A D V A N C E S IN IMAGING A N D ELECTRON PHYSICS
VOLUME 137
D O G M A OF THE C O N T I N U U M AND THE CALCULUS OF FINITE D I F F E R E N C E S IN Q U A N T U M PHYSICS
EDITORINCHIEF
PETER W. HAWKES CEMESCNRS Toulouse, France
ASSOCIATE EDITOR
BENJAMIN K A Z A N Palo Alto, California
HONORARY ASSOCIATE EDITOR TOM MULVEY
Advances in
Imaging and Electron Physics Dogma of the Continuum and the Calculus of Finite Differences in Quantum Physics HENNING F. HARMUTH Retired, The Catholic University of America Washington, DC, USA
BEATE MEFFERT Humboldt Universitgit Berlin, Germany
V O L U M E 137
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CONTENTS
PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
FUTURE CONTRIBUTIONS
ix
. . . . . . . . . . . . . . . . . . . . . . . . .
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi .
xvii
FOREWORD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xix
LIST OF FREQUENTLY USED SYMBOLS . . . . . . . . . . . . . . . . . . . .
xxiii
1
Introduction
1.1
Modified Maxwell Equations and Basic Relations . . . . . . . . . . .
1.2
Basic Concepts of the Calculus of Finite Differences . . . . . . . . . .
8
1.3
Lagrange F u n c t i o n and Modified Maxwell Equations . . . . . . . . .
18
1.4
D o g m a of the C o n t i n u u m in Physics
26
1.5
Concept of Space Based on Finite Differences
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
30
Modified KleinGordon Equation
2.1
Differential E q u a t i o n with Magnetic Current Density
. . . . . . . . .
2.2
Modified K l e i n  G o r d o n Difference E q u a t i o n . . . . . . . . . . . . .
52
2.3
Solution of the Difference E q u a t i o n of Wx0 . . . . . . . . . . . . . .
61
2.4
TimeDependent Solution of Wx0(ff, 0) . . . . . . . . . . . . . . . .
70
2.5
H a m i l t o n Function and Quantization . . . . . . . . . . . . . . . .
78
2.6
Plots for FirstOrder A p p r o x i m a t i o n
86
. . . . . . . . . . . . . . . .
44
Equations are numbered consecutively within each of Sections 1.1 to 6.10. Reference to an equation in a different section is made by writing the number of the section in front of the number of the equation, e.g., Eq. (1.145) for Eq. (45) in Section 1.1. Illustrations are numbered consecutively within each section, with the number of the section given first, e.g., Figure 1.21. References are listed by the name of the author(s), the year of publication, and a lowercase Latin letter if more than one reference by the same author(s) is listed for that year.
vi
CONTENTS
3
Inhomogeneous Difference Equation
3.1 3.2
Inhomogeneous Term of Eq. (2.231) . . . . . . . . . . . . . . . . Evaluation of Eq. (3.132) . . . . . . . . . . . . . . . . . . . . .
92 102
3.3 3.4 3.5
Quantization of the Solution for Ax >> h / m o c . . . . . . . . . . . . Evaluation of the Energy Uc . . . . . . . . . . . . . . . . . . . . Plots for SecondOrder Approximation . . . . . . . . . . . . . . .
115
4 4.1 4.2 4.3 4.4
KleinGordon Difference Equation for Small Distances
Evaluation of the Difference Equation for Ax 0 for mo c2 > eCeo, A2 >> A2
~. +~ ~~~Amos), ~+~~=(mo~T/Nh) 2 (T/Nli)2(m20c4 2 + e2c2 Amox) . . 2e 2CeO
~
(2)
We look for a solution excited by an exponential ramp function as boundary condition. The exponential ramp function avoids the 'instant jump of a step function and is thus better suited for equations that describe particles with a finite mass m0. The constant ~ cannot be chosen. It will turn out to be imaginary:
r
o) = r
 ~~o) = o = r
 e ~e)
for0< 0 for 0 > 0
(3)
An initial condition is needed for the time 0 = 0 and all values of r We choose it to be r 1 6 2 o) = o
for o = 0, r > o
(4)
Since Eq.(2) is a secondorder difference equation with respect to 0 we need a second initial condition in addition to Eq.(4). We proceed as follows. Equation (2) requires 9o(~, 0), and 90(r 1) to compute 90((~, 0) for 0 > 2. Since 90(r 0) is defined by Eq.(4) we may choose a second condition 90(r 1 )  d '
(5)
2.3 S O L U T I O N OF T H E D I F F E R E N C E EQUATION OF ~I/xo
63
but the connection with 0 9 o ( ~ , 0 ) / 0 0 for/9 = 0 of the differential theory becomes clear if we use the "right" difference quotient of Eq.(1.31) instead of Eq.(5)" 9o(r 1)  90(~, 0) = d
(6)
We use the following ansatz (Habermann 1987) to find a general solution from a steadystate solution F(r multiplied by ( 1  e ~~ plus a deviation u ( ~ , O ) from ( 1  e  " ~ 9
Vo(r o) = Voo[(1  ~  ' ~ 1 6 2 Substitution of 900(1  e  ~ ~ 1 6 2
9 o(ff,0) = ~oo(1  e  ' ~
(7)
+ ~(r 0)], r 0 >__0
into Eq.(2) produces the following relations
~o({,0 + 1) = ~oo(1  e  ~ ( ~ 1 6 2
90(~ + 1 , 0 ) = 900(1  e  ' e ) F ( r
+ 1)
and we obtain:
(1  e  ~ ~
+ 1)  2 F ( { )
 iA1 { (1  e  ~ ~ 1 6 2
e~)F({) A3e~~  ~  e~)F(r
+ F ( ~  t)] + e  ~ ~ + 1)  F ( r  1 ) 1 
~  2 +
 ~(1
~~~162
= 0
(8)
We collect all terms multiplied with 1  e ~~ on the left side and the others on the right side:
(1  e~~ {F(r + 1)  2 F ( ~ ) + F ( r
1)  i ) ~ l [ F ( r = e~~
+ 1 )  F ( r  1 ) ]  ,k2F(r }
~  2 + e ~ + iA1A3(e ~  e~)lF(r
This equation must hold for all values of/9, which is possible only if both sides are zero:
F ( r + 1)  2F(r
+ F(r
1)  i A I [ F ( r + 1)  F ( r
e ~  2 + e ~ + i)~1)~3(e ~  e ~) = 0
For the solution of Eq.(9) we may use the ansatz
1 ) ]  )~2F(r = 0
(9)
(10)
64
2 MODIFIED KLEINGORDON EQUATION
F(r = Alv ~, F(~ • 1) = Alv ~+~
(11)
and obtain an equation for v:
(1  i/~ 1)v 2  (2 + ~2)V + 1 + iA1 = 0
I+iA1
{ 2 + A 2+[4(A22 . A 2 ) + A 4 ] 1/2}
= 2i; (I+iA1)[I•
2A2)1/2],
A22, A2 < < 1
(12)
vl " (1 + iA1)[1  (A 2  ~2)1/2]  exp[_(A2  A2)l/2]ei~l v2 " (1 + iA1)[1 + (A2
_
(13)
A2)1/2]
 exp[(A2  A~)l/2]e'~l We obtain for F(r
(14)
in Eq.(ll)"
F(~)=Aloexp{[()~2)~21)1/2+iA1]r
exp{[(A22A2)l/2+iA1]r
(15)
According to Eq.(2) the difference A22 A2 will be positive except for extremely large values of Ceo. 2 We restrict the calculation to this case A2  A2 > 0. In addition, we want F(0) to be 1, which is achieved by the choice All  1 and Alo  0 . T h e function F(r becomes:
F(r
= e x p [  ( A 2  A2)l/2r162
)%2_A12>0, F ( 0 ) = I ,
Alo=0,
Al1=1
(16)
E q u a t i o n (10) yields a quadratic equation for e ~"
(1 
iA1A3)e 2~  2e ~ + 1 + iA1A3 = 0
e~_ (I+iA1A3)(I+iA1A3) 1+22 ~1,~3
. I+i(A1A3+X1A3)
(17)
A trivial solution e ~  1, L  0 and a nontrivial solution are obtained" e ~ = 1 + 2iA1A3, ~ " 2iA1A3
(18)
2.3 SOLUTION OF THE DIFFERENCE EQUATION OF ~x0 Equation (7) must satisfy the boundary condition of Eq.(3). yields F(0) = 1 we get:
65
Since Eq.(16)
0>0
'I.'oo[1  e ~~ + u(O, O)] = ~oo(1  e~e),
(19)
~(0, e) = 0
Hence, u(~,0) has a homogeneous boundary condition, which is the goal of this method of solution. The initial conditions of Eqs.(4) and (6) yield with
F(0) = l: Vo(r o) = Voo~(r o) = o,
9 oo[(1  ~  ~ ) F ( r
r
(20)
> o
+ ~(r 1)]  ~ o o ~ ( ~ , o) = o
~(~, 1)  ~(~, o) =  ( 1  ~  ~ ) F ( r
(21)
r > 0
Substitution of u(~,O) into Eq.(2) yields the same equation with ~0 replaced by u:
[~,,(~+ 1, o )  2~,(r o)+ ~(r ~, o)] [~(r o + 1)2~(r o)+~(~, o  1)]
iA~{[u((+l,0)u(r162162162
(22)
Particular solutions u~(~', O) of this equation can be obtained by the extension of Bernoulli's product method for the separation of variables from partial differential equations to partial difference equations: ~(r
= r162162
(23)
= r162162
We observe that the notation r162 instead of r162162 is a generally made simplification that will be acceptable in Chapter 2. The substitution of u~(r 0) for u(r 0) in Eq.(22) yields:
[r162+ 1 ) r
 2r162162
+ r162  1)r
 [ r 1 6 2 1 6 2 + 1)  2 r 1 6 2 1 6 2 + r 1 6 2 1 6 2  1)]
iA1 {[r162+ 1)r
 r162 1)r
+ Aa[r
+ 1)  r162162  1)1}  ~r162
= 0
(24)
In analogy to the procedure used for partial differential equations we multiply with 1/r162162 and separate the variables:
66
2 MODIFIED KLEINGORDON EQUATION
1
r162 (2zrp,~/N)2 = 4sin2(9~/2)
4 
(m20c4 2C 2 Ce0 2 Jre2 C2A2 .[Im0x) > (27rp,~/N) 2
(6)
We may rewrite these equations as follows:
q0~=
27rN ( ~2 _ /~1~ 322)1/2 N t~o
x e~
sin r
(23)
The summation is symmetric over negative and positive values of ~, while the differential theory always yields nonsymmetric sums over positive values of ~. Substitution of Eq.(23) for u(~,0) and u((, 1) in Eqs. (2. 320) and (2.321) yields: t~o > h/moc and Ax > h/moc
1
( h /~
~ iAt ~ + ~r L~xl,+ ~Lc~l~ iAt _.2z~2( h ) 2 Ol m0cA~ (AeozAmoy AeoyAmoz)
X(_~A '~'~'~r ~ ieAt =2Ze2( a
XO +
~r
)]
~o(r O)
~ iceAt~Amox=AZ~O /~ ~ce2(At)2 ~) /I~A h 2 Ce0Amo ~0(~ ,0)
h )2
mocAx
(AeozAmoy
(1 {[~o(~'+1 0 + 1 )  ~ o ( r
AeoyAmoz) 01)]+[~o(~'1 0 + 1 )  ~ o ( ~  1
01)]}
1 ieAt
+ ffTCeO[~O(~ + 1 , 0 )  ~ o ( <  1,0)] 1
2
ce2(At)2
ieAXAmox [~o.~ ( 0+1)~o(~,01)]+ ~r162 h ' h2
)
(17)
98
3 INHOMOGENEOUS D I F F E R E N C E EQUATION
Ax > Ac  h / m o c
[v~(r
o)2v~(r o)+v1(r 1, o)] [v~(r o+1)2v~(r o)+v~(r o 1)] ~
iAl{[~l(~+1,0)~l(~l,0)]+A3[~l(~,O+l)~l(r162
2
102
3 INHOMOGENEOUS DIFFERENCE EQUATION
_ 4AcAe(~',_eO) [~o(( + 1, O)  2~o((, 0 ) + ~ o ( ~  1, O) eAx .
3
/ eAx .
 i  ~ A m 0 ~ [ ~ 0 ( ~ + 1,0)  ~o(~  1, 0)1 + L~Am~
\2 ~0(~, O)] 4
2Ze 2 + am2oC2 (AeozAmoy  AeouAmoz) X
~ (1{[~o(~.+1
5
~0 + 1 )  ~ o ( ~ + 1 , 0  1 ) ] + [ ~ o ( (  1
_ 1)]}
~0 + 1 )  ~ o ( ~  1 , 0
1 ieAt ~ ~ T C e o [ l I / o ( ~ + 1, O )  ~ o ( ( 
1, 0)]
7
ce2(At)2  ieA~XAmox.~o_(,O[ ( + 1)  ~ o ( ~ , 0  1)] + ~ r
2h
. mo(AX)
h2
'
O)
)
8
2
 z o z h , A ~ Vcx31(~, O) ~1.~o_(,0 + 1) 
6
mo(Ax) 2
(1
i 2o/hA~Ccx51~[lI/0(~, 0 ~ 1) 
e('~)2 Amlx(~'O)( h [q2~

r
, 0  1)] + ~r
q~o(~, 0  1)] ' O)~~
, {9)
ieAt
+ TCeOlI/o(~,
0)
9
)
10
,
11
h
+ 2 l A x [~0(( + 1, O)Amlx(~ + 1, 0)  ~0(~  1, O)nmlx(~  1,0)]
12
 2eAmox.Amix(~, 0)~o(~, 0)] 13 .J
e
C2
(
r
+ '2iAt[qlo(~,O + 1)r
~[~I'0(r
0+1) ~0(~', 01)] 1t eCe01I/0(~,{9)
O + 1)  ~I/o((, O  1)r + 2eCe0r
O  1)]
(~, 0)~I/0 (~, 0)/
J
)
14 15
16
(32)
3.2 EVALUATION OF EQ.(3.132) Equation (3.132) needs to be simplified before attempting to find a solution. A first step is to make as many as possible of the constant components of Ae, Am, Ce, and Cm in Eq.(3.14) zero. The two constants Amox and Ce0 must be retained to keep Eq.(2.32) valid but the following components may be chosen equal to zero: Ae0x = Ae0y = Aeoz = 0,
Amoy  Amoz 
0,
Cm0 = 0
(1)
103
3.2 EVALUATION OF EQ.(3.132)
The first two lines of Eq.(3.132) remain unchanged. We note that they are equal to the left side of Eq.(2.32) except that ~0 is replaced by ~1. Lines 3 and 4 are multiplied by Ae((~,0), which according to Eqs.(3.14) becomes ~Ael (~, 0). These two lines are of order O((~), they are left out. Lines 58 are zero because of Eq.(t). Line 9 is zero according to nqs.(3.125) and (1). Line t0 remains unchanged. We shorten the coefficient Ccx51 to Ccx: Cr = Cr
(2)
The remaining lines 11 to 16 do not permit any significant change. Hence, our first simplification of Eq.(3.132) leads to the elimination only of lines 3 to 9:
[~ ((+ 1, 0) 2~t (~, 0)+ ~ (r 1,0)] [~ (r 0+ 1 )  2 ~ ((, 0)+ ~t ((, 01)]  iA~ { [~t(r + 1,0) ~ t ( r 1, 0)] + &3[~1 (r 0 + 1)  ~ (~, 0  1)] }  &2~1((, 0) = G~(r o) (3)
c~(r
(2 [~o( 0 or ~I'I(C, 1) = 0
(6)
for00, ~>0
(7)
The ansatz of Eq.(2.37) is also used once more but we write 5(~, 0) to emphasize the difference with u(r 8) in Eq.(2.37) or (2.423)" ~1 (r 8) = ~00[(1 The determination of F(r to (2.318):
e~~162 + fi(r 8)]
(8)
and~ follows the calculation from Eq.(2.38)
F ( ( ) = exp[(A 2  A~)1/2r
(9) (10)
L = 2iA1A3
The boundary and initial conditions for ~(~,0) are the same as for u(~,0) in Eqs.(2.319) to (2.321)'
~(0,0) = 0 ~(~, 0) = 0 5(~, 1)  5(~, 0) =  ( 1 
e'~)F(~)
8>_0 ~> 0 ~" > 0
(11) (12) (13)
Substitution of ~2(r of Eq.(8) into Eq.(3) yields the same equation with ~1 (~, 8) replaced by ~(r 0):
[~t(( + 1,8)  2~,(r 8) + ~(r  1 , 0 ) ]  [~(~', 0 + 1)  2~(r 0) + fi(r 0  1)] iA1{[4(r + 1,8)4(~1,8)] + A3[~(r + 1 )  ~(C~, 0  1)]}  A2fi(r 0) = G1 ((, 8) (14) We write the solution of Eq.(14) in the form ~(r 0) = u(r 0) + ~3(r 0)
(15)
where u(~, 0) is the solution of the homogeneous equation" and 73(~,0) is a particular solution of the inhomogeneous equation. The homogeneous equation equals Eq.(2.322). We solved it with Bernoulli's product method:
~.(r o) = r162162 which yielded Eq.(2.336) for r
r162
(16)
3.2 EVALUATION OF EQ.(3.132)
r
= ei)'~r
105
iv'~; + Aa~e ivan) = ei)'~;(A32 cos ~0,~ + iAaa sin q),~) ~
= ~,~/u,
and Eq.(2.420) for r
,~ = o, + 1 ,
...,
•
(lZ)
r = eia~'~~
r
~'~~ + A~oe  ~ ' ~
(18)
Using the approximation/3~ = ~ of Fig.2.41 in the interval 7r/2 < ~o~ < 7r/2 we may rewrite Eq.(18) in the form
r
 ei)'~)~~
i~~ tA4oe i~~
= e~)'~~
cos (p,~0 + iA43 sin (p,:0)
(19)
We turn to the inhomogeneous solution ~3(~, 0). Let ~(~, 0) be substituted for ~,(~, 0) in Eq.(14). Writing in some detail we obtain in analogy to Eqs.(16)(19)"
9,r
0) = r162
 e/'X~r
cos qo,~ + iA33 sin qo,~)
x e i'xl"x3~(A42 cos q)~0 + i443 sin (p~0) = ei,x~,xao (A.32 A42 cos qo,~0 + ifi32A43 sin (p,,0) e i'xl ~ cos qo,~ ei)~ ~30 (A33A43 sin ~0~0  iA33A42 cos ~o,~0)ei'x~r sin qo~r
_
In the spirit of the method of the variation of the constant we make the constants A42 and Aa3 functions of ~ and write ~3~(4, 0) in a shorter form:
~(r
0) = s~(o)~ ~ r cos
27r~r 27r~ N + T'c(O)ei)'~r sin
(20)
For a more general solution 1 ~3({, 0) we may sum over all values of ~. To shorten a number of formulas we use the notation r162 and Cs(~) for e ixl; cos(27r~4/N) and e i~1~ s i n ( 2 ~ { / N ) " '~; sin 27r~r + Gr
(31)
Multiplication with 2 N  l e ~>'1r sin(2zrv(/N)or 2 N  l e i>'~; cos(27rv~/N) and integration over the orthogonality interval 0 < ~ < N yields Gs~(0,~) and
a~(o, ,~). N
2 f al(r o)e  ~ a~(o,,~) = ~
cos.21r~r N de for v  ec
(32)
0 N
G~,~(O, ~) = ~
a l (~, 0)e ~)'~ sin 2zr~;~d~ g
for v = t~
(33)
0
Equations (30) and (31) yield for every component ~ of the two sums the following two equations:
 ( 1 + i/~1/~3)S~(0 ~ 1) +
2(cosA1 + A1 sinA1) cos ~2 ~ _ ~ ) s~(0) 27rt~
 ( 1  iA~A3)S~(O  1)2i(sin A~A~ cos A1) sin ~T~(O)=Gr

(1 + iA1Aa)T,~(O + 1) +
(34)
2~,~ _ ~)T~(O) 2(cosA1 + A1 sinA1) cos~
27rt~  ( 1  iAIA3)T,~(O  I)2i(sin A1A1 cos A1) sin NS,~(O)=G.~,~(O,~,)
(35)
The following substitutions and simplifications are made:
A~ = ecTAmo~/Nh, cosA~ = 1 + O(At) 2, A~ sinA1 = O(At) 2 2 A 2 = (T/Nh)2(m20 c4  eCe20q e 2 c 2 ~Am0x) = O(At) 2
~l)k 322
_..
