l dlre..: +: Ilkm .. OltHe m/.. \\.- uh ..pm up. II) \urpo� an i."lmrun "

1\ 10

pto .lnd tlJI,lmcnlllm ..hgo,.'d In iI fr.lme mm 109 III 1000 mI.. In tht ';'z dir«1lon rd,dne hI the lil'l. the ek,:!ron I" 01(1\ mg al 'iOO m/.. III Ihe flt'.I((JJH r dlr(\:lItlfl .ru " 'rIO up, �l thoIl I" ..pm and 1I111menium are anl.allgned. The �I

Ipm and momtnlUm v( tht.' \.lIl1(' t"Ie\:lron �'an � cllher aligned or anriallgned. dependmg lIn tht- 'rJ� 01 n:il'I\'OCC A mtt.. ,le .... P'lnu.:le. on (he Other hand. hllI\L', at c 10 JIl) IrJffit'. 8el"'IU'c II �'anrll)1 Ix' mcn..J..cn. If II .. 'Pin and rnllflienlUll1 Jrt' aJlgnt.'d 10 one Inll1lt.'. Ihe� mu", be ah,gned 10 all (rames. In the \tant!JlJ Model. neulnno' .1n.' mil"Ie". '0 tht.'y \\.-ould h.ne I.Iclimte hellellle,

I"h,' nt'urnno Iwuld 011"01)' h,ne I" 'pm opp(1\Ile II' momentum-referred 10 Idf.h;lmkJ "hu.:h e\pIJIO.. \\h) tht.' dec'l) in figurt' 1 2 . J .f(b) would not

.h

C\4:,'ur Tht' rl('utnnu \\ould halt" an allllpilnll'le the IInllOelllrino--",hlCh IHlulrJ tx- n!!hl-h.mrJed. I\-Ith 11\ 'pm pardllel to 11\ momentulll, In Frgull' 12, l'i, IIl l'thKh panldt', hll\C Oe-cn c;hangcd 10 anllp;tnicle\. the panicle pro

rJUl't'd

'Min tht" dCl"twlI l\tlulrJ hale 10 Pc the antlOeutrino. Thus. Figurt

1 2 l'ita) \\llulJ nOI bt- oh.."r.cd, am.! Figure 12, 15(h) \l.ould be. Ont" pwNt'm '" IIh the c\planallon

Jfld"

Ihal ncutnno.. apparent I) do ha\�

r\

The ('Ildence -':\1,"('.. Imm n(lulrino o')Cillnlions. Ihe ah.llly of one 1..lOd

tll neutrino lu (·hang.... 1II1{) anlilher Th" phenomenon was li"l documented at

tnc: Su�r KanullJ..ande facility. loe.lIed

Japan and opcrntcd Joinlly

III

l'nlloo SIJle' A, noled earher. the nculnllo mlernCI.. only

"'ii

I't

Ilh lhe

the \l.eaJ.. force.

v.hll'h male, II t''(lrelllel) dltilcult 10 catch ('cc E�ercl ..e -'9). Located I lm

unJt"!J:round III ,hleld II from CO,IOIC ray'. Ihe Supcr KamioJ..Jnde I� es'itn11e or ma\� aloot'. CUI"\t' A, Uoooid "" paRd (lll\ l� ",r, It.. brgrnnrng Uoould be Imer than a ron'talll" peN e\pan\IClO III loday\ nile "oold 5ugge..l, A nl\mOlogKal nm"wlI. I:ul"\e B. �'an an:eierale e\pansion B SmJII. oonJ'cro

1

(o_moIO�H;al

t'on,!anl

•

Con_!,anl

,

\Cloclly clp,an\lOn

Cnh(al dell�. IQ the lab. Assume that

the lotal mas� of the (stat,...,}, sion is M. Show th3l lhe th!'.:

24. Exercise 23 discusses the l:l1 ticles of mass III in a coltidir;",

duce a final stalionary ma�

instead a stationary-target Yj'

energy is needed to produc�

.t'

that the threshold energy i� (,\

I

'drticles after the colli-

:('

'nergy is (M - 2m)2. ;d energy for two par;n accelerator to pro-

:he accelerator is

I;)ore initi31 kinetic une final m ass. Show

/. '11 -

2m)c2. The

calculation is greatly streamll)' ! by using the momen tum-energy invariant, discus: ,I i n Section

2.10 and

"-_

117

. The main point I� tha\ ,L Exerci..e :!. 1 1 2 me q

(£'u&b _ II1II .... .....

only about 10-41 rrl-.

\here in a column IhnIuah _.. _ on will octuaIIy "bin

_ .... 11

n·

.' ...... .. .. , __

If we multipy both sides orequltion (1 2-7) by R2(t) whole equatIon, men caDCC!1 dR(I)/dt, the result is

_ ,&7 •

.., ... JI ... .... ... aiIicaI(O.. - I). � tr 0), "' ''' 01, a_MP:aI a.a.I (II• • 0). - _ (12.'1. be ....... .,. •

'!""'" (1 - Ir"·

, _ _ lI(� oI (, II . ... if_ (Il.n _ . ..........-etecf waivase. . _ _ _ _ 01 _ " _ ....... ...... JIN"iIII "'" ... .., 110 wdb die diI·

Equation (5·35) comes from the Schlixhnger equallon' which says thaI

.... ... . ..... .. ..... 0I _ 1IId Ndi.

d',,(x)

.. \111)' ...... dw .. oIdw uaiwne. MaDer dmsify is

......, - po.,.._ .. ... ....... -

,,,�. ..... II.. .. die ..... deDaify now. bdidiOll

.....1)'. bowewr. would be propwtioc&I lO IIH'. (Nol oaJy doll the volume incrc&le. bul al50 all wavelengtl!s .. IINIdIed in proportion 10 R. lowerinJ the enetBy "1)' by Ihe un (aelor'.) Thil denJ.iry drops !tiler • die ualvene pows, bul it al50 flOWS III(ft qukkly in ... bKtwIrd bme direction. In adler wonb. IonJ aao. Ibt aiwnc would have been r.hatioa domirwc:d. $bow tbII i( tile function IPed (or matter deMiIY in .,...uon ( 1 2·7) is replaced by one appropri* 10 �. but retainin, lhe &IIUlDption 1M! K' and 0A are boctJ O. Chen the scale r.:tor R would grow as ,Ill 11 SI»ow !.bat cbe ptUCnCe oIa positive cosmological CODICaDt in Friedmann equation (12·7) m.w. as R bcoma very 1Itat. lead 10 an exponential expansion 0I1he urUYCnC. 51 You _ • pomisina: theomicaJ physiciSi who does nOl: bdievc IhaI pviry is • ctistinct fundamc:nral force oot is iIIIIIIId ... 10 Che ocher fon:a by an all-encompusing rellciviltic. quanfUm·mcchan:ical rbeofy. In particular. you do noc believe Nt the universal gravitational CODIIIftC G i. really one of naIU/'e's elire set of funda IDItlItaI constants. You believe lba' G can be thrived from more-buic constanl5: the fundamental conSlanl of qaaatwn mechanic•• P1anck's COIlStalll It: the funda· IDIIllII speed limib"l the pro".,ation of any force. the .".ed of IiJhl c: and one othcr--.l fundamenral length I. imponanr CO me cae wlified force. Usina simply dimen tioaa1 ....ysit:. . 1iAd . formula for G. [ben an order·of·

...._ .. qloe "" '" - 1enJdI·

Computational Exercises M. HeN we solve die Friodmaa.a equatioa numerically _ loot II some iaIueaIiaa: IpeciaI cues. (Review _ '.10 aacI die Ouidoliaes . ... bqinaia8 of .... Chapter .!i COIDpIIatiooaI E1cttises.) The first step II 10 COIMI1 &be Pried"""n IDqUIlioa co • scco..ordcr od cti&rutiaJ OfIUIlioa lib daoIe in earlier cbaplen.

'

cate I bme derivative or lhc

-----;J;2- =

-

,,"(E - U(z)) 112

Adapting (5·35) 10 our problem thus gives

R{t + j,)

- 2R(t) - R(t - 61) +

(

!/I(x)

J\,

0"

AR I 2R2( 1) - O ( I1r2

This would allow u� 10 plol all values of R(/) from a starting poinl li the presenl age. whose lime we choose IS I We would also like 10 look backward in time, and the modification is straightforward.

R(t - .1t) •

2R(t) - R(I + .6.t) +

( �M

ZR · (,)

�

- n\R(')� '" .

LeI us usc: 0.001 for .6./, Together with our choices in Section 12.7 that both R and ils time rale of change are I at present. we thus have R(I) = I, R(I + Jot) "" I ,001, and R(I - .11) = 0.999. Now we have e\'erything needed to plol R(I) both forward and backward from / = I However, with an R(/) in the denominatot above. a 0 value could cause trouble. So incorporate inlo your program instructions that (I) will assign 0 to any "new" \'alue of R(I) that the equation might cause 10 be less than O. and (2) will check to see if a prtvi(}uJ value j� 0, and if so. simply assign 0 10 the new value. With thest instructions, your computer will never actually try to divide by O. If your universe ever shrinks to O. it will Just stay there. The values we now input and check are thost of OM and llA' Ploltlng from I :: I to t = - 2 and from I :: I to t 18 (except as noted). consider the follOWing values oflllM. ll.�I. and in each case di

-

[0

V

40,! I 40,l>

C+ii2)

£20 ! IEve20.nt 2!. 2O.l> ) C+.§> gl+2m@j> o g&J�) Event 3 F§t> G§> ill [

HO

Event 4

+60

I

Event 4

301 1+

Ap pe nd ix

c:

P l a n ck's B l a ckb od y Ra d i ati o n Law

is defined as an object from which electro ..\ bI llc kbOciy magnetic radiation he ther ly I mOll' on of ,Its les sole due to t ma ((Il1lJ\1i charges-it reflec •ion 3 . 1 , an b' ts no light . S ect 0 Jec! 0r temper I M l'lotCd in ature

'r

T has an interior cav . . . ity. the cl1 radiatIOn e Xl. t.mg the cav' ity through a 1romagn c h:c Small hole approximate� e . ody qUite weli . Expen'ment eSlabl'IShcd that the electromagnetic ene a plackb rgy " , a blackbody or cavity attalOs a ma)nm by d ,1te um U r! at a certain frequency and e . her 51'de 0 the maxim on Cit ' um. Classical the � wward zero , o�' howe",, ' ass the on um d pti bas on e tha s t electromagnellc Whl.Ch L radiation is strictly a ' " I ' failed miserably. t 'I ndIcated that the energy should be walC, a monotonically req 0 tion uen , d" func e)' lvergl g as the inCreasing frequency approaches infin � : ause the theory a?reed WI expenment at low Bec ', i frequeocies but failed at enC freq ies, the let u d lscrepaocy was tenned the ultraviolet Callas bI)Igh/Ultravio h e Ofcourse, there was no real catastrophe. Classic al wave theory is sim Ma x Planck solved the problem. equate. As we outline his work, we will apply some principles of statistical mechanics (the topic of Chapter 9) . unfamiliar, but we Will still dearly see tha ay be the most important point theory differs from classical wave w ne e theory. b o Our goal is to calculate the spectral energy density: the elec tromagnetic de a cavity in a narrow frequency range be energy insi tween/and/ + df.

r

.

rr

�

u'O� d

�n:o ::

dU(j)

=

energy in frequency rangeftof+ df

1\o7n ? • d t pi "'nlc' p. .. Ibe importanl factor. known as • • p • •'IIJ II I � 01 _ objec:u-often IIOms or

L

...::

.... .. II _ _ _ yotic oocillitioas of floquellC)' I--o _ .. . .... .111 '''''· _ .... 0IICrJJ E i.

1'(£)

oc

.-E/"T

utd T i, the ..mpera..... We obtain the I possib/� value £ times multiplying .... ....,. fJI • •'.... by the pt' •., fil E!? 'p',. tIIIIl value. summing over all values, then dividing by

.... t., II .. ...

en ........1

... ... ..' . wNHIr.

Of CICIIIIIe. . 10111 probIbiIity should be I . bul expressing the a\'crage as a quo.

.... EE3.... . 1O..aid cIeaI.iq

with the proportionality constant in the Boltz

_

_ poWJilily. See AppeDdi. J for further discussion. ' At ... paiIII. the classical theory and Planck s theory diverge. Before pur ..... Ibem. we .. ..... PIaock oriainall)' proposed that his new quantization cIIim be IIfPIied DOC eo the elccrromagnetic radiation in the cavity. as we will do. 111: 10 tbe � 0ICi11atiq in the walls. These "resonators," as he called

..... coaId chIqe eneqy oaly in discrete jumps. Einstein later showed that b Jlva the UIUIIlPtion of tbennaI equili rium between walls and radiation In the cavity. such . view is equivalent to the quantization of the radialion, which is our maiD iDIereIt. Now, 10 clarify the differences. let us follow the classical utd _ 1pIII'OICIIa side-by·,ide. ......' lAy.'i rll", 'II.,ufKIor. .., .... .... .. ..,...,.01' 5' h iii dill COMi._ an to infinity. TiI ___ . .......