(eTr
= O(At) 2,
T = NAt
1 + iA1A3 " e iA1A3, 1  iA1A3 " e iA1A3,
(36) (37)
(38)
3.2 EVALUATION OF EQ.(3.132) 2 (cos A1 + A1 sin ~1
) COS
2r~
109
 ,~22 " 2 COS
27[t~
2(sin A1  A1 cos A1) sin  7
~2r~
(39)
= O(At)
(40)
The small value of Eq.(40) tempts one to leave out the fourth term with T~(0) in Eq.(34) and with S~(0) in EQ.(35). This is not justified mathematically since we do not know how large the sum of the remaining four termsincluding the inhomogeneous termis. From the standpoint of physics one would eliminate the coupling of the two processes represented by Eqs.(34) and (35). We may solve Eq.(34) for T,~(O) and substitute T,~(O), T~(O:hl) into Eq.(35) to obtain an ordinary, inhomogeneous difference equation for S.(8):
e 2v'~ ~'~S,~(8 + 2)  4 cos
 4 cos
2~e'~3S~(0+ 1) + 2
27r~e~'~x~S,~(O
1) +
e~~S,~(O
27rt~
= e~
4~~) 2 + cos ~
+ 1,~) + 2cos ~Gc~(8, ~)

S~(8)
2)
e~

1,~)
(41)
Similarly, we may solve Eq.(35) for S,~(0) and substitute S,~(8), S,~(0 4 1) into Eq.(34) to obtain an ordinary, inhomogeneous difference equation for T~(O):
 ~ T,~(0) e2i~'X3T,~(O + 2)  4cos 27rtCei'X~'X3T,~(O+ 1) + 2 2 + cos 47rtr
2~~X~T~(O 4 cos if= e~
1) + e2~T~(O
2)
27[g
+ 1,~) + 2cos~Gs~(8, ~) e~
1,~)
(42)
The homogeneous parts of Eqs.(41) and (42) are equal; their only differences are the subscripts c and s of the inhomogeneous terms. For the solution of the homogeneous Eqs.(41) and (42) we make the usual
ansatz
0
or
T,~(O)=d,~v~
(43)
and obtain an equation for v~:
e2i~1~3v~2 _ 4 cos ~
+ 2 2 + cos 
27[~s iAIA3 4 cos Nev~ 1 +
e2iAiA3
v~2 = 0
(44)
110
3 INHOMOGENEOUS DIFFERENCE EQUATION
This is a fourthorder equation. Because of its symmetry it can be reduced to two equations of second order:
27r/~ (ei)~l)~3v~ + e  i x ~ v 2 1 ) 2  4cos ff(e i x ' ~ v~ + e i~' ~ v21)
+2(1 +cos~
v~ + e i~l~av~l) = 2 cos ~ 27rg
= 2cos "N
•
~) = 0
(45)
4 cos 2 if  2 1 + cos double root
(46)
The second quadratic equation
ei~'IX3v,~ + eiXl)'3v= 1 
27rt~
2cos if = 0
27r~ ~ e2iX~x3v~  2 cos if e i x~v~ + 1 = 0
(47)
yields two single roots
2~'~ I cos 2 if2~'~ 111/2
e ~1~3v~ = cos if • v~l = e i~lx3
(
v,~2 = e i9'1"~3(
cos 2r~ 7 + i sin 27rS ~ )
= e_i~,i X3e2,~i,~/N
(48)
2zrtr 27rtC)=e_i,Xl,X3e_2~ri,~/g cos   ~  i sin if
(49)
and we obtain for v~0 in Eq.(43) the first two solutions:
o V~l
=ei(2,,~/N~l~3)O=cos
0 V~2
=ei(2,~,~/N+X~3)O=cos
~,kl)~30+isin "~'+)~1)~30isin
~,~l,~a 'ff+)~1,~3
The double root of Eq.(46) calls for two more solutions:
(50)
0 0
(51)
3.2 EVALUATION OF EQ.(3.132)
111
OV01 = Oei(2~r~/N~l)~3)O
: 0[ o:
1
(52)
Ov02 : Oei(2~r~c/N+AiA3)O
 0[COS ( 27r~ ~ + A1A3) 0  i.sin ( 27r~ if+ The general solution S,~(O) in
AIA3)0]
(53)
Eq.(43) becomes:
02 + c,~30v,r01 + C~40V02 S,~ (0) = c,r v,r01 + c,~2v,~ = Ctclei(27r~c/NAIA3)O _}. C,r "~ C~30ei(2r~
For T,r
A3)O Jr" C~40e i(27r~/N+k1A3 )O
(54)
we obtain from Eq.(43)" 0 + d,c4Ov,r02 T,~(O)  d,c~v,r01 + d,~2v,~02 + d, c3OV,r  d~l ei(27r~r AxAs)0 + d~2e  i(2~r~/N+Jk11k3)0 (55)
'} d~3Oei(2~r~'/N)~lAz)O + d~r
We turn to the method for this task available books 1 but c~ and d~, with i c~i(O) and d,~,(O)'
inhomogeneous solution of Eqs.(41) and (42). A general goes back to Lagrange (17361813). It is discussed in the we present it in Section 6.3 in more detail. The constants t, 2, 3, 4, in Eqs.(54) and (55) are replaced by variables
s~(0) = ~(0)~o~ + ~(0)~o~ + ~ ( 0 ) 0 ~ o + ~(0)0v% T,r
0
0
0
0
 d,r (O)v,r 1 + d~2(0) v~2 "+"d,c3(O)Ov,ct + d,c4(O)Ov,r
(5.6) (57)
Equations (6.323) and (6.330) define d,~i(O) and c,~(O) in a form suited for computer use. The solution derived here for Eq.(3) is extremely general. The only significant requirement is that the functioa Gt (~, 0) of Eq.(4) permits a cosine and sine transformation in terms of r according to Eqs.(32) and (33). Convergence of the Fourier series of Eq.(31) is not required since ~ does not approach infinity but only i ( N / 2 1). 1N6rlund 1924, p. 396; 1929, p. 22, 125; MilneThomson 1951, p. 374.
112
3 INHOMOGENEOUS DIFFERENCE EQUATION
Substitution of Eqs.(56) and (57) into Eq.(21) yields the particular solution of the inhomogeneous equation (14). We must still satisfy the boundary and initial conditions of Eqs.(ll) to (13). Using Eqs.(2.319) to (2.321) and Eq.(15) we obtain the following boundary and initial conditions for 9(r 0) from Eqs.(ll) to (13):
9(0,0)=0
forO>_O
(58)
~3(~,0)=0
forr
(59)
for(>O
(60)
~(~,1)~3(~,0)=0
The boundary condition of Eq.(58) is satisfied if we discard the first term in Eq.(21)" N/2 92
9 O) =
E ,~=N/2+l
(61)
T,~(8)ei)~; sin 27r~r N
The initial condition of Eq.(59) is satisfied for the third and fourth term of T~ (0) in Eq.(57) due to the factor 0. The first and second term have the coefficients d~i(0) for 0 = 0' N/21
~(r 0) =
27r~r
Idol (0) + d~2(0)]e ' ~ s i n N
E
= 0
(62)
,~=N/2+l
From Eqs.(57) and (61) we get with Eqs.(48), (49), (60), and (59) the relation
N/21
'b(#, 1) =
E
{[d~1(1) + d~3(1)lval + [d~2(1) + da4(1)lv~2}
~=N/2+1
x e ixlr sin 27r~ = 0
N
e
N/2t ( ~
[d~ (1) + d,~2(1) + d,~3(1) + d~4(1)] cos
27r~N
~=N/2+1
+ i[d~i(1)  d~2(1) + d~3(1)  d~4(1)] sin 2N~)eiXl~ sin We multiply Eq.(62) with Nle thogonality interval 0 > h/moc
119
~oo 0~(~,00 0) 0~(~,000) _. ~20 [ _ 2iAIA3e2i'x~'xa0 x
Nt21 ~ [JR(O,~) + iJI(O, t~)]exp[(A~  A2)1/2~] sin 27r~_~ N ~r N[21 
2A1Aaei'X~x3(~ (
Z
IT(t~/N)sin fl,~ (A1A3sin fl,,O
toN/2+ 1
 ifl~ cos fl,~O)[JR(O,~) + iJI(O ~)] sin 2 27rt~r '
N/21,~tv
+
N
N/21
~
Z
tcN/2+l v=N/2+l
Iw(x~/N) [JR(O, v) + iJi(O, v)] sin fl~
2~rt~r 27rvr x (A1A3sinfl~0  ifl~ cosfl~0) sin 'N sin N JR(O, u) = JR(O, ~), Ji(O, v).= J~(O,~) for t~ ~ v
(22)
As in Eq.(20) we integrate this expression over r
N
/ 0 ~ (~, 0)&)(~, 0) ~oo. O0 ~ d ~ = ~ 2 o
(2
N/21
AIA3 Z
0
{[JR(O,~)+iJI(O,~)]sin2A1A30
~:=N/2+1
+ [Ji(O tc)iJR(O ~)]cos2A1A30} (27r~/N){1exp[(A2A2)l/2N]} , , A2_ A~ + (2zr~/N) 2 gl2a  A1AaN Z IT(x/N) {[JR(0, t~)+ iJi(O,t~)] sin fl,, ~r X [A1A3sinfl~OcosA1A3(O  1) kfl~ cosfl~OsinAtA3(O  1)] 

x [AIA3sinfl~0sin A1A3(0 1 )  fl~ cosfl~0 cos A1A3(0 1)]}~ /
(23)
Turning to Eq.(10) we recognize the need for the term 0~/0(. Equation (t7) yields:
120
3 INHOMOGENEOUS DIFFERENCE EQUATION
0r
0)
ei)~(~)'3~
0~
N/21 E [Jr(0, ~) + iJi(O,m)] ~;=N/2+l
x
cos N
+iAlsin 27r~N)
(24)
The term (0~/0~)(0~/0~) in Eq.(10) is obtained with the help of Eqs.(2.57) and (2.58):
~ooOV~(r o~(r o) : or
or
N/21
kI/020( _ 2[/~1 _ i(,~2 __ ~2)1/2]
27rt~exp[_(A2_A2)l/2r ] cos 27r~r g [J,.(0, t~) + iJi(O,~)] if
~=N/2+1
+ iAlexp[(A 2  A2)l/2r
+ 2iA1A3ei)~)~3
sin/3,,
• A2 sin2 N/21,ytu
E
sinA1A30
N/21 Z IT(t~/N) [Jr(0, ~)+ iJi(O,t~)]sini~O ,~=N/2+I
+ 2A1A3eixla3
g
+
cos 2
N/21
IT(~/N) [Jr(O u) + iJi(O, u)] sin/3~0
E
~=N/2+1 u=N/2+l
sin/3~
x ( A1 sin 27r~ N + ir;~ 2~~ iv cos 2~~{) N (~_~ cos 27ru4 N + i A1 sin 2~u() N )
(25)
This expression is integrated over r following Eqs.(20) and (23)' N
~oo
f ov~(r o) ~o~(r d ~ " o) 0r
0r
= ~20 { 2A2 sin A1A30
0
N/2t
• ~
~'U/2+l
(27rt~/N){ 1 exp[(A 2  A2)1/2N] } /~2 __ A2 + (2~~/N)2
[J,(O,,~)~Jr(O,,~)]
N/21
NA1A3
E ~=N/2+l
 i[Jr(O, ~) cos A1A3+ Ji(0, ~) sin A1A3]} Iw(~/N)sin/3~sin/3~0}
(26)
3.3 QUANTIZATION OF THE SOLUTION FOR Ax >> h / m o c
121
Let us turn to the text following Eq.(2.511). In order to allow for the generalization of the terms of Eq.(2.53) by Eqs.(8)(10) we define the energy U by the sum of the following three components
0
= ~1 + 02 + 03 = 0c +
Ov(O)
(27)
where 01 to 03 are obtained by the substitution of Eqs.(8)(10) into Eq.(2.53). We use the notation U to Uv(0) to distinguish the terms from the approximations U to Uv(0) in Eq.(2.512): N
N
L2 m2c4(At)2 ] qJ*~dr = L2 m2c4(At)2 ((1 + 2 a ) / ~ ; ~ o d { U1 = cat ri,2 cat li2 0
0 N
+ 2c~oo / ~Re(~?)d~)
(28)
0 N
N
02
L 2 / 0~* 0~  o0   d eo0 = ~
= ~
(1 + 2c~)
00
0O
0
0
N
+ 2C~$oo/ J~e( 0 ~ 00 oo
(29)
0
L2 /
N
0a:~~
09* 09
~~dr
or or
0
L2
N
= ~~ (1 + 2c~) f 0 9 ; 0~I,_____sd~ or or 0
N + 2olli/00/:~e (01I/~ 0~
(30)
0
As in Eqs.(2.513)(2.515) we want the timeinvariant part ~rc " brcl + ~rc2 ~ gc3
(31)
of U1, U2, U3 and we ignore the timevariable part Uv(0). Again we write Uc to U~3 to distinguish the terms from the approximations Ur to U~3 in Eq.(2.516). When deriving Eqs.(2.513)(2.515) we simply left out terms containing sin ~ 0 or cos fl~0. The time variations of Eqs.(28), (29), (30) are not so obvious and we must write integrals over 0. Using Eqs.(2.513)(2.515) we may write the time invariant part of Eqs.(28)(30) as follows:
122
3 INHOMOGENEOUS DIFFERENCE EQUATION
0~1 =
UldO = cAt;
h2
(1 + 2ol)~2oNA2A32
0 N
U:2 =
j
sinDI.
N
,,>
,:.( Iw(~lN)si):n/7, 0
"{
U2dO= ~
0
Z ~=Nt2+l
(1 + 2,)tIs020NA~A2
0
(~,,X,'' + n,)'
Z
~N/2+.l
N
N
(33) 0
i
"(
0~= 0 03dO=7A7 (I+2~)@~176
0
"~ '" N
0
tIT(a/N)si7[n/7"~+= ("'71 N N
0
We recognize on the right side of Eqs.(32)(34) the energies Ucl, Uc2, Uc3 of Eqs.(2.513)(2.515) multiplied by 1 + 2c~ ' 1.0146. This difference of 1.5% is barely visible in a plot. The interesting terms are the integrals multiplied by 2o~00. We shall analyze them in the following Section 3.4. A comparison of Eqs.(31) and (2.516) shows that the only difference is the symbol ^. Hence, we may use the results of Section 2.5 from Eq.(2.516) on. The energies E~ of Eqs.(2.532), (2.534), and (2.548) are obtained. One could write F.~ instead of E~ and obtain the result F_~  E~, but there is little incentive to do so. 3.4 EVALUATION OF THE E N E R G Y Uc
We start from the energies De1, Uc2, Uc3 in Eqs.(3.332)(3.334). Their first terms equal the energies Ur Ur Ur in Eqs.(2.513)(2.515), except for the factor 1 + 2c~. The results of Chapter 2 apply to these terms. Here we are interested in the second terms that are multiplied by 2a"
L2 O~ = 2a~oo~~
( m~
2
N
N
(1) 0
0
3.4
0~2 =
123
E V A L U A T I O N OF T H E E N E R G Y 0 c N
N
0 N
0 N
0
0
2~9oo~
O0 O0
9~e
L2
O~ o~)d~]dO
(3)
The sum Uac of these terms is ultimately wanted as function U ~ ( a ) of as shown by Eq.(2.516) N/21
0~=0~i
+0~2+0~3
=
~
0.~(~)
(4)
,r
N/21
ooi: ~
N/21
N/21
a~(~), oo~: ~
.= N/2§ ~
o~(~),
FI0.3.48. P l o t of 9~2~(~) according to Eq.(39) for N  100, r 1OAt, AI = 0.i, A2 = 40, As = i0000. 40
20
,
I
0
,
20
,
2
40
,
9
4
 10A1A3, Aim 
,
9
o9
9
~o
9
9
v
o ........ ..