If/) -

L·

lIP ... ...

. 's

... ... ,..... (C3).

" is an mteger and h I COMtan\.

-"c: . o--

1

,."

.�r�

Carrying OUI !ht summation.)

(C· 2)

•

rMliltion of frequency lis allowed only

thoK energies given by £ ... nhl. where

..-

dk

(0-1)

lei us concentrate on the integral in brackets.

J "

ei(k -k')xdx

==

ei(k- k')b + e -i(k- k'}b k')

i(k -

�

2 sin[(k - k')bJ k - k'

(0-2)

-,

Note that we have used the relationship sin z =

K).

from the Euler formula (Appendix To obtain our final result, we will let

(e+h - e-iZ)/2i that follows

b approach infinity. but the behavior

of the function on the right side of (0-2) is a bit peculiar. Figure 0.1 shows Ihis fUllction plotted versus k for two different values of In either case, it is just

b.

+

21(k-k') times a sine that oscillates back and fonh between I and I ; but the larger the h, the faster the oscillation. Also note thai as increases, the function beComes more sharply peaked at k

=

b

�

k'. ln fact, by I'Hopital's rule, its value at

b

k k' is 2b, as indicated in the figure, so it becomes infinitely tall there as of the right side of (0-2) over approaches 00. Now, it happens that the all k-the area under the curve-is always 21f, independent of or k' (chang ing k' just "slides" it right or left). Thus. the infinitely tall peak at k = k' muSt ==

integral

b

effectively become infinit�imally narrow. At all other regions on the k-axis, it oscillates infinitely rapidly between

+ I and - I . When mUltiplying any reason

ably smooth but otherwi5c arbitrary function-such as A(k}-il would cause

the product to average 10 zero. [n other words, in equation (0-1), it really doesn't mailer whal A(k) is anywhere bUI aI k = k'-its values elsewhere will be

0-1 -

-

-

....,. � -----� ,-- �

Flg...,. 0.1

• = k'

An arbitrary �mooth function A(k). and one sharply peaked at

A, b becomes large. 2 sin[(k - k')bJ/( k - k') becomes a narrow

but tall peak. with rapidly o�cillatlng "wings."

2b\---'''-... --j .

2

k-k'

Arbill1ll)' function

I�e b -..... '. 2 sln({k-k')b] k-k'

A(k)

\mall h·....··"

,

averaged to zero. II might as well be just the constant value A(k'). which can be side of (0-1) pulled out of the integra1! Therefore. in the limit b -+ co the becomes

+� -00;

A(l) !;m

2 s;n[(k-k')b] k-k'

b.....OO

dk � A(k') !;m

b-OO

� A(k')27T

+00 -;00

right

2 s;n[(k-k' )b] k - k'

dk (D-l)

where we have used the fact, as noted, that the integral is 21T. Having allowed b to go (0 infinity on the right side of (0-1), we do the same on the left side, arriving al

+00 ;-00",(x)e-'"x,,-,

� 27TA(k') or A(k') �

+00 � ; ",(x)e-iKX dx _00

2

(D-4)

Now that the earlier k is gone, we can choose to omit the prime, giving us equation (4-22). Having shown what we sel out to show, it is worthwhile to take a second look at our "peculiar" function. In equation (0-3), it might be a function of any arbitrary variable-for instance. z rather than k-and the function it accompanies in the integral would also be a function of z. Dividing both sides of (D-3) by 21T. we can thus say that

+;00 -00

{

A(,)

Hm

b_oo

s;n [(, _ " )b]

l7(Z - , ) ,

}

d,

�

A(z')

(D-5)

The function in braces has an unusual property-it "sifts" out the value ofA(z) al the point z' only. It does this because it goes to infinity al Z = " but is zero

App.ndi. 0

\

This may sound rather pathological. and indeed it is. but it is , f)'where �IS is in fact the way we represent a particle whose 0 p ySiCS and ,\c! used 1 nonzero only at the point z == z'. ycisel (I1ucn . lOwn pre It is called the IS }u . . . We leave further study of thiS 'tion n tio wen d ne function to a fu

I"'" pirac

delta

....el course. ef-Ie

. n �,g

.

Solving for the Fourier Transform

0-3

Appendi x E:

The M o m e ntum Operator

of the momemum operator rests on the assertion that the derivation

A(k) is to wave number Ie what .p(x) is to position x: a prob '!be er traJlSfonn Fo� aJllplitude of finding a value-of either x or k. The symmelr)' in the abli t iy and k appear in equations (4-21) and (4-22) is certainly suggeslive. ,

",ay x

,(. zoo, the fml teon on the

riabl is O. When we reinsert the second term inlO (E·2), we see that k cancels.

or

(E-3) Now the integration o..'er k may be carried QUI, Writing it explicitly as a limit. we have

/ t'+&\l' ••

lim

b_1

"

-lj elk ::: 11m

b_OO

-.

- I ff

Thus. (E·) becomes +00+00

k = -

21T

"" (x')

-00-00

{

k

=

'f""['f'''' '(X'){

-00

_00

'"

--'7x'--::::1: x

' (2 sin[(x' -x)b] _ , ,_ - -'_ ") . �) � '" '" .�� ..�) + 1.1) I� JW1 of the tySKnI- with .. mcrg) i, e,cl"wlged and ",,,,- �pnf JIIVP"" ert1C' 4efillt" the conuooa � An) chatF ID the number of paru.:1n tII :t pvao oQIc f1IYibD _ ot8SS 1u capture. SF §lands for spontaneous fiSSIOn. s for elecuun st.aIld EC t

\-1

.. .... ",, . f ..... ", .. -

z

27.976927 28.976495

28 29

29.97377

3D

IS 16

3/

p

PhMphOrus

J2 J2

s

Sulfl.lf

JJ 34

"

36 17

CI

Clllorine

"

J7 36

Argon

III

"

40

20

Scandium

So

22

Ti!anium

Ti

24

25 26

27

Cllromium

Manganese

Iron

Cobalt

Mo

F,

Co

0.02 75.77

34.968852

24.23

36.965903 35.967545

0.3]7

37.962732

0.06]

39.962384

99.600

0.01 17

41

40.961825

6.7302

"

v

Vanadium

34.969031 35.96708

9].2581

40

34.99523 39.962591

41

40.962278

42

41.958618

96.941 0.647

43

42.958766

0.135

44

43.95548

2.086

45.953689

0.004

48

47.952533

0.18 7

"

44.95591

100

46

45.952629

8.0

47

46.951764

7.3

"

47,947947

73.8

49

48.947871

5.5

49.944792

5.4

50 23

0.75 4.21

33.967866

38.963707

46

21

3 1.97207 32.97146

39.963999

39

C,

Calcium

14.28 d (fJ-) 95,02

'"

K

pO!lI5Sium

19

30.973762

31.973907

50

49.947161

0.250

51

50,943962

99.750

50

49.946046

"

51.940509

4.345 8].79

53

52.940651

9.50

54

53.938882

2.365

55

54.938047

100

54

53.939612

5.9

56

55.934939

91.72

57

56.935396

2.1

"

57.933277

0.28

"

58.933 198

60

59.93381 9

100

1.26 Gyr (P-. r r , Eq 50 ms

(fJ+)

0.103 Myr (EC)

Atlp·ndta l PIopIdia ol 1L11II'

8Jeldent

t

� �

57.935346 59_930788 60.93 1058 61 .928346 63.927968

63 65

62.939598 64.927793

69.\7

64 66 67 68 70

63.929145 65.926304 66.927 129 67.924846 69.925325

48.6 27.9 4.1 18.8 0.6

'"

G,

69 71

68.92558 70.9247

Gallium

)1

60.108 39.892

G,

70 71

69.92425 7 1 .922079 12.923463

21.24

Germanium

)2

p.rsenic

Jl

Selenium

}I

Bromine

)5

Cu

Zn

Zinc

J1 18

)9

40

Rubidium

Strontium

Yttrium Zirconium

68.077 26.223 1.140 3.634 0.926

30.83

73.921177

76

75.921401

7.44

A,

75

74.921594

S,

74

73.922475

0.89

76

75.919212

77

76.919912

9.36 7.63

78

77.917308

23.77

80

79.91625

49.61

81

81.916698

8.74

79

78.918336

50.69

81

80.916289

49.61

78

77.9204

80

79.91638

81

81 .913482

1 1 .6

83

82.914135

11.5

84

83.9 1 1 507

57.0

86

85.910616

17.3

91

9 1.926270

Rb

85

84.9 11794

72.17

87

86.909187

27.83

S,

84

83.91343

0.56

86

85.909267

9.86

87

86.908884

7.00

88

87.905619

82.58

94

93.915367

Y

89

88.905849

Z,

90

89.904703

51.45

91

90.905644

1 1 .22

92

91 .905039

17.15

B,

-......

(Ilo-\l1l1Tl

l e lill n 1elln r.

�

Symbol

A

Atomic Mus

'" Nalural AbuDCb.oc:e.

S.

2.3

263 . 1 1 82

8h

262

262.1231

H,

265

265. 1300

Mt

266

266.1318

......ur. lDecay

MocI-.)t

O.M , \Sf. Q)

O.tO '\Q} 2 m\{al "'3.4 m\ \0)

AP pe ndix

J:

Pro b a b i l ity, M e a n , Sta n d a rd Deviati o n , a n d N u m b e rs of Ways

I lighl the main ideas arising. whenever we speak of probabililie� we h'gh , I fiere most equatIon numbers have an n, b. c, or d. The ad physics. Note that 10 [11

�m

not 10 sneak in more equations, but to emphasize the fact that. .... ,.a50n 1S ' Ih'mgs. we realty deaI Wit ' h ingly endIess ways 0f reexpressmg , Ihe seem

despite t.s. few basic concep ' " ' . . onIY a The first concept is �robablltty Itself A quantity Q IS the focus. �nd. 11 k . various possIble values-Q t _ Q2' Q3' and so on-each wIth liS 1a e on . P2' Pl, and so on. For example, the quantity Q might be the an bility-P 1 � Prob3 . ' · '· . of a pet house cat. AI an arbIl rary InSlant, It may be at a Window, Q I ; iocal1on . ' ' a probabl'I'lIy. e sleeping location. Q2 : or Its f00d d'ISh . Q3' each wIth its fa\'oril P, = 0.065. And there may be some P., = 0.780, 0.040, P = ce _ s.an L ' For ,m where the probability is 0 unless an eXlernal agent intervenes, such locaU'ons ' ' ht have a theory Ihat predicts W mIg the probabl'I"Illes. or he pet washtub. e observation, experimenlal but in either case, one thing i s on may rest i n: The sum of all the probabilities-the total probability of finding the a

�t ce:

cat somewhere in the house-must be I .

L , p,

�

I

(j· 1 aJ

Often probabilitic� are based on a collection of data. such as N exams. where the quantity Q is the Score. or N repetitions of an air-quality experi ment. where Q is a fluctuating pollulant leveJ. If value Q; turns up Ni limes out of a tolal of N. then the probability is simply

p,

=

N, N

which filS perfectly.

N, I I L, P. � L , - � - L N � - N � N N " N

On the other hand. Ni is often a theoretical number of ways of obtaining a

particular value. For example, suppose we have an office with four identical

cubicles and four workers. We define Q as the number of workers in cubicle I . As Figure 1.1 shows. of all the possible ways of distributing the workers. one

subset has three worker� i n cubicle I , which we designate Q3' and can be done

J·1

FIgun J.t ,·ubldes.

WilYi Q( arranging

r� �r . r

w()(k�r'i a. b. c. and d in four

Cubide J

J

� [�;-�

(j

::�

,-

(j "" .., , (jj) :(jj)

- - -,- - - -

-

, (j :@� -------

, (jj)

:(jj)� (j : -,-- . -

,

12 ways. Another subse[ has all four [here, designated

Q4' and this can be

done only one way. Wha[ are the probabilities? We need a bit more to go on. The mo.�t common addi[ional assumption is thai any way of arranging specific obJecb In specific loca[ions is equally likely. If [his were [0 apply to the work· ers (0 dubious assumptjon for real coworkers), then all 1 3 ways shown in Figure J.I and all the other ways not shown are equally likely. If we were to do an experiment. each way should tum up as often as any other. Therefore,

Q),

In which fhree workers (independent of their identities) occupy cubicle I , $hould [urn up 12 times as oflen as

Q4' simply because there are 1 2 times as

many ways to do it. II happens Ihat the ,otal number of ways is 256 (thai is,

44), so !.he probabilities

are

121256 and 1/256. Again,

Pi

=

NJN. where N is

now a number of ways.