. .................... 
.,.,..
2 4
40
 0
0
20
40
F I c . 3 . 4  9 . P l o t of 9~3~(~) according to Eq.(45) for N = 100, r IOA1, A1 = 0.1, A2 = 40, A3 = 10000.
= 10At),z, Axm =
3.5 PLOTS FOR SECONDORDER APPROXIMATION
135
replacing an infinite interval by an arbitrarily large but finite interval. For instance, sinusoidal functions that fit into a finite interval have the form sin 2~nO or cos 2~n0 with n = 1, 2, ... , while there is no restriction on n in an open or half open interval. A restriction of eigenvalues is of great interest in physics, but it is too early to tell whether this particular restriction leads to results of practical interest. We shall explore the matter some more in Section 3.5 with additional plots but we refrain from drawing conclusions at this time. 3.5 PLOTS FOR SECONDORDER APPROXIMATION We are presenting 18 more plots to show the effect of variations of r Aim, A1, ,k2, and A3 on 9{1~(~) to 9{3~(~). The parameter N  100 remains unchanged from Figs.3.41 to 3.49. Figures 3.41 to 3.43 are shown with the electric potential r reversed from +10A1A3 to  1 0 A l ~ 3 in Figs.3.51 to 3.53. We obtain essentially the amplitude reversed plots, but the vertical scales of Figs.3.42 and 3.52 are also changed. In Figs.3.54 to 3.56 we see the plots of Figs.3.41 to 3.43 with the magnetic potential Aim reversed from +10A1 to 10A1. We obtain essentially the same plots as in Figs.3.41 to 3.43. Again, the vertical scales of Figs.3.42 and 3.55 are not the same. A closer inspection reveals other differences between the two plots. In Figs.3.57 to 3.59 both r and Aim have a reversed amplitude compared with Figs.3.41 to 3.43. The plots of Figs.3.57 to 3.59 appear at first glance to be amplitude reversed plots of Figs.3.41 to 3.43 but a closer inspection of Fig.3.58 shows differences for + n = 1 5 , . . . , 20. In Figs.3.510 to 3.512 the parameters ~1 = 0.05 and ~3  10000 are changed compared with Figs.3.41 to 3.43. The plots are changed more drastically than by the changes of r and Aim. In Figs.3.513 to 3.515 the parameter A1 is reduced from 0.05 to 0.02 compared with Figs.3.510 to 3.512. The plots of Figs.3.513 and 3.515 look more like those of Figs.3.41 and 3.43 rather than those of Figs.3.510 and 3.512. If we look at the products A1A3 = 100 in Figs.3.41 to 3.43, A1A3 = 500 in Figs.3.510 to 3.512, and A1A3 = 200 in Figs.3.513 to 3.515 we recognize that the product )~1A3 is important for the similarity of the illustrations, as suggested by Eq. (2.413 ). Finally, we show in Figs.3.516 to 3.518 the plots of Figs.3.41 to 3.43 with the changes A1 = 0.01 and )~3 = 10000 instead of A1 = 0.1 and A3 = 1000. The product A1A3 = 100 is the same in both cases and the illustrations look very similar. However, a closer inspection shows differences in Figs.3.42 and 3.517 around + ~ = 20. We must once more ask the reader for patience until Section 4.4 for an explanation why we show so many plots here without deriving any conclusions from them.
136
3 INHOMOGENEOUS DIFFERENCE EQUATION 40
20
0
20
40
i0 T
5
"J
0
'.,
I  IRBB~OBLDB
9
9 ....
:__==__
_
.o"
0
_= ....
5 I0
'4o
" ~o
o
2'o
4'o
FIG.3.51. Plot of ~ 1 ~ ( ; ' ; ) according to Eq.(3.428) for N  100, r Aim = 10A1, A1 = 0 . 1 , A2 = 10, A3  1 0 0 0 . 40 ,

20
0
,
20
,
,
.

10A1A3,

10)~1)~3,
=
I0~3,
40 ,
60000
40000
T
20000
0
9
9
.
~
...;....

20000 40000 60000
~o FIG.3.52. Aim
~o
o
;o
~o
Plot of 9f2~(~) according to Eq.(3.439) for N  100, r
 lOA1, A 1 = 0.i, A2 = I0, A3  I000. 40 150
T
,
20 ,
0 ,
20
40
,
.
i00 50 .o. 9 ~
000.0
%~
,*~
50 i00
F~c.3.53.
~o
~o
~
I
Plot of ~3~(~) according to Eq.(3.445) for
Aim = 10A1, A1 = 0 1 , A2 = 10, A3 = 1000

~o
~o N
=
I
100, r
3.5 PLOTS FOR SECONDORDER APPROXIMATION 40
20
,
0
,
20
,
137
40
,
,
io
v
T
5 ;;. o,,,,o,,
9 9
~5
::::::: .... 9
.o
o
""" 9
9
i0 I
K;
I
i
I
~
Fic.3.54. Plot of ~ 1 ~ ( ~ ) according to Eq.(3.428) for N = 100, r A~m = 10A1, A1 = 0.1, A2 = 10, A3 = 1000. 40
20
,
60000
0
,
20
,
= 10A1A3,
40
,
.
40000 I
20000
"~
o
99 0000 9 Og 9 ,
,"
20000
gO 9
9
9
~
ODOOoQIo
,
,
.
9
,
",
40000 60000
ko
o K;
Fio.3.55. Alm
!
i
20
'~
P l o t of 9~2~(~) according to Eq.(3.439) for N = 100, r
= 10A~A3,
= 10At, AI = 0.I, A2 = I0, As = i000.
i00
f
40
20
0
20
40
5O o
".~
0 9 ~.~vOg U 9 vIVlOo0 O
e.Oto
50
I00 150
.. 9 ,,..
o'" 9
9
i
20
o
!
20
4o
K,     ~
FIG.3.56. Plot of 9~3~(~) according to Eq.(3.445) for N = 100, r Aim = 10A1, A1 = 0.1, A2 = 10, A3 = 1000.
= 10A1A3,
138
3 INHOMOGENEOUS DIFFERENCE EQUATION 40
20
0
20
4O
i0 5
o
,
o
9..
::'........
9
.
0 9
.. 9
~5 i0 i
io
' 20
;
9
2'0
i
40
FI0.3.57. Plot of 9Q~(~) according to Eq.(3.428) for N = 100, r Aim = 10A1, A1 = 0.1, A2 = 10, A3 = 1000. 40
20
,
0
,
20
,
=
10/~1,~3,
40
,
,
40000 9
T
20000
.
9
9
9
9
.
. o
".. 9 9
....
_::__:
:::__:::
9
..
.
....
,
. 9
9
. 9
.,,
2oooo 40000
;,o
;o
2'o
o
/,;, .)
4o
FIG.3.58. Plot of 9C2~(~) according to Eq.(3.439) for N = 100, r Aim =  1 0 A 1 , A1 = 0.1, A2 = 10, A3 = 1000. 4O
20
0
20
40
,
,
l
,
= 10A1A3,
150
T ~, N
I00 ".o"
50
~
o
o**
50 I00
;,o
~o
o
~'o
;o
FIG.3.59. Plot of 9C3~(~) according to Eq.(3.445) for N = 100, r Aim = 10A1, A1 = 0.1, A2 = 10, A3  1000.
= 10A1A3,
3.5
PLOTS
FOR
SECONDORDER
40 .
.
.
.
20 ..
0
.
139
APPROXIMATION 20 ,
40
.
,
1500 i i000 50O ~
N
9
0
9
......
9
9
9
9149
....
::
,
500 I000
 4 0
&
 2 0
III
210
4I0
K; ~
FIG.3.510. Plot of 9Q~(P+) according to Eq.(3.428) for N  100, r Aim = 10At, A1  0 . 0 5 , A2 = 10, A3 = 10000. 40 .
T
20
.
0
.
20
.
= 10A1A3,
40
2x 10 8 9
9
9
2x I08
9
. . . . . . . . . __+
9
,, 9
.
o
9
9
4x I08 I
t
K; *
FIG.3.511. Plot of 9~2~(P+) according to Eq.(3.439) for N = 100, r Aim
= 10A1A3,
 10At, AI = 0.05, A2  I0, A3 : i0000. 40 ,
20 ,
0 ,
20
,
,
40 ,
500
t
o,
500 i000 ,,,
FIG.3.512. Plot of Aim :
4i
..... 0
~C3~(g)
10A1, A1 : 0.05, A2 :
,
m
20
0

20
,
according to Eq.(3.445) for N : 10, A3 :
10000.
l
40
100,
r

10]~1]~3,
140
3 INHOMOGENEOUS 40
20
..
,
DIFFERENCE
0
,
EQUATION
20
,
40
,
,
2O
T
$
0
v
........
o9 9
9 9
9
20
o **,* 9
,
,
,,o
9
40
~o
~o
o I
/~
+
20
40 i
F i o . 3 . 5  1 3 . P l o t of ~~I~(E) according to Eq.(3.428) for N A~m  10A1, A1 = 0.02, A2  10, A3 = 10000.
20
40 i.
20
 10AIA3,
40
5x 10 6 Ix
I
0
100, r
10'
500000 o
,,.,,,.
.
500000 lx
10 6
I. 5x
106
. . . . . . . .
.,.,,.,.
9
9
9
~o
o
9
~o
o
~o
.o
Fz0.3.514. P l o t of 9C2~(~) according to Eq.(3.439) for N  100, r Aim  10A1, A1 = 0.02, A2 = 10, A3 = 10000. 40
20
,
,
0
20
40
,
,
,

10A~A3,
100 l
9
9
" 9149149149149
N 100
.
.
" "..
9149 .o*'"
9149
9
200
;o
;_o
!
i
,
o
~o
~o
F i e . 3 . 5  1 5 . P l o t of ~3,~(~) according to Eq.(3.445) for N = 100, r Aim = 10A1, A1 = 0.02, A2 = 10, A3 = 10000.
= 10A1,%,
3.5 PLOTS FOR SECONDORDER APPROXIMATION 40 ,
20
.
0
,
20
,
,
141
40
.
,
I0 5 o
~
= =j.o 9 o Q ~ 1 4 9 1 4 9 ...
.
.
"%
9
[g
9
5
9
I0
9
9
I
i
! 40
I 20
&II
2[ 0
4 0
H;      ~
FIG.3.516. Plot of 9 Q , ( , ; ) according to Eq.(3.428) for N = 100, r Aim = 10A1, .kl = 0.01, A2 = 10, As = 10000. 40
20
0
20
= 10A~A3,
40
40000 20000
T
o"
9
.
"0
U,
e 9
20000
40000 I
40
2o
o
/~ +
I
2o
40
FIC.3.517. Plot of 9(2,(~) according to Eq.(3.439) for N = 100, r Aim  10A1, A1 0.01, A2 = 10, As = 10000.
i00
T
40
20
,
0
,
.
20
= 10A~A3,
40
,
50 ~ Q90O 999
00000 9 ~ o.O"
50 I00 9
150
~o F~G.3.518. Aim
9
~o
i
o
I
20
I
40
K; "~
Plot of 9[a,~(~) according to Eq.(3.445) for N = 100, r
= 10A1, A1 0.01, A2  I0, A3 = I0000.
= 10A1Aa,
142
3 INHOMOGENEOUS DIFFERENCE EQUATION 4
3 .....dj
a
4 ", ~
~ 2
"" 1
"~" 2 "~
'
'
o
(~
+ 89
"'"" "'~
1[
L
!
I
J
~+1
~+2 ~ g:+l ~+2 ~+3 ~ .4 ~ ~ FIG.3.519. First order (a) and second order (b) polynomial approximation of a physically not defined amplitude 2/0 of 9{(~) = 2 / ( ~  ~) at ~ = ~ . The plots in Figs.3.41 to 3.49 and 3.51 to 3.518 assume values tC/O = 11/0 for certain values of ~. An infinite amplitude is no more observable than an infinite distance or time. According to Section 1.4 an infinite amplitude implies infinite information, which cannot be observed, processed, or transmitted. A result 1/0 must be treated as physically not defined just as a result 0/0 must be treated as mathematically not defined. Consider the function 9{(~) = 2 / ( ~  ~) that equals 2/0 for ~  ~. We may replace 2/0 by 3 obtained by the firstorder polynomial approximation of Fig.3.519a, or by 11/3 obtained by the secondorder approximation of Fig.3.519b. Going to higher order approximations one gets larger and larger results for 9{(~). But for ~ :> 0 we have N / 2  1 values ~ = 1, 2, . . . , N / 2  1. Let k of them have the value 1/0. A polynomial expansion of order N / 2  1  k will yield k finite values instead of 1/0 from the N / 2  1  k physically defined values. We see here that a finite number N = T / A t of space intervals Ax or time intervals At produces a finite amplitude g{(~,) if the physically not defined value t / 0 is made defined by a polynomial approximation. O b t a i n i n g the equivalent of the constants 3 and 11/3 in Figs.3.519a, b for polynomials of high order is not difficult for a computer. Hence, we have found a very general and powerful method to avoid infinite amplitudes in physics if the calculus of finite differences is used. We do not want to discuss it any further until information about the limitations and drawbacks of the method becomes available.
ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 137
4 KleinGordon
4.1
Difference
Equation
EVALUATION OF THE DIFFERENCE
for Small
EQUATION
FOR
Distances
Ax ~
h/moc
In Chapter 3 we solved a secondorder difference equation based on Eq.(3.110) that held for Ax large compared with the Compton wavelength h/moc. The physically more interesting but mathematically more difficult case is Ax '>h/moc and Ax h/moc. Here we shall derive the corresponding equation for Ax 0
(37)
fore>0 for r > 0
(38) (39)
The boundary condition of Eq.(37) is satisfied if we discard the first term in Eq.(19)"
V(~, O) =
N/21 ~
9 27r~ T~(0)e '~1~ sin
(40)
,~=N/2+I
The initial condition of Eq.(38) is satisfied for the third and fourth term of T~(0) in Eq_(36) due to the factor 0. The first and second terms have the coefficients d,r for 0 = 0:
v(r 0) 
N/21
E Idol(0) + d~2(0)]e ix1 r sin tc=N/2+l
27r~{ = 0 N
(41)
From Eqs.(36) and (40) we get with Eqs.(3.248), (3.249), (39), and (38) the relation
154
4
KLEINGORDON DIFFERENCE EQUATION FOR SMALL DISTANCES
N/21
9(~, 1) =
E
{[&1(1) + &3(1)lv~1 + [&2(1)+ &4(1)]v~2}
~=N/2+1 x e i'xlr sin 2~'n~ = 0 N N/21
~ 27rn [d~l(1) + &2(1) + &3(1) + &4(1)] cos N
__ eiA1A3
~=N/2+1
+ i[d~1(1)  &2(1) + d~3(1)  d~a(1)] sin ~  ~ ) e ~'x'r sin 2~n~ N = 0
(42)
Since Eqs.(41) and (42) equal Eqs.(3.262) and (3.263) if ~) and d~i are replaced by ~ and d~i we may skip from Eq.(3.263) to Eq.(3.272) and replace Eqs.(3.272) to (3.279) by the following equations: &2 =  & l
d~1(1)
n0~l(0 ) [ 0~1
=
=
(43)
b~1(1____.~)+ D~o(1)
d~3(1) = Ad~3(0)+ d~3  /)~3(1) ~

d~3 = d~l 
+
o~1
(44)
d~a
(45)
b ~ l ( 1 ) + b~3(1)
D~o(1)
(46)
d~2(1) = Ad~2(0) + d~2 = b~2(1___~)+ s D~o(1)
(47)
d~4(1) = Ad~4(0)+ d~4 = b~4(1) d~4 (1~ D~o +
(48)
=


d~4 = d ~ l 
D~o(1) b~2(1) + / ) ~ 4 ( 1 )
D~o (1)
(49) (50)
We may now write the solution ~1(~,0) of Eq.(2). Starting with Eqs.(10), (3.29), (3.210), and (15) we obtain the following expressions:
4.3 QUANTIZATION OF THE SOLUTION FOR A x ~ h / m o c
~1 (r O) = ~oo{ (1

e 2~176 exp[(A 2

155
A2)l/2~]ei~; + u(4, O) + v(r O) } (51)
N/ 21 u(r 0) = 2iAIA3 e'*xl((+,xa0)
'
27~ar I T ( a / N ) sin
E
sin fl~
~=N/2+1
IT(a/N)
see Eq.(2.429);
~(~, 0) =
fl,~Osin
N
(52)
A1, A2, A3 see Eqs.(27),(28)
g/21 27r~r E T~(0)e i~1r sin ,~=N/2+1
(53) ~
(54)
v~l, v~2 see Eqs.(3.248), (3.249) d~i(0)
see Eqs.(6.411), (6.428)(6.431)
(~2, d~3, (~4
see Eqs.(43), (46), (50)
Comparison of Eq.(51) with Eq.(2.432) shows that ~. 1(~,0) can also be written in the following form'
~,~(r o) = ~o(r o) + ~,oo~(r o) N/21
~(r
=
([~1(o) + os
Z
'(~/~~'~~
~=N/2+l
+ [d,~2(0) + Od,~4(O)]e'(2~'~/N+~'~'3)~ e v'~r sin 2 ~ r N
4.3 QUANTIZATION OF THE SOLUTION FOR A X ~
(55)
h/moc
We follow Section 3.3 but are careful to put a tilde  or replace a hat ^ by a bar  where appropriate. Equation (3.31) becomes:
Equations (3.32) to (3.34) assume the following form:
~,*~, = ( ~ + ~ , ; ) ( r
+ ~,1) = r
+ ~(r
+ ~,;r
+ o(~ =) (2)
156
4 KLEINGORDON DIFFERENCE EQUATION FOR SMALL DISTANCES
o~,* o~, O0 O0
o~,; OVo)
or30Vo O0 O0 F a
o~,* o~,
o ~ O~o
or or
or or
00
O0
O0 O0
( o~; o~
(3)
+O(~
o~; O~o) or or +~
(4)
The function ~o(~, 0) of Eq.(3.35) remains unchanged
~o(~, 0) =~oo[(1  e2ix~~
+ u((, 0)]
=~oo ((1  e 2/x~xa~ exp[(A~  )~12)1/2~]e iAz~
N/21
IT(x/N)sin/3~0 sin 2 ~Nr )
2iA1A3eiA1)'3e~)~l(ei)'l~a~ E
~N/2+l
sin/3~
(5)
but ~1 (r 0) of Eq.(3.36) becomes ~1 (~, 0) according to Eq.(4.255)"
~1(~, O) ~ t~O(~, O) "[~00~)(~, O) N/21
~(~,o) =
~
(6)
([a~(o) + o ~ ( o ) ] a ( ~ . ~ / N  ~ ) o
aN/2+l + [(~g2(0) } Odgn(O)]ei(2~r~/N+)~13)O)e iAzr sin
N
(7)
We may use Eq.(6) to rewrite Eqs.(2)(4). The notation :Re(... ) is used for the real part of the expression in parentheses:
~'*~' = ~;~o + ~[2V;r + ~,oo(V;~ + ~*Vo)] = ~;~o + 2 ~ [ ~ o + ~oom~(v;~)]
o~,* o~, F2a
00
00
F~oog~e
00 00
(9)
= 0~ 0~ ~2a
0(
0r
~oo~e
0r 0~
(10)
00 00 = 00
o~* o~ 0r or
(8)
00
We have again the terms ~ o , (Og2~/O0)(O~o/O0),as well as ( 0 ~ / 0 r x (0~o/0r obtained in Eqs.(2.54), (2.59) and (2.511). The terms ~ , (0~;/00)(0~/00), and (0~;/0~)(0~/0r Eqs.(8)(10) must be calculated with the help of Eqs.(5) and (7). First we show how ~(4,0) depends on G2(r 0) of Eq.(4.22). This starts with/~s~(0, ~) of Eq.(6.41) and (~s~(0, ~) of Eq.(4.224)"
4.3 QUANTIZATION OF THE SOLUTION FOR A x > ZecrOAe~/i ~ and
(1 
V2/C2) 1/2
>> Z ecr sin 0 ~A~o / § (16)
For small values of ~' but not rO and r sin ~ ~b we require alternate conditions for Eq. (8): Amr >> ZrOAev/c
Amr >> ZersinO~Ae#/C
or
(17)
Equation (16) states in essence that the energy due to the potential Ae should be small compared with the energy moc2/(1  v2/c2) 1/2 whereas Eq.(17) demands that the magnitude of A~ should be small compared with the magnitude of Am. With these simplifying assumptions we obtain from Eqs.(8)(10): m0§ +eAmr (1  v2/c2) 1/2
=
Pr
(18)
m0r0
Po = (1V2/C2) 1/2 + eAmo m0r sin 0 ~b
P~o =
v2/c2) 1/2
(1 
+ eAm~o
(19) (20)
Solution of these equations for § r0, and r sin 0 ~b yields" + =
(1 
v2/c2)1/2
(p  eArn),.