Mean No matler how probability arises. !.he concept of a mean, or average, is always the same. Suppose a quantity can take on only two values,

Q , = - 60 and Q = 2

+60. and that PI is 213, so !.hat P2 is 1/3. With twice the probability, it is logical

thaI we should give

QJ twice the weighting. A mean of -20 would indeed

be half as far from -60 as from +60 and would follow from the general prescription: To find the mean

Q, multiply each value by its probability and add.

(l-2a)

u.t' u� P,

•

NJN. thi,

a....\ume...

_Q=

a common fonn thai may be familiar�

N" �Q,N, Q .... . ' N = N

'"

(l·2b)

value Qj

limes the number of tllne!Jwa)'\ nu' �)... that the mean is a possible obtained, summed over all values. then dl\llded by the total \\ tit)' dle quan times/ways. number of

Qulle often It is relatIVely easy to obtain something thai is. pmfHJrrionai to

�abililY, but actually expres..mg the proportionality constant i� difficult or �sy. In these cases. II IS common to write the mean differently. Suppose P' i� proportional to the actual probability P: P, = A P,' ,

"bere A is a constant. The total probability must be 1 . so

�'Pr =

I =>

L,A P; = A L, P.' = I � A =

I

� p' ""

,

Thus

(1-2c)

Again. in this form,

P'

need only be proportional to P and the proportionality

Q.

constant disappears. the prob Finally, if we wish to average something that is ajunction of ability of obtainingj{Qi) is the same as obtaining QI' and the probabilities stlU add to I , so the mean is

KQ)

=

k,j(Q, )P,

(1-3,)

Standard Deviation Our nex.t concept is a way of quantifying how much the values of Q deviate from the mean. Many recipes are pOSSible, but the most common is the e standard d viation. How do we come to choose it? A given value deviates from the mean by deviation: Qj - Q Some values of

Q i will be above the mean, some below. The deviation is a

function of Qj' which we can average via (J-3a). The result of the average IS logical, but not very helpful.

U

_I.. J

....1.., . . ..... .s..IInf � ... Nambm ol W ...P

I1IQII ofdo'......

1:.( e,

- O)P, = 1:. e, p. - Q ,£, P, = Q - Q I

= 0

.

So Ikt don', '\'cragt the dC'\'ration' Ho"'ever. if we a\'erage i rs squOfl'. rhe sum ,an fant no ntg.O\'C' ".Jut.. and 'he farther tha! 0, vaJues stray from the

mttn. lht larger lht a\tnge �houJd be.

deviation. 1:,(Q; Q)lp, To Yield ..omelhing that ha,� the same dimemiom as Q. we rake [he square mean oflhe ..quare of the

RJOt. Siloing

u� !he root-mean-square deviation.

_

known as the standard devia-

1IC)n, Of the many symbols u.�ed for this important concepl. we choose a della.

vr,(e, - Q)'p, 0-4.) only if Pi i zero whenever Q, - Q is nonzero, Note thai tlllS can be ;\1;1lhemalically expressed. Q, Q � P, = 0 In ocher words. the value IIQ ·

..

:no

0,

:I-

=

Q I� Ihe only one ever obcained, Combined with ilie fael ilia[ 6Q does

'pread a., de\'iacion,� increase. we see Ihal slandard deviation is a very logical definition, 51andard deviation i� often not presented or calculated in form 0-4a), but

1O\leJd in a closely related fonn. whose derivation is a good exercise in sorting

oul the \'anous quanlities we have discussed thus far.

�Q = vr,(Q, - Q )'p, = vr,( Qf - 2Q, Q + Q')p, = V"i, Q�P, - 2Q Li Ql, + Q'2 L/ P,

We have used the fact that

Q is not a function of summnlion index i, but sim

ply a number thai can be broughl outside a summation, In the second term

Q.

imide the radical. we now recognize the definition of Q, and in the third, a unit lolal probability. In the firs!. we have the mean of Ihe square of

IIQ = VQ' - 2QQ + QI

or

IIQ = VQ' - Q'

(J-4b)

While obscuring the fact that the radical's argument is necessarily nonnegative. !hie; fonn makes a simple point: To ca1culate standard deviation. we need only find the mean of the square.

Q2. and the square of the mean. Q2,

JUSI as equation (1-2b) follows from (J-2a), an alternative fonn for stan dard deviation follows from (J-4a).

-

( J 4c )

� Differe

nt Route

deviation can hi! el(p�,�d In a seemingly different form and ..tandard t a. matter of redefinitlon_ If the sum.. are nm over Pl l....!Oible .. rtall) ju ..

�

�5

Q

y but Instead over a.n "tnals"-aU mdi"idual mstam:c!f. of of the quantit we will uc;,c J rather than l-then a sum over I With an N In hlch - 'ut'-- for .... I all '''' " the same as a sum over j alone. For example, If in a �ne!io of SIX is lbe sum - 1..5. and e'( riments we obtain only three values-Q \ = \,2. al� pe t .8. 1.5, 1.2. 1.2. \.8. the following are cqui"alent: 1.2. ce �uen

Q,

Q�

�8-ln

Q=

�:"'I Qj N, N

=

Q= With this

=

-'-'----'-----""-;-'----==-=6 1 .2 · 3 + L5 · l + 1 .8 ' 2

1 .2

+

1 .8 + 1.5 + 1 .2 + 1.2 + 1.8

6

redefmition. a\l N, would disappear from sums. and (J·2b) and (J.4c)

\\ould appear as

/l. Q =

Q=

�'Zj (

)( QJ - Q)'

....

N

We avoid this route because it tends to obscure the role of probability and clut ters the otherwise seamless transition to continuous quantities.

Continuo us Quantities The number of workers in cubicles is a discrete quantity. its values being nonnegative integers. Except on average, we don't obtain 1 .3 restricted to �'orkers in a cubicle. Some quantities are inherently continuous, meaning that

from one value to the nex.t is an infinitesimal change. An example would be

the locations of a swinging pendulum. In such a case, a sum naturally becomes the probabilities of being at particular point locations must an integral. and

beCome infinitesimal; otherwise. summing over the infmity of locations could not yield a unit total probability. The basic formulas translate in a straightfor ward way.