mo r~
=
(1 
v2/c2) ~/2 (p
 eAm)o
(22)
v2/c2) 1/2'" (p  eAm)~
(23)
m0
r sin 0 ~ = "(1

(21)
mo
Squaring and summing § tO, r sin ~ ~ yields:
§ + (r0)2 + (r sin ~ ~b)2 = v 2 = 1 (p  eArn) 2 =
m2ov2 1  v2/c 2 =m~

v2/c 2 m2 (P  earn) 2
( 1  v 21/ c 2  1 )
m 0c2
mo C2
(1  v 2 / c 2 ) 1/2
1  (+2 + r2~2 + r 2 sin 2 ~ ~2)/c2 ] 1/2 = c[(p  eArn) 2 + m2c2] 1/2
(24)
5.2 RELATIVISTIC MASS VARIATION
193
The last line of Eq.(24) is substituted into Eqs.(12)(14):
~I~r  c[(p  eArn) 2 b m2c2] 1/2 "~ eCe  ~'~cr
(25)
~ 0  c [ ( p  eArn) 2 + m2c2]1/2 4ceCe  ~cz9
(26)
~
(27)
= ~[(p  ~ A ~ ) ~ +  ~ ] ~ z ~
+ ~r
 ~c~
If we leave out the correcting terms Lcr, Lc#, Lc~ we have the conventional relativistic Hamilton function for an electrically charged particle in an EM field. The assumption we had to make to obtain Eqs.(25)(27) was that A~ must be sufficiently small. If one wants to leave out the correcting terms Lc~, Lc~, Lc~o one gets more complicated conditions than Eq.(17) since Eqs.(1.333)(1.335) contain Cm and its derivatives in addition to A~. Let us turn to the solution of Eqs.(8)(10) for § r0, rsin0~b without simplifications. The term (1  v2/c2) 1/2 = [1  (§ + r2~2 + r2sin 2 ~(02)/c211/2 makes these equations nonlinear while the corresponding Eqs.(5.12)(5.14) of the nonrelativistic theory were linear. There is no standard method for the solution of a system of nonlinear equations and we must find a method suitable for the case at hand. As a first step we ignore that v2.is a function of § r0, rsin0~b and treat Eqs.(8)(10) as a system of linear equations. We note that Eqs. (5.12)(5.14) are transformed into Eqs.(8)(10) by the substitution .~ ~
t o o ~ ( 1  v 2 / c 2 ) ~/2
and we get our 'first step of solution' of Eqs.(8)(10) by making the same substitution in Eqs.(5.16)(5.19). The common denominator D of EQ.(5.16) assumes the following form:
D
he=
__
ZecA~ "~0~ ~ '
I (v2)
m~ 1 + c ~2 1 ~~ (1 v2/c2) 3/2 a~
(V2) 1
1/2 ~
=
ZecA~ , ~ 0 ~ / ( 1  v21c2) li2
(28)
(29)
The constant oLe represents the ratio of the energy due to the electric vector potential Ae and the rest energy of the particle. The reference energy should actually be moc2/(1 v2/c2) 1/2 rather than moc 2 but we shall need v as an explicit variable. The term C~e has no physical dimension but it is a variable due to its component Ae. The solution of ~, r~, r sin 0 ~b as functions of Pr, P#, P~o has the following form if the factor ae of Eq.(29) is used:
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES
194
(1v2/c2)1/2 mo
1/2 [Ae • (p  eArn)It
[ (v2) (p  eAm)~ 1 + ae 1  ~
( 7) v2
21
+~~
rO  (1v2/c2)1/2 mo
A~(p  eAm)~
~  p  ~Am)~
][
[
1/2 [Ae • (p  eAm)]9
A~rA~.(p
eArn)
(v2)
(p  eAm)o 1 + O/e 1  ~ 2 1 _ v2
+OLe
~
mo
[
( )
V2 1/2 A~vA ~ (p
+O/e2 1  ~
7~
1 + a~2 1 _
Ae2(p eAm)~
(v2)
(p  eAm)~o 1 + ae
1
~e
(30)
A~(p  eAm)d
A ~ A ~ . ( p  eArn)
rsin~9~~ (1v2/c2)1/2
2
1+
1/2[Ae x (peAm)]~
1  ~
eAm)
Ae(p  eAm)~o 2
1+
Ae2(p  eAm)~
(31)
(
O/e 1  ~'~
(32)
Squaring and summing Eqs.(30)(32) yields: §
2sin 2 ~ @ 2 = v 2 _ 1  v 2 / c 2  rn~ (p  eArn) 2 1 ( +2ae
1~
v 2 ) 1/2 [Ae 9(p  eAm)] 2 Ae2(p_eAm) 2
2 /~ 1 _ V2'~ [ t e x ( p  eArn)]2 + 2 [ t e  ( p +c% ~~) Ae2(p _ eArn) fi
(
A3(p _ eAm) 2
+ 2a 3 1  ~
(
v2)2[Ae'(peAm)]2
4 1 + c% c'~
eArn)] 2
A~(p

[
eArn) 2
(
1+21 ae

v2)] 2
(33)
To find an approximate solution of this equation for ae(1 
v2/c2)1/2 0
h
= 0
(41)
for 0 < 0
= 900 sin
EAtO
for 0 > 0 (42) h T h e b o u n d a r y condition of Eq.(41) starts at 0 = 0 like a step function while Eq.(42) yields a linear increase. More complicated b o u n d a r y conditions can be represented by making E = En a function of n = 0, 1, 2, ... and using a Fourier series expansion. 5.5 SPATIAL DIFFERENCE EQUATION In the differential theory the equation corresponding to Eq.(5.439) can be written in the following form (Schiff 1949, p. 309; Messiah 1962, p. 884) if er is replaced by  Z e 2 / p and t~ is written instead of p. We have already defined p = r / A r in Eq.(5.44) and A r has no place in a differential theory. Hence, some other normalization factor has to be used and this creates a different normalized variable iS:
d
2 d
A
1
l(1 + 1 )  ' ~ 2 / /
moc(
Ze 2
=~,~=~j,~=2 F
u(/~) : 0
E2 )1/2, i = h ~2E~, z = l ,
1 .~4
2,...
(1)
T h e difference equation (5.439) can be brought into a similar form. Note that the hat ^ is replaced by a tilde ~. 1
[%,(p + 1)  2%,(p) + %,(p  1)] + p[%,(p + 1)  %,(p  1)] .
.
.
p
.
4 Ar
h/moc ( a=
2 1
2E~'
E2 ) 1/2 m2oc 4
'
l(1 + p2 1) ~2). u~, (p) ~ = Z Ze2
h/moc' ~_
~= moc25z '
7 = 1, 2, ... ; Z " 376.730 [V/A]; a = ~z d
= 0
  ~ = 4;rZa 2Ar[~
hc
Ar
' h/moc
__ 1
A~
(~ h/moc
fine structure constant
(2)
212
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES
The terms 5  A~l(lilmoc) = (~Arl(hlmoc), which express the length A ~  5 A r in relation to the Compton wavelength, are an important feature of Eq.(2). We cannot choose A~ = li/moc since we have seen in Sections 3 and 4 that Ax >> h/moc and, Ax 2  ~  2
A~ )2] hlmoc < " 0.69662
3
(20)
5 . 5 SPATIAL DIFFERENCE EQUATION
+i
217
+i
']
= 2
....
"
~
a
b
FIG.5.52. The convergence circle of Fig.5.51b around s3 reaches both 0 and s2 for g = 3/~/'2 (a). For smaller values of g the singular point s2 becomes unimportant since the circle of convergence is limited by sl = 0 (b). we have convergence of the power series along the integration line l of Fig.5.5la. In essence, as long as the spatial resolution A~ is larger t h a n the C o m p t o n wavelength of the particle we have convergence. For smaller values of AP t h a n defined by Eq.(20) we m a y use a second way t h a t requires conformal m a p p i n g to achieve convergence. T h e circle of convergence of the power series of Eq.(15) according to Fig.5.51b is m a p p e d onto a loop t h a t runs t h r o u g h the origin. Except for s = s3 and s = 0 there must be no singularity either inside or on the loop. T h e factorial series obtained in either one of these two ways do not necessarily converge in the whole interval 0 _ p < N unless certain conditions are satisfied. For a convergent power series we must have the following relations according to Eq.(20) and Fig. 5. 52a:
[2 )112
A~ &At .A moc( h/moc = h/moc = ~ , , r   ~ 1  m~c4 hr
hlmoc
>
0.69962
2
= 0.34981
> 0.69662
for [ ~ m0c 2
(21)
We thus get in Eq.(16) for p the values c~  7.297 • 10 3, P_/m0c 2 ~ 1, [m2c41(m~c4 [2)]2 9 1, and [Arl(hlmoc)] 2 > 0.122. Hence, p is a negative n u m b e r with small magnitude, IPl 0.69662
(23)
The notation O ( A r ) or O(Ar) 2 indicates terms multiplied with Ar or (Ar) 2 as well as with higher powers of Ar. These terms do not need to be small compared with the other terms of Eqs.(22) and (23). The recursion formula of Eq.(17) may be rewritten as a system of linear equations. Cramer's rule yields the coefficient q3(# + 1): q3(# + 1) q3(O)
(  1 ) "+1
H ~,1(. + 1) v'0
a3,0(O) 013,1 (0)
C~3,1(1) 013,0(1)
~,~(2)
a3,_l(v  1)
~,0(~)
(24)
a3,1 (v + 1)
a3,l(t~  2)
~ , 0 ( ~  1) ha,1 (t~  1)
~,,(,) a3,0(#)
All the terms aa,l(v + 1) for v = 0, 1, ... , # contain terms of order O(Ar) or higher. If we ignore these terms we obtain the determinant without terms of order O ( A r ) "
5.5
SPATIAL DIFFERENCE
219
EQUATION
It
q3(# + 1)
(1)"+1 H [(Po + v)(po + ~' + 1)  l ( 1 + 1) + ,~2] ~=0
q3(0)
H as,l(~ + 1)
tt
(25)
v~0
Let the last factor for u = # in the numerator of Eq.(25) be zero: (P0 + #)(P0 + # +
(26)
1)  l ( 1 + 1) + ~2 = 0
The # + 1 coefficients q3(0), Arq3(1), ... , (Ar)t'q3(#) are then of order O(1), while the coefficients ( A r ) ~ q 3 ( u ) for u > # contain only terms of order O(Ar). In this case the factorial series of Eq.(19) looks like a polynomial if one ignores terms of order O ( A r ) . The power series of Eq.(15) converges for A~ > 0.69962 according to Eq.(20) and the factorial series of Eq.(19) converges too. The finite upper summation limit N in Eq.(19) is of little consequence if N is made large enough. If the condition of Eq.(20) is not satisfied we get a divergent power series in Eq. (15). This does not mean that the factorial series of Eq. (19) diverges too but only that we are bumping against a limitation of the well workedout part of the calculus of finite differences. It is known that functions defined by certain divergent but summable power series are also defined by convergent factorial series (NSrlund 1924, Ch. 9, w3). With the change of notation #  n 1  1
(27)
we obtain from Eq.(26):
p0 =  n + 9
P01
1+ ~ •
~2
 n + 21 +~,
Let # run from 0 to n 
l+ .
P02
_ ~2 ~2
nt~2l+l,
n' = n
21  1
(28)
1. The index l then assumes the values 10,
1, . . . , n  1
We use the wave number ~ introduced in Eq.(23)
(29)
220
5 DIFFERENCEEQUATIONIN SPHERICALCOORDINATES
?Tt0C( _ ~E2/1/2 ._ ~77~0c ~ = ~
1
m~c4
83 
2
1
[
li '
( t~~0~a/2]1/2
F_ = moc 2 1 
~ mOc A r = 1  aAr, 2 h
83
(30)
" e teAr
and rewrite Eq.(16):
P = Po + O(Ar),
Po = "Y~ 1 
(31)
moc
Substitution into Eq.(28) yields with the help of Eq.(23):
( ~1 ~2
nh
1+ ~
9 Zym~ n'h
2 21 + 1
1
+
~ (
E.  moc 2
p2o
2l+1
"2
~/2 )  1 / 2 1  p2 + Zy2
E1 " moc 2 1
2n 2
n4
2l+1
8
[ ,2 ,~4 ( n, 3)] E2 " moc 2 1  ~ + ~  7 ~ 2/+1 + n'= n 21 1
(32)
The values shown in Eq.(32) for ~ and E are the same as obtained from the differential KleinGordon equation (1). The positive sign of the square root in Eq.(28) and thus the values ~2 and E2 are usually not considered in order to avoid poles in the eigenfunctions of the differential equation. The recursion formula of Eq.(17) yields the following coefficients q3(u) for the terms not containing Ar
q3(1)_ 1 (Po(Po+l)l(l+l)+~2 q3(0)  n a t 2(po + 1)
) + O(Ar)
5.5 SPATIAL DIFFERENCE EQUATION
221
q3(2) 1 /' (P0 + 1)(po + 2)  l(l + 1) + ~2 q3(0) = (~Ar) 2 ~, 4(p0 + 2) x
Po(Po + 1)  l ( 1 + 1') + #2 2(po + 1)
) + O(Ar),
(33)
etc. From Stirling's formula for the Gamma function one obtains the relation (MilneThomson 1951, pp. 254,255)
r(p + ~) r(p + y)
= pzy,
p >> 1
which yields
lim F ( p  n + / + 1) (no+n0 p>>l F ( p + p 0 + l ) = pP(p) = ( p  1 ) . . . ( p  n + l + 1 ) P ( p  n + 1 + 1) We get from Eq.(28): (P0 + n  l ) =
(34)
1 [( :/, ]1> 1" (p  1)(p  2 ) . . . (p  n + l + 1) = pnl1 + O(pnl2)
(36)
The factorial series of Eq.(19) becomes for p >> 1:
)
_
= ~,(oI~"~, ~176 = q3~
q(0,1>p,0+l (I  ~o(~o + I12~  2,
In 2 In 2 P >  2 In 82 " 2 n A r
(3)
5.7 CONVERGENCE FOR SMALL VALUES OF A r splane
227
(plane ,
~'0 s l

/s2
~ g = 2.0635 s3 = 0.77777 s2 = 1.28573
._ ~ i
i
a
b
FIG.5.71.Transformation of Fig.5.51b from the splane (a) to the Cplane (b) according to Eq. (2). is satisfied. In Eq.(5.44) we introduced the relation p = r / A r to connect the normalized variable p with the nonnormalized distance r. We introduce in analogy a distance rp: p = r__p_p Ar
(4)
One recognizes from Eqs.(3) and (4) that there must be a m i n i m u m distance rp for which one can obtain convergence. In order to d e t e r m i n e this distance we express s2 with the help of Eqs.(5.514) and (5.530): s2=
1 83
" l+~Ar
" e ~Ar
(5)
Equations (3), (4), and (5) yield:
rp > 
ln2 ~
 In 2
h m0c
(
1
E2 ) 1/2 m02c 4
(6)
For small values of F_ we obtain in essence the result t h a t rp must be larger than the C o m p t o n wavelength. But for large values of t! or n we can obtain much larger values of rp a c c o r d i n g t o Eq.(5.532). The inverse transformation of Eq.(2) s = ssr lIP
(7)
maps the circle I ~  1] = 1 of Fig.5.71b into a loop in the splane. The loop is shown in Fig.5.72.
228
5
DIFFERENCE
E Q U A T I O N IN S P H E R I C A L C O O R D I N A T E S
0
0 5 FIG.5.72. The circle I r 11  1 in the ~plane of Fig.5.71b mapped into a loop in the splane by means of nq.(7); s3  0.77777, P = ln2/(21ns2)  1.3791, s2  1/s3. The power series in the point s2 may be mapped in the same manner as the one in point s3. A complex number P must be used. This solution varies for large values of the distance r like e +~r rather than like e ~r as the solution in point s3. One may infer this from p = r / A r in Eq.(5.44), s~ in Eq.(5.519), and s2 in Eq.(5). We will not discuss the solution in point s2 any further. For a convergent solution according to Fig.5.71a we start with the differential equation (5.512) and write it for the point s3 by the substitution s = ( s  s 3 ) + s3. The terms ( s  s3) '~ are rewritten as s~(s/s3  1) n and s/s3 is replaced according to Eq.(2) by s/s3 = ~l/P. We now have a differential equation in the r plane that still has to be transformed some more to make it a Fuchstype equation. Once we have done that we may solve it with a power expansion in r 1 and obtain the solution OO
w(r 1 / P  1 ) = E
q ( ~ ) ( ~  1)P+~
(8)
v'O
The calculation is shown in Section 6.9 from Eq.(6.91) to Eq.(6.924). The final step is to transform w(~ 1/P  1) into the splane by means of the substitution  (s/s3) P. This is done in nqs.(6.925) to (6.950) and the function w p ( s  s 3 ) is obtained:
N
oo 
=
=

v=O
"

(9)
v=O
We observe that the upper limit co in Eq.(9) is replaced by a finite number N since an arbitrarily large but finite interval can be subdivided only into a finite number of arbitrarily small but finite subintervals. In other words, there is only a finite number N + 1 of linearly independent functions. The subscript P is used for w and q to distinguish them from w3 and q3 in Eq.(5.515), which
5.7 CONVERGENCE FOR SMALL VALUES OF Ar
229
holds for P = I only. We note that the exponent i5 in Eqs.(8) and (9) holds both in the ~plane and the splane, but the coefficients ~(v) and qp(v) are different. The series expansions in Section 6.9 are the bane of the Laplace transform method of solution of difference equations. A simpler and faster method is urgently needed. The operational methods of MilneThomson (1951, Ch. XIV) may bring this simplification. The series expansions required here are worked out in Section 6.9. Equation (5.515) was transformed into a factorial series Utt3(p) in Eq.(5.519). Replacing ug3(p), q3(v), and p there by up(p), qp(v), and i5 yields the factorial series associated with Eq.(9):
r(p) N (~ + 1)... (i5 + v) up(p) = s~r(p +~ + 1) ~(1)~qP(v)= o (p +15 + 1)... (p + i 5 + v)
= P
83  82
1 + l = P ( p  1 ) + l,
Eq.(6.916)
~p(1) = _ ( ~ [ s 3 P  l ( ~  l + 2 P )  ( s 3  s 2 ) ( ~  l ) ( P  1 ) / 2 P ]  P [ l ( l + Pls3(s3  s2)(i5 + 1) 0P(O) s3
2!
'
1 )  ~ 2]
Eq. (6.954)
s2, s3: see Eq.(5.514); ~, A, 5: see Eq.(5.52); p: see Eq.(5.516)
(10)
For P = 1 we obtain i5 = p of Eq.(5.516) and ~p(1)/(tp(O) = q3(1)/q3(O) of Eqs.(5.519) and (5.518). We want to work out a few numerical values. From Eqs.(5.523) and (5.530) we obtain the relation:
Ar = 83(82  s3) 2~
~3(~  ~3) ~
2
[2 )  1 / 2
1
moc
83(82  83)
h
47r
moc
m20c4 1 m2c 4
(11)
For the limit case represented by Fig.5.52a with s3 = 1/v/2 and s2 = x/~ we get Ar > 0"03979~hm0c(1 From Eq.(5.532) we obtain:
~2 I  1 / 2
re]c4
(12)
230
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES
E2
1
1/2
m~c4 )
__ ,~2
= [1(1
1/2
~n~n2 )]
. x/~n_15.416 Ar > 0.6134.
forn
1, Z = l
h
(13)
moc
Hence, the smallest value for Ar is about 61% of the Compton wavelength. If we use s2 = 1.2857 and s3 = 0.7777 of Fig.5.71a we obtain E2 )  1 / 2
Ar >_ 0"03144hm0c (1
m2c4
h " 0"48467m0 c
We still have to derive the numerical value of P. Equation (3) yields the relation P> 
In 2 2Ins2
(14)
which yields P _ 1.3791 for 8 2 ~ 1.2857, 8 3    0 . 7 7 7 7 in Figs.5.71 and 5.72. For plotting it is often more convenient to start with a numerical value for P. One obtains then from Eqs.(3) and (5.523)" 82 >_ e In 2/2P
(15)
83 ~_ e  l n 2 / 2 P
(16)
s3(s2  s3) = 2~Ar = e l"2/v  1
(17)
The sign _< in Eq.(16) can readily be replaced by = if the exact value of s3 requires more decimal digits than used for plotting. 5.8 PLOTS FOR SECTIONS 5.5 AND 5.7
We start with certain constants obtained in Section 5.5 that are generally applicable. From nqs.(5.52) and (5.532) we obtain ~
Ar
~ = 2~~
( ~ 2 1 27tAr = 2~ i  ~ h/,~o~'
= 4~r~/ 1  ~ ~=47r7a,
k,
E1
" F~2
(1)
k= h/moc
a = 7.297 535 x 1 0 a,
z = 1,
2,...
(2)
5 . 8 PLOTS FOR SECTIONS 5 . 5 AND 5 . 7
231
Equation (5.516) yields: It
P = 2?rZ~
(I
\
It2 (I m20c4) [ 1~ \
It2
2
]
2 1/2
m2c4) ( Ar
(3)
(1 moc2
mo2C4
(5)
We see from Eqs.(5.712) and (5.713) that the choice k  1 will yield convergence for any value of P > 1. But for smaller values of k we must be careful that P is chosen large enough for convergence. For tt we obtain from Eq.(5.532)
mOc 2
=
m o c2
=
moc 2
= 1
(6)
2n 2
if terms of order ~4 are ignored. At this point we can choose a value P _ 1. Equation (5.710) yields:
= P(p
1)+ 1
(7)
Further, we obtain from Eqs.(5.715) and (5.716) with >__ and 6 ~'Ar
> e~/3 r 2i 6 s~  s3
7r 1 6 ]Jm(s3)]
m
(s)
is satisfied [~]m(s3) = imaginary part of s3]. In Fig.5.102 we have ~'Ar =7r/12,
P = 2
(9)
The inverse transformation of Eq.(7) P  2
8   S3~ 1 / P " s
(10)
maps the circle I r 1]  1 of Fig.5.102b into a loop in the splane that is shown in Fig.5.103. For a convergent solution according to Fig.5.102a we start with the differential equation (5.512). It is written for the point s3 by the substitution 8 = (883) + 83 aS in Section 5.7. The terms ( s  s3) '~ are rewritten as s ' ~ ( s / s 3  1) '~ and s / s 3 is replaced according to Eq.(7) by s / s 3  r We now have a differential equation in the Cplane. It requires some additional transformations that bring it into a form that permits a solution by a power expansion in r 1: CO
~(r
_ 1)= ~ v0
~(.)(r
1)~+~
(11)
5 . 1 0 UNBOUNDED BOSONS IN A COULOMB FIELD splane
+i
+i
P Cplane
.... m___+ __r
,_+2
82
'
~
)
247
2
g =  2 cos ~r/12 83 ~ e  i ~ t / 1 2 82 ~ e i~r/ 1 2 a
FIG.5.102. Transformation from the splane to the (:plane according to Eq.(7) in analogy to Fig.5.71; P = 2, ~'/Xr = n//12.
0.5i
o.~
~
i
4
~'.~
0.5i
FI0.5.103. The circle I ~  11 = 1 in the ~plane in Fig.5.102b is m a p p e d into a loop in the splane by means of Eq.(10); s3  e in/12  1//82, P  2, g  2cosTr/12. T h e c a l c u l a t i o n is carried o u t in Section 6.10. T h e final s t e p is to t r a n s f o r m w(r 1/P  1) into t h e s  p l a n e by m e a n s of t h e s u b s t i t u t i o n ~ = (s/s3) P. T h i s is d o n e in Section 6.10 t o o a n d t h e f u n c t i o n w p ( s  s3) is obtained" c~
N
uO
u=O
~ ( ~  ~) = ~ ( ~ ) = ~ ~ ( ~ ) ( ~  ~)~+~ 9~ ~ ( . ) ( ~  ~)~+~
(12)
All this is as in Section 5.7 except t h a t s3 is no longer a real n u m b e r b u t is now a c o m p l e x n u m b e r .
248
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES
Equation (5.515) was transformed into a factorial series u~3(p) in Eq.(5.519), and the same transformation produced the factorial series of Eq.(5.710) from the power series of Eq.(5.79). If we change ~p(v), i6 to ~p(v), i5 in Eq.(5.79) we obtain Eq.(12), and the same change transforms the factorial series of Eq.(5.710) into the one associated with Eq.(12):
r(p) N (i5 + 1 ) . . . (i5 + v) up(p) = s~ r(p + p + 1) ~=o(1)'~P(~) (p +/~ + 1) ... ( p + / ~ + u) =P
( ) A5
83

1 + 1 = P ( i p '  1) + 1
Eq.(6.101)
8~
~p(1) (~[s3Pl(~l+2P)(s3s~)(p1)(P1)/2P]P[l(l+l)72] ~p(0) = Pls3(s3 s~)(~ + 1) 
+ i5 P  1 ) s3 2! ' s3, p'
see Eq.(1),
~, A, 5
Eq.(6.1014)
see Eq.(5.52)
(13)
As in Section 5.7 we work out a few numerical values. From Eq.(1) we obtain for the spatial resolution Ar"
Ar
s~s3 =  2i~'  =
s~s3 h (E 2 i ~47r moc m~c4
1
)1/2
(14)
From the limit case for convergence shown for P = 1 by Fig.5.101b we see the condition s 3  s3 = s2  s3 _ i
(15)
and obtain
1 h ( Ar > 47r moc
E2
rn2c4
1
= 0.079577 h
?7%0 C
?7%0 2 C4
1
(16)
which is in line with Eq.(5.712). The lowest possible energy E for an unbounded boson is m0 c2, which yields Ar ~ oo. A localization of such a boson is completely impossible. For larger values of E the irresolvable spatial interval decreases to an arbitrarily small but finite one, which implies that a boson with very high energy is not much disturbed by the Compton effect. The equivalent formula for Eq.(5.714) is expressed by Eq.(8). If we choose a value of P we obtain an equation for the magnitude of the imaginary part of 83
5.10
r
UNBOUNDED
T
B O S O N S IN A C O U L O M B
35
/
1.6
"/ :'/
.,.
15
"~ I0
/
,,'Y
5 2

i
25
4
249
/
I
P=l.8
r 20 "2 ~a,
FI(3.5.104. P l o t s o f
// //
FIELD
1.4
1.2 6
p+
[um(plu~l(p)] 1/2 a c c o r d i n g
8
i0
to E q . ( 2 a ) for Z = 1,
A~l(hlmo~ )
k = 1, a n d P = 1, 1.2, 1.4, 1.6, 1.8 for p = 1, 2, . . . , 10.
14
~ 0."
12
P=1.8
.." ." .' ..
10 ~I~,
.'
~, ,4
.." .'"
.a. 4
...
2
j:" 4
P l o t s of [upl(p)u~,l(p)l
= k = 0.5, a n d P  
1.2
.............
2
FIG.5.105.
.,,.~"
6
p+
1/2 a c c o r d i n g
1
7
8
i0
to Eq.(23) for Z = 1,
Ar/(h/moc)
1, 1.2, 1.4, 1.6, 1.8 for p = 1, 2, . . . , 10.
....'"
I 2.5
....."'" 9
2
7 ~. ... .~ ' " ..............
..
. . .
.....
,m 1.5
9. ' " ' "
1.6 . . . . . . . . . .

. . . . . .
1.4  ........
"'0.5
1 i
,
I
2
,
1.2
"" " " . . . . 
i
4
,
i
p,
"2.52K2.5"2. ,
!
6
,
!
,
i
8
,
!
,
I
I0
FIG.5.106. P l o t s of [upl(p)u~,x(p)]1/2 a c c o r d i n g to Eq.(23) for Z = 1,  k  0.2, a n d P   1, 1.2, 1.4, 1.6, 1.8 for p  1, 2, . . . , 10.
Ar/(h/moc)
250
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES
:oo :~;~1/7,6i 8o
~_
:a,
i
/
,
/,,' 1.6
.
/:.4
S ~~ / 2o
~
1.2
........................................... I0
15
....
p, 20
25
FIG.5.107. Plots of [up: (p)u~: (p)] :/2 according to Eq.(23) for Z = 1, = k = 1, and P  1, 1.2, 1.4, 1.6, 1.8 for p  5, 6, ... , 25. 25
:' f
2O
."
Ar/(h/moc)
./" P=l.8
s.s"
,.'"
~ 15
t~
~
5
i.
1.4

 ............................
I0
15
1.2
1
:,~..,._ ,
p, 2O
25
FIG.5.108. Plots of [up: (p)u~,l (p)] :/2 according to Eq.(23) for Z = 1, = k = 0.5, and P = 1, 1.2, 1.4, 1.6, 1.8 for p = 5, 6, . . . , 25.
5
......
:4
.....
9. . . . .
Ar/(h/moc)
.'"
F ....... '""
......'"
~3
....'"
~2
1.6 1.4 .....
,..___;
.
i0
. . . . :. ,
. . . . ~ "7~~
15
O'*
1.2 , ~
t
20
., , 
1
J
,, 25
FIe.5.109. Plots of [up: (p)u~: (p)] :/2 according to Eq.(23) for Z = 1, = k = 0.2, and P = 1, 1.2, 1.4, 1.6, 1.8 for p  5, 6, . . . , 25.
Ar/(h/moc)
5.10 UNBOUNDED BOSONS IN A COULOMB FIELD
251
I x 1024
T
5x1023
9
,4
"~
_
I'2
5x 102a
14
'
.
,
!
16
,

9
L
18
,
"
20 '
'
9
22 '
'
p+
1024 _
.
,
9
1 . 5x 1024  2 x 1024
FIG.5.1010. P l o t s of [upl(p)u~,l(p)] 1/2 according to Eq.(23) for Z = 1, =k=l,P=5, andp=ll, 12,...,22.
T
IxlO 9 j
6
,
.
j

8
,
i
9
,
~
,
i0 !
'
.
,
i
,
"
i
,
I
Ar/(h/moc)
J
14
12.
~~., _ l x l 09 p, .1~.,
2xl
0
9
3xlO 9
4x 109 5xlO
9
FIG.5.1011. P l o t s of [up1 (p)u~,l (p)]1/2 according to Eq.(23) for =k0.5, P5, andp=4,5,..., 14.
Z
= 1,
Ar/(h/moc)
250
r
4
T
250
~.
500
,4
I000
!
3
.
.
.
.
6 a
,
,
,
9
!
5
.
.
.
i
.
.
.
.

.
.
.
.
7
,
8
p
750 n~
1250
FIG.5.1012. P l o t s of [upl(p)u~,l(p)] 1/2 according to Eq.(23) for Z = k = 0.2, P  5, a n d p  2, 3, . . . , 8.
1,
Arl(himoc)
252
5 D I F F E R E N C E E Q U A T I O N IN S P H E R I C A L C O O R D I N A T E S
IJ,~(~)l = ~_1
(17)
6P
which is the equivalent of Eq.(5.717). The real part of s3 is determined by the imaginary part since s3 is always on the unit circle according to Fig.5.101b. We restrict at this time the location of s3 to the interval  2 < g < 0 but we shall investigate larger values of g in the following section. In analogy to Eq.(5.810) we want to plot the first term upl(p) of up(p) in Eq.(13):
r(p) upl(~)=SgF(p+iS+l),
P=I' 2,..., N
(18)
First we determine i5 with the help of Eq.(5.81)"
Ar E Ar A5 = 2 E 7 ~  = 4n~moc2 h / m o c = 4 r S e ~ k = 47rZa, e, = E / m o c 2, k = A r / ( h / m o c ) a = 7.297 535
x
103, Z

I, 2,...
(19)
From Eqs.(13) and (17) we obtain:
i5 = P
X5 s3  s~
= 3p2As
1 P+
+ 1= P
A5 2JJm(sa)l
1)+1 (20)
1
71"
Since i5 is real the only complex term in Eq.(18) is s3. W i t h the relations
s3 = x 
iy,
71"
y = I:Jm(s3)l = ~fi,
X2
+
y2
= 1
(21)
we obtain: s3 = [1  (~/6p)2] 1/2  i n / 6 P
(22)
In order to eliminate the complex term s~ in Eq.(18) we may plot up1 (p)u~, 1 (p). The square root of this expression yields a function that is easier to compare with Eq.(5.810):
[~p~ (p)~;~(p)]~/2 =
r(p)
r ( p + t + i)
(23)
5.10
UNBOUNDED
2
BOSONS
IN A C O U L O M B
253
:' ",
.'
,

","
,
1.2
. ".