'Z , P, = \ Q = L; Q, P , KQ) = 'Z;!(Q,)P,

AQ " V}; (Q, - Qj'P, ,

--->

--. ---> --->

j dP(Q) = \ Q = j Q dP( Q) KQ) = jJ( Q)dP( Q) /l.Q = Vj(Q - Q)'dP(Q)

Note that equation (JAb) 1� unchanged, as its fonn is independent of whether

a sum or integral is involved.

" - , ,1. '' . ,. .,,1 a J

.. 7 7•5 .... .... " .. ... .. . _ . ... _ .. _

FiuUy. I« «JIIIIDuous varillb� iI i� u:o.ually mo� t:onH'ment 10 lkaJ noc

..uti , dilf'crtati&I probebilil)' bur with a prolNlbUUy de-Mi.y. a probabllil), �r _

Q. __ u follow

,

U(Q) D(QI - -- ,,, .... dP(Q) - D((J)dQ dQ

With thi� delimllon. formullL\

0·1a), {J-1al. (J-Ja). IlI1d (1-4a) become

/D(Q)dQ - 1 0 - /QD(Q)dQ iW) - /M) D((J)dQ � Q - "rrrQ - OrD(Q)dQ

N,'IIe thai In quantum rnt\:h4nit:,. the quanfJIy

(J- I b)

(J-2dl (J-3b)

(J-4d1

Q might be location x and the

prohabiliry drn..il)' comt, from me wa ..,C' function,

D(x) = 'I/I(x)i2.

EXAMPLE J . 1

I

,

I

I I/'

VatU(' CJ, �lImbn of Tunn. .\

I

J

,

,

.

.

"

•

4

7 1

Yo'hat 0Itt W mean and \land.lnJ de\iatlon" SOLlJTION The \Ioa)' in whi,h the data are gncn make� _

Q-

(J-2b) quickest for the mean.

" 1 + 2 ,4 + 3 ' 9 + 4 ' 1 2 + 5 , 9 + 6 ' 4 + 7 " . � _ . • -

40

= 4

hlr the: \,andard dC\'ia!lon, we can u\c (J-k) directly.

.lQ

-

1 0,].1)

/.

2 ,

/(1- '1' 1 " ( 2--- ')"

1-------

\

( J -

+ ( 7 - 4)' 1

+

'0

=

1.30

An...'lIher way i'i equation 0-4b). Thj� requires caku/aling the mean of the square. for whll'h equation (J-Jal i, appropriate. the " funclion" being ju�t the square. Equation

Thus,

i� ex�s�d in

N, IN puIS �J{Q,)N,

term, of probability. but P, •

JlQ)

�

N

II in the form

NoW usin, (J-4b)

�Q

=

\/17.7

- (4)' - \.30

We see that 30 of the 40 "alues are withm t standard �iation of the mean, Standud df:Yiation usually cover-, the majority of \la1UC!i.

EXA MPLE J . 2 "'"""' 1be probability density-probabihty per unit ht:ight- for finding a gi\len object at 1 tw:iJht )' is given by

and applies to all values of)' from 0 to +00, Find the mean and standard de.\liation of " in terms of the constant The following Integral will be useful-

b.

fooo Y"�-b"dy

""

SOLU1 c'...

Apparently our resuh should not depend on naturally using y in place of Q_

m!jb'".. t

A, We can use (l·lb) to detennine it,

Thus,

{Nott: In quantum mechanics, the process of ensuring unit probability i!i called nor· malization.} Now using (1·2d),

The result is sensible, If

b increases, the exponential faUs off faster. and we would

expect the average height to be smaller. Moreover, its dimensions are correct. The

argument of the exponential must be dimensionless. so the dimensions of b must be

one over length.

For the standard deviation. we could use (J.4d) directly, but let uS instead use

(J-4b) after finding the mean of the square via (1·3b). As in Example 1 . 1 . the func·

tion here is just the square,

y' = fo

-

00

y' be-b'dy =

b fo

00

y'e-b'dy =

Now inserting in (J-4b)

t.y =

I

\/(211)')

b(2!/b')

(I/bl' =

= 2/b'

l ib

Not only does this have the correct dimensions of length. but it also happens to equal

the mean. We conclude that most values obtained should be in the fange of heights between zero to twice the mean.

...... J� "'-)"101 JWl.. two cl .. c.- I. two ,.,t". 1If*ft.

FH ', r·n;

-r

;' .. ' f, ,

,

; .. f

/' I

r,

1

-

Factorials-Numbers of Ways

1

Suppose we h.t,-e Nd.ffel'C'nl can. and N md.,iduaJ parking splices. How many d,fft'ft'nl .. a)\ L"iU1 we arrange' Ihe' c� m Ihe space�? Any of the N cars could

be in � tim: 'paL't', and an) Ilf th� I'C'maJnlng N - I could be m the second Thu�. tilhng Ju�t the' tiN IWO ,p"ccs. Ihere are N(N - I ) ways. In the nd c. and �rec �pace�. these six possible Ways \pc.:.aJ ca.'iC' of three J.:aJ"\. O. b. .. _ are �hOwn in Figure 1.2. Conllnumg, any 01 the remaining N - 2 cars might be 'pao.'C'.

In the thiN 'pace, �o to thi� point we would have N(N - I )(N - 2) ways. Of cotJ.r>C. in lhe thfU-car case. N - 2 i� I. and no more ways are added. as there

i_' onl)' Oil(: choJ('c for the la\1 car. For arbitrnry N. the number of ways would 2. then N - J. and so on. until again there is only one car lefl So the 10lal number of ways is N(N - I HN - 2) . . . 3·2· 1 . which is the delinJlion of Nf and j� referred 10 a� "N factorial." Nov.- let us de,ignale a group of NI parking �paces as region i. as depicted be multiplied by N

,..".. J,J A �'''Il .." hln ..hJ�·h

�,"anpffiC'nlll' lotoJn.:h " da:l�

IrrrkW'allI

in Figurt J3. By the aOO\'e arguments, there are Ni! ways of rearranging cars among lhe-.e ..paces alone. without affecting caTS elsewhere. What happens if

�e om.. declare thaI it doc�o't maner where the N, cars are in this region. and we will con�ider it as anI)" one way of arranging all N cars? For (lll)' previous

,in,gle way of parking all car; in all individual �paces. there II'ould hUl'e been N, ' different ....ays . (including Ihal single way) thaI would leave the cars not in

region i exactly where ,hey were before. and we have now declared these N '

�f

fomlerly different ways as jusl one way. Thus. the previous lotal number ....ap. . N'. j, simply divided by N/. the number of rearrangements within region j thlll change nothing out�ide. Repeating the process. we choose

another group of parking spaces from the N - N; not in region i, calling this

ne� group of N �paces region j. and again declaring that rearrangemeOls

l

"ilhin il are irre e\'ant. By the same arguments. we must divide the existing

number of ways by

N ! to obtain the new number of ways. If in the end we N spaces into M regions. the number of ways W to park

1

have broken the line 0

cat\ in !!tpaces. where rearrangement within any given region is not considered J different WIl)·. is

(l-5) Consider the limits. If each space were a "region:' there would be N regions. each with one cat-SO that NI '"" I for all i-and W would be simply N! (divided by I ! 10 the power N). as we expect. At the other extreme, if there were just one region encompassing all

�nsible.