,/
.
.
p__+
.
.
.
.
.
.
~ ,  2 2  2 ~  = ~ _. _ _ ( _ . . :
.
4
1.6
~,
':
100~,p2 (p)
 2
a.,
FIELD
'.
"~ P
1.8
6 %:
8 ;i ::
10
F i c . 5 . 1 0  1 3 . . P l o t s of ~P2(p) = [ue2(p)@2(p)] 1/2 according to Eq.(25) for I = O, Z = 1, k = A r / ( h / m o c ) = 1, and P = 1, 1.2, 1.4, 1.6, 1.8; p = 1, 2 , . . . , 10. T h e plot for P = 1 is multiplied by 100.
'. . P = 1, 1 0 ~ P 2 ( P ) x'~ ........ i
T
i
r
2
I
,
1.4
]""
;"
P = 1.2, lO0~p2(p) I""
  I  
4
6
I,
,
,,
p ____~ 8
i0
1.6
., ,~
3 4
"......................"'... P 
1.8
"'...
5
P l o t s of ~Pu(P) = [up2(p)u;2(P)] 1/2 according to Eq.(25) for l = 0, 7_  1, k  A r / ( h / m o c )   0 . 5 , and P  1, 1.2, 1.4, 1.6, 1.8; p  1, 2, . . . , 10. T h e plot for P  1 is multiplied by 10, t h e plot for P  1.2 by 100. FIG.5.1014.
1
.
4
~
.
.
.
.
.
_~
_ _~ i
T
2 .........
...,.. 9""
r .........
~ .........
p ~
~.........
i
i0
 o . 2
0.4
~ ........
0.6 0.8
/
P l o t s of ~P2(p) = [up2(p)u~2(p)] 1/2 according to Eq.(25) for 1 = 0, Z = 1, k = A r / ( h / m o c ) = 0.2, and P = 1, 1.2, 1.4, 1.6, 1.8; p  1, 2 , . . . , 10.
FIC.5.1015.
254
5 DIFFERENCE EQUATION IN SPHERICAL COORDINATES p 4, *
,.
~
,
,
,
,
,
J
10
"::: ..........
,

,


15
o.s ~   ' ~


i
J
i,
25
L:i.2 : 2~;:2(;,5
'.~
"~=,,~uu~tp)~":" ~ s
I
,~

20
'"
% "'..... .:. ~ 0.~ "'....
1.5
2
~
"..
~b ...:~%
.....
Fie.5.1016.
Plots of ~P~(O) = [,~P~(p),4~(0)l 1/2 according to Eq.(25) for l = 0, = 1, and P = 1, 1.2, 1.4, 1.6, 1.8; p = s, 6 , . . . , 25. T h e plots for P = 1 to P = 1.8 are multiplied by 200 to 0.0002. 7 =
1, k = A r / ( h / m o c )
1.5
\ x
T 0.5
""" " ".......
P=
I0 0.5
.
15
20
.
.
"
1
500~p2(p)
1.2,
p\
25
:
P=l.6
.,.
..
FIC.5.1017.
Plots of ~P2(P)  [ u p 2 ( p ) u ~ , 2 ( p ) ] 1/2 according to Eq.(25) for l  0,  0.5, and P  1, 1.2, 1.4, 1.6, 1.8; p  5, 6, . . . , 25. T h e plots for P  1, 1.2, 1.8 are multiplied by 100, 500, 0.1. Z   1, k 
Ar/(h/moc)
0.02 0.01
l
.
.
.
.
.
.
10
c~  0 . 0 1
,,

,
. ..~
p+ 15
.
..
_,:_7._,:=:__v..=c_~ 20 25
0.02 0.03
0.04
FIG.5.1018.
o.O~,. ,.'" i" ],]
Plots of ~P2(p) = [ u P 2 ( p ) u ~ 2 ( p ) ] 1/2 according to Eq.(25) for l = 0, = 0.2, and P = 1, 1.2, 1.4, 1.6, 1.8; p = 5, 6 , . . . , 25. T h e plot for P = 1 is multiplied by 0.1. Z_ =
1, k = A r / ( h / m o c )
5.10 U N B O U N D E D BOSONS IN A C O U L O M B F I E L D
255
i
ix I026i
l
,
t
.
i
.


12
,
i
,

14
,
,

16
,
!
,

18
20
p,
~GS Ix 1 026 2x 1026
F I o . 5 . 1 0  1 9 . P l o t o f ~ P 2 ( p ) = [up2(p)u~,2(p)] 1/2 a c c o r d i n g t o E q . ( 2 5 ) for l = 0, Z = 1, k = A r / ( h / m o c ) = 1, P = 5, a n d p = 10, 11, . . . , 20.
2 . 5 x 101~
l
I
~~.  2 . 5 x l
,
!
9
8 !
6
01r
.
..
"
IO
i
I
,
l
,
I
9
p .,
12
O..,
~
5x 101~  7 . 5 x 101~  I x l O zz
F I G . 5 . 1 0  2 0 . P l o t o f ~P2(P) = [up2(p)u~2(P)] 1/2 a c c o r d i n g t o E q . ( 2 5 ) for I = 0, Z = 1, k = A r / ( h / m o c ) = 0.5, P = 5, a n d p = 4, 5, . . . , 12.
30000
T
20000
~" e,l
I0000
~L
4
. . . .
J
3 I0000
9
.
.

,
9
.
,
!
5
p+
.
,
,
i
6
.
.
,
t,
7
F I G . 5 . 1 0  2 1 . P l o t o f ~P2(P) = [up2(p)u~2(P)] 1/2 a c c o r d i n g t o E q . ( 2 5 ) for I  0, Z k  A r / ( h / m o c )   0.2, P = 5, a n d p   2, 3 , . . . , 7.
1,
256
5 D I F F E R E N C E EQUATION IN SPHERICAL COORDINATES
In Eq.(5.86) we used E / m o c 2 = 1  ~ 2 / 2 n 2 with n = 1. To obtain c o m p a r a b l e values we choose E / m o c 2  1 + ~2/2 in Eq.(19) since E must now be larger t h a n moc 2. As previously, we choose initially k = A r / ( h / m o c )  1 in Eq.(19), but reduce it to k  0.5 and k  0.2 rather t h a n to 10 2, 10 3, and 10 4 as in Figs.5.89 to 5.811. This implies t h a t the transition from nonr a n d o m to ( p s e u d o ) r a n d o m numbers will be much faster for u n b o u n d e d bosons t h a n for b o u n d e d ones. In line with this result we shall also use P = 1, 1.2, 1.4, 1.6, 1.8 instead of the larger values P = 1, . . . , 5 in Figs.5.85 to 5.811. C o m p a r i n g the plots of Eqs.(23) in Figs.5.104 to 5.109 with those in Figs.5.85 to 5.811 shows t h a t large values of the plots no longer occur close to p  0. This is to be expected since the eigenfunctions t h a t we derived are used for probability density functions for the location of a boson and a b o u n d boson is more likely to be close to the center of the Coulomb field t h a n an u n b o u n d boson. T h e rapid change of the plots when k is reduced from I to smaller values is conspicuous. One would expect t h a t the location of an u n b o u n d boson is changed more by the C o m p t o n effect than that of a bound boson. In line with this observation are the plots of Figs.5.1010 to 5.1012 t h a t show the r a n d o m values obtained from Eq.(23) for P = 5 instead of for P = 10 as in Fig.5.88. The narrow and varying ranges of p in Figs.5.1010 to 5.1012 are due to the enormous variation of [up1 (p)U~l (p)]l/2 for P = 5. We t u r n to the second term up2 of the factorial series of Eq.(13) and write it in the following form:
r(p) Op(1) ~ + 1 ~'~(P) =  ~ r(p + p + 1) p + p +f Op(1) see Eq.(13) for Op(0) = 1; s3 see Eq.(22); i5 see Eq.(20)
(24)
In order to obtain for Up2(p ) a formula of the form of Eq.(23) we observe t h a t ~p(1) is a complex number. Hence, we write
[up2(p)u~,2(P)] 1/2  UP2(P) = [qp(1)q~,(1)] 1/2
r(p)
~+ 1
r(p +~ + 1)p+~+ 1
(25)
Plots of Eq.(25) are shown for the range p  1, 2, . . . , 10 in Figs.5.1013 to 5.1015. T h e values of P run from 1 to 1.8 as in Figs.5.104 to 5.106. The value of k = A r / ( h / m o c ) varies from 1 in Fig.5.1013 to 0.5 in Fig.5.1014 and 0.2 in Fig.5.1015. It is evident t h a t the plots for P = 1.6 and 1.8 are quite different from those for P  1, 1.2, 1.4 and erratic. ;i'his behavior is also recognizable in Figs.5.1016 to 5.1018, which hold for the range p  5, 6, ... , 25. For P = 5 and k = 1, 0.5, 0.2 we obtain the r a n d o m plots of points of Figs.5.1019 to 5.1021. They are very similar to the plots of Figs.5.1010 to 5.1012.
5.11 ANTIPARTICLES
257
~ =a//2~
~=~
a
~'~
b
FIG.5.111. Plot of the left halfplane of Fig.5.101 in analogy to Fig.5.52. The values g   3 / v / 2 and g   ( 2 + 1/6) are replaced by g  3/x/~ and g  2 + 1/6. The points s3 and s2 of Fig.5.52 become s2 =  s 3 and s3 =  s 2 in this figure. (a) holds for the limit when the circle of convergence around s2 goes through s3 and s = 0, while (b) holds for smaller absolute values of s2. 5.11 ANTIPARTICLES In Fig.5.101 we have shown the loci of s2(g) and s3(9) b o t h for positive and negative real values, while only the half for positive real values was shown in Figs.5.51, 5.52, and 5.71. We extend here our investigation to negative real values of the splane. Figure 5.111 is the equivalent of Fig.5.52 for negative real values. We see t h a t g is replaced by  9 , s3 by  s 2 , and s2 by  s 3 . We s u b s t i t u t e s  ( s  s2) + s2 into Eq.(5.512) in order to solve the differential equation with a power series in the point s2. T h e relation s~ + gs2 k 1 = 0
according to Eq.(5.513) is needed to obtain:
[(~ ~2) 2 + (2~  ~ ) ( ~  ~2) 2 + ~2(~2  ~s)(~ ~2)]~"(~ ~2)
[z(z + 1) ~]~(~ ~) = 0 (1) For the solution of this differential equation we need a power series according to Eq.(5.515): N
~(~
~)= ~ q~(~)(~ ~)~+~
(2)
v0
W h e n we compare Eqs.(1) and (2) with Eqs.(6.81) and (6.82), which hold for Fig.5.52, we recognize that the substitutions
258
5 DIFFERENCE
EQUATION
IN S P H E R I C A L
COORDINATES
~ ~ ~ , ~ ~ ~ , p~ ~ p~, q~(.) ~ q~(~)
(3)
transform Eqs.(6.81) and (6.82)into Eqs.(1) and (2), which hold for Fig.5.111. Hence, we can take the results of Section 6.8 and make the substitutions of Eq.(3) to obtain the corresponding results for Eqs.(1) and (2). Let us start with Eq.(6.83):
P3 P
A5
8382 < 0 according to Fig.5.52
8 3  8 2
A5 8 2  8 3
A5 . . . .
8 3  S 2
P3,
s 3  s 2 < 0 according to Fig.5.111
(4)
For the further evaluation of P3 and p~ we turn to Eq.(5.516)'
[ (1
p ' = +27rZa
1+
= 2~'Za
hlmoc) h/moc)
= p~ + o ( ~ ) p~ =  2 r Z a
E (1 too 02
E2
mo2~4 )
(5)
The change of sign for p~  pt in Eq.(4) permits us to write  m 0 instead of m0 in Eq.(5). We postpone a discussion and turn to the rewriting of Eqs.(6.85) and (6.88). Since we wrote p3  p and p~ = p' in Eq.(4) we must be careful to replace P3 in Eqs.(6.85) and (6.88) by p' rather than p~' (3~I 3,1 (0)q~(1)
I + 33, 0 (0)q~ (0) = 0
OL3 , 1 ( 0 )     8 2 ( 8 2  8 3 ) ( p t b 1)
a~,0(0) = s2p'(p' + 1 )  l(l + 1 ) + ~2
(6)
Finally, we rewrite Eq.(6.88)"
' (v)q'3(v + 1) + a3,0(v)q'3(v) ' 33,1 + a'3,1 (v)q'3(v 1) = 0 !
33,1 (v) = s2(s2  s3)(p' + v + 1)(v + 1)
~ , o ( ~ ) = ( 2 ~  ~ ) ( p ' + ~)2 + [~ _ p , ( ~ _ ~)](p, + ~) = (p' + ~)(p' + ~  1)
O/t~,_~(~)
(~)
5.11 ANTIPARTICLES
259
From Eq.(5.523) we obtain with the substitutions of Eq.(3) for ~':
s3 (s3  s2) _" moc ~1 2Ar h \ t~t
s2(s3  s2) . moc (
82(8283).
=
2Ar
E2 / 1/2 m 20c4
 
2At

h
E2 /
1/2
m2c4
1
(8)
With ~' we obtain from Eq.(5.532):
, mock[ 2(n2l + 1 21)] nh 1+ ~
~1
,.moc~/[ ~2  n, h
, 9 F_I
~/2 ( n ' 1) 1   ~ 2l+1 + "2
[ moc 2 1
"~2
"~4( n
2n 2
n4
2l + 1 3
E " moc 2 1
2 7 + ~
21+1
(9)
From Eq.(2) we get in analogy to the transition from Eq.(5.515) to (5.519) a factorial series:
r(p) N ( p ' + 1)... (p' + v) ~(P) = ~ r ( p + p,+ 1 ) ~ (1)~q~(v)= 0 ( p + p ' + 1)... (p + p ' + v)
q~(1) _ _a~,o(0) q~(0)  ~,,(0)
(10)
The first term of Eq.(10) cannot be plotted like Eq.(5.810) since s2 is negative. However, we can plot according to Eq.(5.1023):
[( = =
r(p) S~F(p §
I~t"
1)
r(p) 9 r(p+p'+l)'
)(~
r(p) )*]~/~ F(p+ p' ~ 1)
p=l, 2,..., N
(11)
We essentially obtain Eq.(5.810) except that i5 is replaced by p~. This difference disappears if we substitute E~/(moc 2) " E'/(moc 2) or E~2/(moc 2) " E'/(moc 2) from Eq.(9):
260
5
DIFFERENCE
E
_
_mOC
2

E Q U A T I O N IN S P H E R I C A L C O O R D I N A T E S
E~
_
_moc
2
E~
 1 
,~2
+ O(~ 4)
(12)
_moc 2 

This substitution produces for p' of Eq.(5) the same result as the substitution of Eq.(5.86) produces for p in Eq.(5.516):
p ' = p =  2 7rZ a
1
~n 2
1
x
1
1+
"~ = 47rZa, Z = 1, 2 . . . , n = 1, 2 , . . . ,
2}
~~n2
1
~/2
(2~k) 2
1~n~n2
a = Z e 2 / 2 h = 7.297 535
x 10  3
k = Ar/(h/moC)
(13)
Equations (5.810) for a particle and (11) for its antiparticle are equal. Consider the second term of the factorial series of Eq.(10). W i t h the help of Eq.(6) we obtain:
q~(1)_ q~(O) 
~'3,0 (0) = p, (p' + 1) a' 3,1(0) (s3
[l(l + 1) s2)(p'+
 ~2]/s2 1)
(14)
From Eq.(6.85) we obtain the result: q3(1) q3(0) =
a3,0(0) p ( p + 1)  If(1 + 1)  ~2]/s3 a3,1 (0) = (s2  s3)(p + 1)
(15)
Equations (14) and (15) are different. Even though we again have p' = p we recognize in Fig.5.52a the values s3 = 1 / v ~ ,
s2  s3 = x / 2 / 2 , g <  2
(16)
which must be used with Eq.(15). But Fig.5.111a shows the different values s2 =  1 / v ~ ,
sa
s2 =  v / 2 / 2 ,
g > +2
(17)
Equations (17) and (16) bring Eqs.(14) and (15) into the following form: q3(1) _ q~(1) q3(0) = q~(0) = x / 2 p  2 [ l ( / + 1 )  ,~2]
(18)
The points g = 0 in Figs.5.101 and 5.111 are of interest since a particle changes to an antiparticle or viceversa at these points. We obtain for g = 0 from Eq. (5.53):
5.11 ANTIPARTICLES
g= 
Arc) 2
h/m0
2 + 4
h/moc [2
261
=0 Ar
2
m2oc4)(h/moc) =  8
= 4 (1
moc 2
h/moc
The energy E at g = 0 becomes strictly an effect of the resolution A r of the observation and says nothing about the observed particle. If one approaches the points g = 0 in Fig.5.101 from the particle side g < 0 one obtains
[
moc2 =+ 1+2
h/moc
but if one approaches g  0 from the antiparticle side g > 0 one obtains
moc ~
=+
[
1+2
h/moc
(21)
Hence, the finite but otherwise undetermined resolution A r permits a transition between particle and antiparticle without a q u a n t u m jump. This is a result characteristic for a difference theory using A r rather than dr. We note that [Ar/(h/(moc)] 2 in Eq.(21) is not changed if we replace it with { A r / [ h / (   m o c ) ] } 2.
ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 137
6 Appendix
6.1 DIFFERENCE OPERATORS OF HIGHER ORDER
The first and secondorder difference operators have been introduced by Eqs.(1.21), (1.24), (1.25), and (1.27). For difference operators of higher order we must revisit the definitions. Difficulties are created by the use of symmetric difference operators in Eqs. (1.21) and (1.27). The nonsymmetric difference operators of Eqs.(1.24) and (1.25) are easier to extend; this extension is used by mathematicians (Nhrlund 1924; MilneThomson 1951; Spiegel 1994). There are a number of reasons to prefer the symmetric difference operators to the nonsymmetric ones in physics (Harmuth 1989, Sec.8.1 and 8.2). Starting with Eq.(1.21) we redefine the symmetric difference operator of first order: ,~A(O) A(O + A0/2)  A(O  AO/2) ,~0 = A0
(1)
The second, third and fourthorder difference operators become: /~2A(0) A02
{',~A(O)'~
= ,50 \
Ao
A A(O + A0/2)  A(O  A0/2)
) = ,~~
,,,o
A(O + Ae)  2A(0) + A(O  A0)
(2)
(/xo)~ AaA(0) Ao~
A (/~2A(0))
= Ao /~o ~A(O + 3 A 0 / 2 )  3A(O + A0/2) + 3A(O  A 0 / 2 1  A(O  3A0/2)
(/xo)~
(3)
A'A(O) _ A ( A3A(O)) A(O + 2A0)

4A(O + A0) + 6A(0) (Ao)~

4A(O

A0) + A(O

2A0)
(4)
262 ISSN 10765670/05 DOI: 10.1016/S 10765670(05)370066
Copyright 2005, Elsevier Inc. All rights reserved.
6.1 DIFFERENCE OPERATORS OF HIGHER ORDER
1
1
0
1
2
1
O
O
1
3
1
4
15
0
 3 A 0 2A0
0
A0
0
0
5
0
20
0
1
0
10
0
6
0
4
0
10
0
1
0
6
0
5
0
3
0
0
1
A/Ao
0
O 0
263
1
0
15 0
A0
0
6
1 z~6//~06
0
2A0
O
3A0
0, FIG.6.11. Arguments and factors of symmetric difference operators from order 1 to 6 using odd and even multiples of A0/2 according to Eqs.(1)(4).
TABLE 6.11 DIFFERENCE OPERATORS FROM FIRST TO SIXTHORDER ACCORDING TO EQS.(1) TO (4).
1 [A(0+ 2~ )
1 1  A ( 0  2~)1 ~~
1 [A(O+AO) 2A(O)+A(OAO)]~ (AO)~ 3AO AO _~_ 3AO [A(O+~)3A(O+~) +3A(0   )  A ( 0    ~ ) 1
1
(A0)g
1 [A(O+2AO)4A(O+AO)+ 6A(O)4A(OAO)+A(O2AO)]~ (2zxo)4
[A(O+5'~)5A(O+3'~')+IOA(O+~ )
10A(0
Z~~O)+5A(O3~)A(O5~)](A10)5
[A(O+azaO)6A(O+2ZXO)+ISA(OAO)2OA(O)+ISA(OAO)6A(O2AO)+A(Oaz~O)]
(zx0)6
The operators of odd order, z~/z~0 and z~a/z~0 3, contain odd multiples of the difference A0/2 while the operators of even order, z~2/z~0 2 and z~4/z~0 4, contain even multiples of the difference A0/2. Figure 6.11 shows what causes this complication. In Eq.(1.21) we had sidestepped it by using the arguments 0 4 A0 and the denominator 2A0. The first six difference operators listed in Table 6.11 make the connection clear. An alternative way to generalize the symmetric difference operators of Eqs.(1.21) and (1.27) is as follows: AA(O) Ao
A(O + A0)  A(O  A0) =
2do
(5)
264
6 APPENDIX
o
o
1 o 1
2o
1
40
o
0 o
1
4o
o
1 0
60
150
3A0 2A0
2o
A2/A02
2o
o
60
40
0
5o
4o
o
1
A~iAo5
200
150
60
1 0
/~6/z~06
0
A0
2A0
3A0
5
A0
1
o
1
/~3/z]0a
1
A4/A04
0~ FIG.6.12. Arguments and factors of symmetric difference operators from order 1 to 6 using integer multiples of A0 according to Eqs.(5)(8).
TABLE 6.12 DIFFERENCE OPERATORS FROM FIRST TO SIXTHORDER ACCORDING TO EQS.(5)
To (8).
[A(O+AO)
A(OAO)]
1
2A0 1
[A(O+AO) 2 A ( O ) + A ( O  A O ) ] ~ (AO)2
[A(O+2AO)2A(O+AO)
1
+2A(OAO)A(O2AO)]2(AO)3
[A(O+2AO)4A(O+AO) + 6A(O)4A(OAO)+A(O2AO)]
[A(O+3AO)4A(O+2AO)+5A(O+AO)
1 (A0)4 1
5A(OAO)+4A(O2AO)A(O3AO)]2(/,,0)5
[A(O+3AO)6A(O+2AO)+15A(OAO)20A(O)+15A(OAO)6A(O2AO)+A(O3AO)]
A2A(0) AO2
AaA(O) AO3
=
AO A(A2A(O) 2
1 (A016
A(O + A0)  2A(0) + A(O + A0)
= 2(A0) 1 3 [f~[A(O +
(6)
2A0)  2A(O + A0) + A(0)]
.[A(0)  2A(O  A0) + A ( O  2A0)] }
= A(O + 2A0)  2A(O + A0) + 2A(O A 0 )  a ( o  2A0) 2(ZX0)3
(7)
6.1 DIFFERENCE OPERATORS OF HIGHER ORDER
4
z~4A(0) _.z~0 
z~0 2A2 (A2A(O))z~O 2 =
2[a(0 + A0) =

(A0) 41
{[A(O+2AO)2A(O+AO)+A(O)]
2A(0) + A(O A0)] + [a(0)
A(O + 2A0)
4A(O + A0)

265
+ 6A(0)

2A(O A0)+ a(o 2A0)] } 4A(O A0) + A(O 2A0) (8) 


(A0) 4
There are no odd and even multiples of A0/2 as in Eqs.(1)(4) but only integer multiples of A0. Figure 6.12 shows the scheme of amplitudes and arguments. For difference operators of even order, ,'~2n/~O2n, we obtain the same binomial numbers as in Fig.6.11, but for difference operators of odd order, z~2'~+l/z~0 2~+1, we obtain different numbers. In analogy to Table 6.11 we list the difference operators up to order 6 in Table 6.12. In order to decide whether to use the operators of Table 6.11 or of Table 6.12 consider a finite set of numbers f(0), f(O + A0), . . . , f(O + mAO). These numbers may have been produced by calculation to be checked by measurements or they may have been obtained by measurements to check a theory that should yield the same numbers by calculation. The operators of Table 6.12 correspond to this condition better than the operators of Table 6.11. One could interpolate the set of numbers by writing
f[O + (2n
+ 1)A0/2] =
1
~{f(O + nAO) + f[O + (n + 1)A0]}
(9)
to obtain values for integer multiples of A0/2. This interpolation would not add any information if the numbers f(O), f(O + A0), ... f(O + mAO) represent physical measurements, it would only create the illusion of an improved resolution. As an example let us write Eq.(3) with the terms A(O + 3A0/2) to A(O3A0/2) interpolated from A(O + 2A0), A(O + A0), ... , A(O 2A0):
A(O + 3A0/2) A(O + A0/2)
1 [A(O+ 2A0)
+
A(O + A0)]
= I[A(0 + A0) + A(0)]
A(O
A0)]
3A0/2) = I [ A ( 0  A0) +
A(O
A(OA(O
=
A 0 / 2 ) =
1 [A(0)+
2A0)]
(10)
If we substitute Eq.(10) into Eq.(3) we obtain Eq.(7). This becomes quite different if f(O),..., f(O + rnAO) represent the values of a defined mathematical function, say the Gamma function F(O). The value of
266
6 APPENDIX
/"(2.5) is not the average of/"(2) and F(3). Hence, the information contained in the set F(0), F(O + AO), ..., F(O + mAO) is increased by specifying the additional values F(O+O.bAO),..., F[O+(m1/2)A0]. Unlimited information can be obtained from many defined mathematical functions by sufficiently fine interpolation. This is completely different from the finite set of numbers that can be produced by or checked by physical measurements. No mathematical manipulations can increase the information contained in them. The transition A0 , dO yields the same result for the difference operators of Tables 6.11 and 6.12, but the inverse transition dO ~ A0 is not unique and we must decide which set of difference operators to use. For a finite set of numbers f(O), f(O + A0), . . . , f(O + mAO) that can be obtained by or for observation in physics the operators according to Fig.6.12 or Table 6.12 are simpler than the ones of Fig.6.11 or Table 6.11 plus substitutions according to Eq.(10); we shall use them. This is a good example of how concepts of pure mathematics must be carefully analyzed before using them in physics. 6.2 EXTENSION OF SECTION 3.1 FOR Ax 1/v/2. We start from Eq.(6.81) and replace the terms ( s  s 3 ) '~ by s ' ~ ( s / s 3  1 ) n. This calls for a change of w ' ( s  s3) and w " ( s  s3) in Eq.(6.81):
w'(s
s3) = d w ( s  s3) dw(s/s3  1)d(s/s3  1) _ 1 w ' ( s / s 3  1) ds ~ d(s/ss1) ds s~
w"(s
s3) ~  ~ 3 w " ( s / s 3  1)
(1)
Equation (6.81) becomes:
83
+
S_l
+(2s3s2)
83
2s3
s 1
s _1
+(s3s2)
+(4s3+gAh)
s  1
s _
w"(s/s31)
+ 2s3 + g  A5 w ' ( s / s 3  1)
 [ l ( l + 1) zy2]w(s/sa  1) = 0
(2)
The substitution s/s3 = ~ l / P transforms the differential equation from the splane to the Cplane:
[~3(r ~/P  1) 3 + (2~3  ~2)(r ~ / P Jr [283(r lI P


1) 2 + (4s3 + g

1) ~ + (~3  ~2)(r ~/P  1)]~"(~ 1/P  1) ,~)(r
_ 1) ~ 283 ~. g  ~ ] W ' ( r lIP  1)   [ l ( t + 1) z)'2]w(r
 1) = 0
(3)
In order to obtain a differential equation of the Fuchstype we must transform the terms ( r 1)n into power series of ~  1 or 1  r This is possible by expanding r _ 1 =  ( 1  ~l/P) in a binomial series:
6.9 CONFORMAL MAPPING FOR SECTION 5.7
[ 1 1~ 1/P 1[1(1~)]I/P = 1 1  ~ ( 1  ~ ) +
ll(p ~
1
301
)
(1r
2 ...
_ 1r X
P P  1 ( 1  ~ ) + ( P  1 ) ( 2 P  1) ( c~ ( P  l ) . . . ( 3 P  1) ( l _ r X = I + 2!P ~. P"2 "1"~2+ 4!P 3 = 1+ b~(1 r
+ b~2(1 _ r
blj
+ ... + b~j(1  r
+
.
.
+ 9
9
9
.
(P1)...(jP1)
(4)
(j+I)!PJ

The series is absolutely convergent in the interval  1 < 1  ~ < +1. We can derive the following powers of (1  { 1 / P ) n from EQ.(4)"
X '~,
(1(1/P)n=
~l/P1
=
X",
n = 1, 2 , . . .
(5)
We obtain for X 2, X 3 , . . . , X 6" X 2 =1 + b21(1  ~) + b22(1  ~)2 + b23(1  ~)3 + b24(1  ()4 + . . . =1 + 2b11(1  ~) + (2b12 + b121)(1  ~)2 + 2(b'13 + b11b12)(1  ~)3 + (2514 + 2511513 + 522)( 1  ~)4 + . . .
b21 = 2b11, b22 = 2b12 + b21, b23 = 2(b13 + b11b12),
(6)
b24 = 2b14 t 2bllb13 + b22
X 3 = 1 + b31(1  () + b32(1  r
+ b33(1  ~)3 + . . .
531  3511, 532 = 3(5121 + 512), 533  6511512 + 3513 + 531
(7)
X 4 = 1 + b41(1  ~) + b42(1  ()2 + . . .
(8)
b41  4b11, b42 = 6b21 + 4b12 X 5 =1 + b51(1  r
+...,
b51 = 5bll
(9)
x ~ =1 + . . .
If we substitute 1  (~ =  ( ( e.g.,
x = 1 b,~(r
(10)
1) into Eq.(4) for X we obtain alternating series,
1 ) + 512(r
1) 2  b , 3 ( r
1) 3 + . . .
(11)
302
6 APPENDIX
Hence, the substitution of (r
_ 1)~ of Eq.(5) into Eq.(3) brings'
{s3p3(r
1)311  b31(r 1) + b32(r 2) 2  . . . ] + (2s3  s 2 ) p  2 ( r 1)211  b 2 1 ( r 1)+ b22(~ 1) 2  . . . ] ~(83  8 2 ) p  1 ( ~ 1)[1  b l l ( ~ 1)+ b12(~ 1) 2  . . . ]}'w"(~ 1/P  1) + { 2 s 3 p  2 ( ~  1)211 b21(~ 1)+ b22(r 1) 2  ...] + (4s3 + g 
~)pl(~_
1)[1  b 1 1 ( r
1) + b12(r
1) 2  . . . ]
+ 2~ + ~  ~ } ~ ' ( r
 1)
 [ l ( l + 1 )  z~2]w(~l/P  1) = 0
(12)
This is the differential equation (2) in the ~plane. Since it is of the Fuchstype we write its solution w(~ 1 / P  1) as a power series in r 1. The notation ~(v) and t5 is used in the Cplane instead of the notation q(v) and p in the splane: oo
W(~l / P  1 ) = E ~(v)(~ 1)#+" v'0 oo
W'(~lIP


1) = ~
0(v)(i5 + v)(r

1)~+'1
v0 oo
Wt'(~l / P  1 ) = E 0(v)(~5 + v)(f + v  1)(~ 1)p+~2
(13)
~0
Substitution into Eq.(12) yields" { s 3 p  3 ( ~  1)311  b31(r  1) + b32(~ 1) 2  . . . ] + (2s3  s2)p2(r  1)211  b21(r 1) + b22(r 1) 2  . . . ] + ( s 3  s 2 ) p  l ( ~  1)[1b11(r 1)+ b12(r 1 ) 2  . . . ] } oo
x E q(v)(15 + v)(15 +
v

1)(r

1) p+~2
~0
+ { 2 s a p  2 ( ~  1)211 b21(~ 1)+ b22(~ 1) 2  ...] + (4s3 + g  ~ ) p  1 ( ( _ 1)[1  b 1 1 ( ~  1) + b12(~ + 2~ + g  ~ } ~
1) 2   . . .
]
q(~)(~ + ~)(r  1) ~+~~
~'0 c~

[l(1 +
1)  ~2] E v0
~(u)(~  1) p+" = 0
(14)
6.9 CONFORMAL MAPPING FOR SECTION 5 . 7
303
The sum of all coefficients of a certain power ( ~  1)f+,~ must be zero. The lowest power in Eq.(14) is ( (  1)f1 for v = 0. There is only the coefficient ~(0) for this power: [(83  82)p1/5(15 1) + (283 + 9  X(~)iS]~(0) = 0
(15)
Using the relation  g = s2 + s3 of Eq.(5.514) we obtain the following value for the initial power 15: /~=P
83  82
1
+1
(16)
For P = 1 we get the same value for i5 as for p in Eq.(5.516). We derive from Eq.(16) an expression for A5 that will simplify some of the following equations: ~(~ = (83  s 2 ) [ p  I ( p 
1 ) + 1]
(17)
The second lowest power in Eq.(14) is ( ~  1) f. It contains the coefficients ~(1) and ~(0). It yields ~(1) if we take ~(0) as the choosable constant:
(283  s2)p2q(0)i5 (/5 1)+ (83  s2)P1 [~(1)(/5 + 1)i5 b11~(0)/5(/5 1)] + (483 + g  i~)plq(O) + (283 + g  X5)~(1)(/5 +
1)
[l(l+l) ' ~ 2 ] ~ ( 0 ) = 0
(18)
We may rewrite this equation with the help of Eq.(17) and eventually obtain the following simpler form'
~p,~(0)~(1) + ~p,0(0)4(0) = 0
(19)
C~p,l(0) = P  1 ( s 3  s2)(p + 1) ctp,0(0) = P  l i 5
(83 P1 (/5\
1 + 2P)
P1
) l(l +
1)+ ~,2
(20)
(21)
For P = 1 and thus 15 = p we obtain:
(22)
ap, l(0) = (83  s2)(p + 1) ap,o(0) =
s3p(p + 1)  l ( 1 + 1) +
~2
(23)
Equation (23) equals aa,0(0) in Eq.(5.518) while Eq.(22) differs by a factor sa from c~3,1(0). We expect that the inverse transformation from the (plane to the splane will provide the missing factor s3.
304
6 APPENDIX
The next step would be to determine the coefficient of ( r t h a t yields in analogy to Eq.(19):
5+1 in Eq.(14)
ap,2(11#(2) + ap, l(11#(1) + ap,o(1)#(O) = 0
(24)
The elaboration of ap,2(1), ap, l(1), ap,0(1) is a challenge but it is not needed here. As ( r 115+1 increases to ( r 115+2, ( r 11P+3, ... the recursion formulas according to Eqs.(191 and (24) increase to 4, 5 , . . . terms. Let us turn to the inverse transformation ~ = (s/s3) P from the Cplane to the splane and see what becomes of t5 and ~(~) of Eqs.(16) and (191 for ~ = 1. We write:
w(# I / P  1)= E O(")(# 1)f+" = E q(") ,,,=0 ____
s

1
u=O
[
(.)(:3)
(8/83)P1 ~ S __1 Eq(u)(1)" 1S/S3 1
(25)
The reason for this notation is that we can change the signs of the first factor:
(~3>~ 11~ (1 1 r~/~3 s/~3 1

(26/

There is no problem with a sign change for integer powers ~ = 1, 2, . . . :
[(~/~3) P  1]" = (11"
[1 (,/~3)P]
(27)
~
We use the binomial series to expand 1  (s/s3)P:
1
~ 83
[ (
=1
1
+
1~
P ( P  1) 2!
83
)] ~
[
=1
(~)
1P
1
s
s
1

3
3!
_~(1_ _ ~) [1 ~_1 ,, (1_;) +
(P
1)(P3!
21
1
s
The new variable Y is defined as follows:
...