N spaces,

IV would be

1,

which is also

We have used cars and parking spaces as the framework, but the arguments

are general. Formula

among

M

(J-5) gives the number of ways of arranging N objeclS

separnte calegories (regions, boxes. energy levels. etc.), where

N

�

panicles are in calegory I-rearrangements within being irrelevant-N2 i category 2. and so on.

N-

al case i!> just two categories, one \\lth rnmon speci ost cO m objects. in which case we ha\'e II e 'flI ),lith \' other

)C'11\ 10g

the

tl

obJe4.:h.

"::N-"-W � _II!(N -! ,,--,

fI)\

ial coefficient and is so common lhllt It hu,

vn as the binom IS kllO\ r to l. fac . 0wn special symbo ""' 1"1� ItS • en

1> .... a IptZ'. ... by which \he fun'

(K-4)

Solution of a first-order equation yields one arbitrary constant, essentially a constant of integration, which. in the above result, is A. It c�n take on any . techmque of sOlving value while still solving the e� uation. The m�thodlcal this basic differential equation IS to rearrange It to dflf = b dx, then integrate both sides.

Second-Order Linear

d'�;)

bf{x)

{A �

=>

f(x)

�

( VIbIx) + B cos ( VIbIx) 0' Ae+iv1il• + Be-iv1il• b Ae+v'bx + Be-v'bx or A sinh ( v'bx ) + B cosh ( Vbx) b sin

Ax + B

(K-5) < 0 > 0

b = O

A second-order equation yields two arbitrary constants (of integration). Note that the sign of the constant b is crucial, for different signs lead to functions whose behaviors are entirely different. If b is negative, the functionJis oscilla_ tory; if b is positive.! is exponential. growing or decaying or some combina_ tion thereof; and if b is O,fis a straight iine.

Useful Integrals Below is a short list of integrals most often needed in the text.

J . (L ) dx = "2 L = ""4 J . ,(mTX) J . (--L ) sm2

x sm -

mT X

x

-

x'

-

d.x

T x, sm2 n7 X dx

=

( -) ( ) (-- \,

2"-,, sin - X L 4mT

L

Lx 4mT

2mrx sin -L -

2mTX Lx' - -- sin 6 4mT L J

x'

LOS

(2mTX) -L

Appendix K

Some Important Math

K-S

An sw e rs to Sel e cted Exercises

Chaple. 2

99. -48 1"

17. 0.1'''"

ll. 43.75 m 23. ]:ller, 0.8 m 15. 60 m. 2.67 x 10-7 ,

lOS.

n. \'!c = 0.78 1 . 1 .04 x 1 0 -1 s, 24.375 m

19. 24 m. I'le = 0.6 31. 0.0067 s behind )5. 9.8 ps earlier

41. 1549 m

.0. (a) - l OO ns: (b) 1 4 1 ns; (c) I OO ns, zero 45. Bob is 60 yr, Anna is 52 yr

47. (a) 32 y" (b) 32 y,

(a) jumps ahead 1 2 8 days; (b) 250 os; (e) behind by same amounts

53. (a) toward. I'le =: 0.25 ; (b) 687 om; (c) 549 nm

55. yes

:/ T : ,= 3k r y m A, a . 1 5 0m 7 2 5 . c

59. O.385c. c_ ", , " (b) (0, c) c 63. (3) 2

II;:

65. (.) (-0.8c. 0.6c): (b) (O. c)

1016 J. 6 X 1 0 1 6 1 , 1 . 5 X 1017 J 71. 9 10-17 kg X 73. 2.5 . 75. (a) 2.19 X 1 0- 26 kg mls; (b) 3.64 X 10-22 kg . mfs; low 40% low. 10-7% X 3 (e) X

77. c/V'i

79. 1.83 X lOS kg/day 8I . 4.71 X 109 kgls

o

83. -c 2

85. 25.1 MV 87. (3) O.759c; (b) 2.07c; (e) O.948c 89. ue = 0.9997 I c.uc = - 1 .62 X 10-Jc. 20.6 MeV, [7.1 keY

91. "'2 = 6.43 kg. ml

I

0

0

0

0 - �,/C

I I I.

J7. c/V'i

V

0 0 -V/l'

0

0

0

0

I

(.) 4mo' and 5"'0"': (b) 0"'uc and

lIS. (.) A \/8/3 and 5A \/8/ 12: (b) bolh

39. it must; lOP passes through (v/e2)Lo cos 80 earlier

49.

I

=

1.43 kg, 1 .93 X 1 017 J

93. Ca) 1.29moc2; (b) 0.795c; (e) 2.57mo' 3.29mo' expo B

2 0"'0'" VSqA/24".o'

Chaple. 3

17. 3.12 eV. 399nm

19. 82.4 nm

21. 6.42

23. 1.22 29. 8.15

X 1031 phorons per sec X 1()6 mls X 107 mls

31. 130.5°,23.9° 35. 0.00659 nm

41. (b) opposite, 1 . 1 8 X 10 14 m 43. 9.1 X 10-27 kg 45, 60°, 5.73 X 10-3 degrees. visible light

47. 7.38

X 10' 10 m. 600

49, 5 X 10-6 Pa, 6.37

51. je.,E'A/h

x

IO� N

53. 2.91 X 10-12 m

55, imoCl, lmo

Chapter 4 11. 40 13. 0.333 nm 15. 728 mls 17. 2.43 X 10-12 m 19. (a) 6.29 nm; (b)D.147 nm

21. (a) 1.46 X IO-I o m to 1.32 X 10-0 m;

(b) 6.23 X 10-9 m 10 2.43 X 10- ro m 23. 2.2 X 10- 10 m 25. cannot be treated classically 27. Ca) 0.14 mm; (b) 1600; (c) 400 29. (a) s/m: (b) 10 s- 1/2v'b; (c) 5 s'I/2 v'b. (25 S - I)b, 25 s - I

AN-1

AIWI

_ .. _ _

71. 1.61 X 1 0 22 kI an.. ' )S x 1 0 11 J. 1.67 )( 10'l7 kI ) 6 )( 10 li mit

.L G. 4I. #7.

10' "'" 10 " )( 23 (11'1 ' 3 )( 10 . J .. (.)0.01 67'. Cb) 1.0:5 )( 10 J I dtpn - 1.83 )( 10 '\) rtd 51. r < 10 ' m

6.l &r

51 (b)

"

�V'A'.'2M'

5'7. .. (4 ....,)A.. 2 2 x 10' mi. 59. c .) O lI4nm. O ' nm 11. 0 1 '. 6J. 9' nm 65. 67.

...

I

2.

Ct

Cbapc.r S

53.

I

I X 10' �

lJ. (a) 2t1m. (b) 2Lln 31. 'a) .boul 9'1 as large 33. (.) 0.080 and 0.16; (b) 5.8 x 1O-17nnd7.1 X 10-1 4.. (cn,4 x 10 1M and 1.8 x 1O- 1l1 •

'" v ph.ouot

0,0-'3 s

partid�

- B.n«' - 1»

+ c, '11, u.... nh U.

�( �: y �

(b) R => k?L2/(4 + .l:1Ll). T "'" 4/(4 + klL2)

Chapter 7

- lAo A �inkL + B cos kL - F�L B 'I n U.l ... o(FrrL �� QL) _

m ji(�:t.J��(mK)I ·.lh I

21.

O,� om

2J. 0.609. 0.196.0.609 25. 107 'TrZfa2/2mL2, at center 27. -mi'!2rr4iffi2nl 29. n, "'" 6. nr' '''' 3, 1 . 1 X 10-6 m 31. n "" 6. 13,2 cV. lOS K

33. 5.5 X 10-8 eV, 5 X 10-9

37. 150°. 125.)0, 106,8°. 90°, 73.2°, 54.7°, 30°

45. -0.85 cV, magnitude of angular momentum: 0, V 2h.,

V"i2JI, l...component of angular momentum:

-Ii, 0, +11. +211, +311

51. (a) yes; (b) no '3. 0.2 l 2

\lH "'It'll v ICf

_

59, lrm

'7. 0.238 59, ellipse

&J.

61. no 67. 13.5 nm

73. (a) plane wa,'(' (1). Dirac df'lta (I); (b) A =< 1I

,,11-;

Ie) B _ 11 75. tb) no� Ie) 1

3�

,

v2b :

X 10 :!O J 77. (al and (bl doOl

..

.....�' 'I

..

I""

... �� . ,

..

"

" ..

,.

. � .. " "

��

.. ,..

" ..

"

"

,W'

"

"

, • � ...no'

.,',"'

.. , ,,,'

.. .. 'U" uU'" ""

,"

..

..

,. ... I""

.... .. .... ' "

" "',..,

,...

"

..

•

" ..

..

. ,

--

•

"."

,�,

"

,...."

.. . ,,,

' .. n.· ..

..

• •

..

,.,.Il

>I .... �'

.-

,. ..

\"0)

�

,,(l,lI

.. -

�

"...�'

.., "

"

••

...

"'.

•

,.

'U'l

'"� .. �'

.. ..

,,'

•

"

IS''»

, ,..... "�'

•

'"''

.

....'�' >I

,

•

..

.. �

, " O. ,W' ,�,

• ..

" ,

,..

"

•• ..

..

..,.n'

H'"

..

•

..

.. .

,�, " ,

•

-

•

,,,

.

,

..

..

'"

..'

..

-

\"..

",

" ,

,.

,,.,

..�

""

0

..

."

.. -

•

.. .. .. , u',. ,..�

..

.. .. , "

� ,,' ,,.,, ..

•

..

..

..

�,

,

,

..

,",. -

..

..

..

,

.. ..

,

•

" . 0 ..�

.

0

. ..-

.. .. • ,� �-

• ,. ,. • .. .. I"" ,,,II ".n

\, .. -

.

,,. C..

" ,

•

-

.. .. " .. . ,... "..",

• • .. lOu.

,.,.

.

,

.-

,. .,..

""

",. ,,,

",

..

.. "

,

, •

....

-

"

.. ,

,.w .,..",.. ' "

.m

..

..

"

..

,,.. ,

,

...."' .. .

.. �

..

"

,

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Modern Physics

Modern physics

Modern Physics

University physics with modern physics

University Physics with Modern Physics

Modern vacuum physics

Modern Teaching of Physics

Modern Vacuum Physics

Ancient and Modern Physics

Mad About Modern Physics

Modern Physics For Engineers

Modern Physics for Engineers

Modern Physics, Fifth Edition

Modern Many-Particle Physics

Concepts of Modern Physics

Modern Teaching of Physics

Mad About Modern Physics

Modern physics (3rd Edition)

Optics and Modern Physics

Schaum's Modern Physics

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Modern physics for engineers

Modern elementary particle physics

Modern Physics, 3rd edition

Modern Introductory Physics

Modern Atomic Physics

Modern Plasma Physics,

Principles of Modern Physics

SSM Modern Physics 3e

Concepts of Modern Physics

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Modern Physics

Modern physics

Modern Physics

University physics with modern physics

University Physics with Modern Physics

Modern vacuum physics

Modern Teaching of Physics

Modern Vacuum Physics

Ancient and Modern Physics

Mad About Modern Physics

Modern Physics For Engineers

Modern Physics for Engineers

Modern Physics, Fifth Edition

Modern Many-Particle Physics

Concepts of Modern Physics

Modern Teaching of Physics

Mad About Modern Physics

Modern physics (3rd Edition)

Optics and Modern Physics

Schaum's Modern Physics

Modern Physics and Vedanta

Modern physics for engineers

Modern elementary particle physics

Modern Physics, 3rd edition

Modern Introductory Physics

Modern Atomic Physics

Modern Plasma Physics,

Principles of Modern Physics

SSM Modern Physics 3e

Concepts of Modern Physics

University Physics with Modern Physics (12th Edition)

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