=P
1
s
Y
...t (281
6.9
Y=I
Pl(s~) 2! 1
+
(P1)(P2)(~3) 3! 1
. . . + (_I)j (P 1)
2
c1=
( ) (
Cl .
.1 .
s
83
c2 1 . . s . . . . .
, . . . , cj
=
s) j
1  s3
( ) 1B s
cj
83
P1 (P1)(P2) 27' c 2 = 3!
+ ...
" " ( P  J) (
(j~: 1)! .  1 .
305
CONFORMAL MAPPING FOR SECTION 5 . 7
+""
J _...
83
(1
)j ( P  l ) . . . ( P  j ) (j+l)!
(29)
The choice of the negative sign for the coefficients cj will become evident presently. Equation (26) may be rewritten:
( (s/s3)P1) ~ (1 (s/s3)P)~ s/s3  1
=
1  s/s3
(30)
= P~Y~
The series of Eq.(29) for Y terminates for positive integer values of P. Hence, the question of convergence is avoided for P = 1, 2, . . . . The expression Y~ in Eq.(30) can again be expanded into a binomial series P ~ Y ~ = P~[1  ( 1  Y)]~
(31)
and Eq. (30) becomes:
1  s/s3
= P ~ Y ~ = P ~ [1/5(1  Y ) +
~(p
1)
2[
(1 
y)2
_
1 [ "'" ]
(32)
The positive integer powers of I  Y will be needed. From Eq.(29) we get:
1 Y =
1 s
c~+c2
1u
s
[c3
1
83
s
+c4
1
s
+
83
...
(33)
The reason for the choice of the signs of the coefficients cj in Eq.(29) becomes clear. For higher powers of 1  Y we obtain:
(1Y)2
=
( 1/  2 [ s  
c 2+2clc2 ( 1I s
83
+ (2cl c4 + 2c2c3) 1 
 
r92~_
+ ~  c ~ c 3 + c . ~
( / 2 s1 
83
+(2clc5+2c3c4+c


83
2) 1  u
+... 83
(34)
306
6 APPENDIX
(1 _ y)3 8 8 ( ~ 3 3 ) 3 1 1  c3+3c2c2 ( 713 /
+(3c2c3+3ClC22) ( ~13 3 )8 2
~ (3Cl2C4 + 6Clc2C3 } C3) 1
(1y)4=
(35)
+ ooo
~
83
( 1  ~s3 ) 4 [ c4 + 4 c 3c2 ( ~133 ) s 22 8 + (4c31c3 ~ 6c lc 2) 1 
(:3)
(1  y ) 5

(1

s s ~33}5[ C l5 .Jr.. ,.~4 . , C l C 2 ( ] 1 _) _3t .
+...
]
(36)
(37)
. .
83
( )0[ ( 1  V)a =
1  ~ 33
]
(38)
c~+...
Substitution of Eqs.(33) to (38) into Eq.(32) produces a very long formula that we bring into a printable form by defining new coefficients dj:
(1(s/s3)P) / = [ (  ~~3) ( s) ( ~3) +"" ] s's3 Pf 1 + dl 1 + d2 1  + d3 1 s 
8 3
=P~ 1 
dl
__s1
s3
+d2
 1
s3
d3
1
+
...
dl /5Cl 1 d2 /5c2 + ~./~(/5 1)c~ 1
d3 " pc3 } ~ . i p ( p 
d4 
1
pc4 ~ ~ . f i ( p 
1 , 1)2clc2 + ,~.i5(/5 1)(i5 2)c 3 1 1 1)(2CLC3 + c2) } ,~/~(/51)(/~ 2)3c2c2 .+ ~.p... (~3)c~ 9
1 1 d5  ~c5 + ~fi(15 1)(2clc4 + 2c2c3) + _.,~.'iS(fi1)(15 2)(3c2c3 
3c~c2)
1_ + ~1_ ; . . . (;~ 3)4~3~ + ~.p... (;~ 4)~
6.9
CONFORMAL
MAPPING FOR SECTION 5.7
1
307
1
6CLC2C3
d6 j6c6 + ~.15(/5 1)(2CLC5 + 2c2c4 + c 2) + _~.15(/5 1 ) ( i 5  2)(3c2c4 + 1~ + c 3) + ~ p . . .
1_ (15  3)(4c3c2 + 6c2c~ + ~ P . . .
+ T h e t e r m s [(s/s3) g Eq.(28) we obtain:

8
1
=(1)"
1
For v 

1
s) y,=1+e,,l(l_ s
(~)
Y=
1
=
1Cl
e~2
(~) 1
1  
e21     2 C l ,
1  2Cl
C2
( ~) 1  
~4(:
1_
(41)
)~ ( s )4 ]
+ (  2 c 2 + cl) 2
(
1  
83
1   
83
+...
2CLC3+ c~

s

13cl
1
s
83
e31     3 C l ,
1
F (   2 C 4 ~ 2CLC3 {C~)
~ : ( )~[ (~) 1
(
83
+ (~
\say
(40)
+...
e22     2 c 2 + c21, e23     2 c 3 + 2CLC2,
e24 =  2 c 4 +
s__~
yv
1
~   c 3 , . . .
+(_2c3 +2ClC2)(1_ _.~)3
(1
s
1
83
83
S3/
~
6 we o b t a i n with the help of Eq.(29):
( ~)~ (~) ~[
\
+
83
e l l   C1~ e12  c2~ r
(1s~
8
=(1)'P

83
1
(39)
83
1, 2 , . . . ,
(~)
P

83
(1
1 ~.,/~... (15 5)c61
1] ~ in Sq.(25) also require a series expansion. Using
P
s
(15  4)5c~c2
+(3Cl23c2)
(42)
( ~)~ 1
s
+ 6~1~  3~) (1  ~
(43)
e32   3C 2  3C2, e33   C 3 + 6CLC2  3C3
 s
)4[ (~) 14gl 1
83
e41  4Cl, e42  6Cl2  4c2
s
+(6c~4c2)
+...
( )~ "*'] 1 s
83
(44)
308
6 APPENDIX
( :,)~ ()~[ 1
w
s
y5
1
 
s
.
.
.
1
.
551 .
83
()] 1
s
+
.
.
.
83
(45)
e51     5 C I
( )~ ( )~ ] 1
s
y6
1 _s
~
1+
~
83
(46)
. . .
83
We m a y rewrite the last sum in Eq.(25) as follows:
[ ()~
E~(v)(1)"
s
1 
u=O
= E~(u)(1)'P"
83
=~(0)P ~
q(1)P

1
+~(2)P 2 ~(3)P 3 

1
1


1

+ ~(4)P 4
s
1
s
1 A e l l
s
1+e21
s
1+e31
s
~(5)P 5(1~3) +~(6)P 6
(;) (:,) (~) (~) 1
1
1 + e41
5 [l+e~l s


1
y.

1+...
( :,)~ ] ( ~)' ] ( ~)~ ] ( )~ ]
+ " ' + e l ~
1 
s
+ " ' " + e24
1
s
 t  " ' "  ~  e33
1 
s
+ e42

(1
( ~)~ ] 1
s

1


s 83
yO
( ~)~[ ( :,)~[ ( ~)~[ (~)'[ 1
1
u0
1  s
s
+...
s
+
...
+
...
s
+ ...
83
~3) +...]
+0(1s/s3)
(47)
7
We collect t e r m s m u l t i p l i e d by the same power (1  s / s 3 ) t' with #  0, 1, 2, . . . , 6. T h e coefficients e,~ used in Eqs.(40) to (45) m u s t be d i s t i n g u i s h e d from t h e new coefficients e~:
E
q(v)(1)~'
[ (~)~] ~ [()~ ] (~) ()~ ( )~ () (~)~ ( )~ 1
s

~(v)
s
u=O
uO
=
eo + e l
1
s
+e2
~ 1
83
1n s
+e3
1n s
83
= e0  el
8
  1 83
+ e2
8

1
+...
83

e3
8 83
 1
+ ...
6.9 CONFORMAL MAPPING FOR SECTION 5.7
309
~o = 4(0)
gl  ~ ( 1 ) P e2   ~ ( 1 ) P e l l + q ( 2 ) P 2 e3 =  ~ ( 1 ) P e 1 2 + q(2)P2e21  q(3)P 3 e4 =  ~ ( 1 ) P e 1 3 + ~(2)p2e22  q(3)p3e31 + q(4)P 4 e5 =  ~ ( 1 ) P e l a + ~(2)p2e23  4(3)P3e32 + q(4)p4e41  q(5) P5 e6 = c](1)Pel~ + c](2)p2e24  4(3)P3e33 + ~(4)P 4e42  ~(5)Pbe51 + q(6)P 6 (48) E q u a t i o n (25) may be written as follows with the help of Eqs.(39) and (48):
Wp




1) 

1
1d1
[
x eoel
 (el + e o d l )
1
+d21
d31
83
~~3I +e2
~31
+
83
.
.
.
e3 ~~3I + . . .
s3
+ (e2 + eldl + eod2) (~3  1 ) 2  (e3 + e2dl + eld2 + eod3) (~3  l ) 3 + (e4 + e3dl F e2d2 + eld3 "+"eod4)
(81)4 s3
 (e5 + e4dl + ead2 + e2d3 + eld4 + eod5)
s _ 1
+ (e6 + esdl + e4d2 + ead3 + e2d4 + eld5 + eod6)
(;)~ s _ 1
+ o(~/sa
 1) 7
(49)
W i t h the final transformation s/s3  1 = s 3 1 ( s  s3) and w(s/s3  1) = w ( s  s3) = w(s) we bring Eq.(49) into the form of Eq.(5.515):
310
6
APPENDIX
c~
N
~ ( ~  ~ ) = ~ ( ~ ) = ~ ~ ( . ) ( ~  ~)~+~ " Y ] 0~(~)(~  ~)~+~ v=O
v'O
~p (0) = + P P s 3 f eo ~p(1) = qp(0)(el + eodl)/s3 qp(2) = +qp(0)(e2 + eldl + eodl)/s 2
9
(50)
The upper limit of the sum was reduced from oe to N since an arbitrarily large but finite interval can have only a finite number of arbitrarily small but finite subintervals. Let us check our result. The constant qp(0) is of little interest since it can be chosen. For qp(1) we get from Eqs.(29), (39), and (48) the components el

q(1)P,
eo = ~(0), dl =/)c1, cl =
P1 2!
(51)
Equations (19)(21)yield:
~(1) ~(0)
~p,o(0) ap, l(0)
plp(s3pI(pI+2P)(s3s2)(p1)P'/)2'
 / ( / + 1) + ~2
(52)
P  l ( s 3  s2)(16 + 1) From Eqs.(50) and (51) we get:
4~(~) = 4~(0)~(0)
(~(1) P ~(65~
~
/hP1) 2~
(53)
~
We reduce the choosable constant qp(0)4(0) to qp(0) by choosing ~(0) = 1: ~p(1) = _ (/5 [s3 P  1(/5  1 + 2P)  (s3  s 2 ) ( / 5  1 ) ( P  1 ) / 2 P ]  P[l(l + 1) _,~21 (~3  s2)(~5 )(f + 1)) ~p(0) \ Pls3(s3 /~ P  1 ) s3 2!
(54)
For P = 1, 15 = p we again obtain Eq.(5.518): qp(1) ~p(0)
s3p(p + 1)  l(l + 1) + ~2 sa(sa  s2)(p + 1)
q3(1) q~(0)
(55)
6.10 CONFORMAL MAPPING FOR SECTION 5.10
311
6.10 CONFORMAL MAPPING FOR SECTION 5.10
According to Fig. 5.101b we must use conformal mapping if a power series in the point s3 is to converge as far as s = 0 for g >  2 c o s T r / 6 = 1.73205, We start from Eq.(6.81) and use the substitutions of Eq.(6.91). In principle we can use all the equations of Section 6.9 that do not specify that s3 is a real number, but we must write i5 and 0 rather than i5 and ~. Furthermore we recognize the relation s2 = s~ from Fig.5.101a and Eq.(5.101). Hence, in Section 6.9 we go directly to Eq.(6.916)
~= p
= P
A5 s3  s~  I
+1,
A5 i2~,A r
+ 1
1
9 82  83,
83  e'
itc'Ar
(1)
and Eq.(6.917)"
(2)
A5 =  2 i t c ' A r [ P  l ( ~  1 ) + 1] Equation (6.918) assumes the form"
(2s3  s~)p2~(0)iS(i 5  1 ) + (s3  s~)Pl[0(1)(i 5 + 1)i5 b11~(0)i5(i5 1)] + (4s3 + g  AS)P10(0) + (2s3 + g  X5)~(1)(i5 + 1) [l(1
+
1) ~2]0(0 ) = 0
(3)
bll   ( P  1)/2!P Using Eq.(2) we may bring this equation into the following form:
OIP,1(0)0(1) + ap,0(0)0(0) = 0
(4)
ap, l(0) = P  l ( s 3  s~)($ + 1) ap,0(0) = P  I ~ ( s 3 p  I ( ~ 
(5)
I + 2P)  P2!P  1 (s3  s~)(i5 1) } 
l(l
+
1) + ~2
(6)
As in Section 6.9 we do not a t t e m p t to work out the formulas for the higher coefficients 0(2), c~(3), . . . . The inverse transformation from the ~plane to the splane follows Eq.(6.925). We have only to replace i5, ~ by i5, 0:
312
6 APPENDIX
w(#1/g 
((~/~) s/s3
~  1)
1)=
1
P\s3__I/(S~
(~(v)(__l)~
1__
v0
(7) 83
For the evaluation of this equation we go from Eq.(6.925) to (6.939)"
(1(s/s3)P) ~ [ ( s ) ( s ) 2 ( s ) 1s/s3 = P P 1  d l ~~31 +d2 ~331  d 3 dl =/3cl d2
3 731
]
+...
see Eq.(6.929) for cl, c 2 , . . .
=/~c2 + ~/~(i~ 1)c21 1
(8) The sum of Eq.(7) is worked out in Eq.(6.948)'
4(v) (  1)" u0
[
1
s
=
83
= eo  el
(~(v)
s s3
1
u0
 1
+ e2
   1
 e3
 t
+...
83
~o = ~ ( o ) el 
O(1)P
(9) For the whole Eq.(7) we obtain from Eq.(6.950)'
c~
N
~(~  ~3) = ~(~) = ~
~(.)(~
 ~3) ~§
u=O
~p(0) =
+PPs3Peo
Op(1)  qp(O)(el +
"~
O~(.)(s  ~3) ~+~
u=O
eodl)/s3 (lO)
The upper limit of the sum was reduced from oo to N since an arbitrarily large but finite interval can have only a finite number of arbitrarily small but finite subintervals.
6.10
CONFORMAL MAPPING FOR SECTION 5.10
313
The constant qp (0) is of little interest since it can be chosen. For gp(1) we get from Eqs.(9), (8), and (6.929)" el =  ~ ( 1 ) P ,
eo = ~(0), dl =15cl, Cl =
P1 2!
(11)
Equations (4)(6) yield:
~(1) ~(0)
,~,o(O) c~p,1(o)
gl~ (s3PI(PI+2P)(s3s;)(P1)P/)2'
/(/+1)+72 (12)
Pl(s3  s~)(~ + 1) From Eqs.(10) and (11) we get: 0(1) P
/~ P~  l ) s3 2!
+
(13)
~
We choose ~(0) = 1 to reduce qp(0)~(0) to qp(0)' ~p(1)
~p(0)
//iS[s3P 1(i5 1 + 2P)  (s3  s~) ( i 5  1 ) ( P  1 ) / 2 P ]  P[l (l + 1) ~2] Pls3(s3  s;)(~ + 1) i5 P  l ) 83
(14)
2!
For P = 1 we obtain the simpler equation:
s3p'(p' + 1 )  l(l + 1 ) + ~2 s3(ss s~)(p'+ 1)
~p(1) Op(0) A5 2~'Ar' s3
*=2i~
 83
1At, s3
" 1i~'Ar,
P=I
(15)
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Index
differenciation 18 Dirac equations 197 direction of time 29, 30 discrete coordinate systems 3643 discrete spherical harmonics 210 dogma of the circle 27 dyadic 21
A Abramovitz 214 Almagest 27 Anastasovski 1 antiparticles 257 Apostle 26 Aristotle 26, 28 Astronomia Nova 27 Ae ")0, g m * 0 197
E
egg of Columbus 243 Einstein 28, 30 Eleatic school 26 electric dipole current 2 electric monopole current 2 electromagnetic pulse 2 elementary particle 143 energyimpulse tensor 78, 222 equilateral triangle 31 Euler Script, Euler Fractur 50 Euclid 31 excitation functions 69 exponential ramp function 62
B
Barrett 1, 3, 4, 5, 7, 18, 28, 47, 74, 189, 197 Becker 83 Berestezki 78 Bernoulli product method 104, 209 binary digits 29 bit 29 boundary condition 65 bounded coordinate system 38 box normalization 74 branch of mathematics 143 byte 29
F factorial series 216 Feshbach 19, 286 finite information 29 fourvector 190 Frobenius 213 Fuchs 213, 228
C causal function 1 causal solution 210 causality law 2, 3, 28, 30 character group 27 charge renormalization 29 clocks 28 closed coordinate system 38 Columbus, egg of 243 commuting 8 Compton effect 172 , random numbers 182 Compton wavelength 239, 297 computer evaluation 124, 171 conformal mapping 226 conservation law of energy 1, 3 contour integral 212 convergence at r = 0 243 Copernicus 27 current densities 4
G Gaussian curvature 34 Gaussian pulse 1 Guldberg 212, 216 gm * 0, Ae ~ 0 197 H
Habermann 63 Hamilton 19 Hamilton function 4, 6, 44, 58, 78, 82, 186, 191, 193, 195 Harmuth 1, 3, 4, 5, 7, 10, 18, 28, 47, 81, 84, 86, 189, 197, 262 Harrison 1 HauptlSsung 13 Heisenberg approach 83 Heitler 83 Hermite polynomials 86
D difference equation, one dimension 62 difference operators p, ~C, ~ , grad, div 53, 204 318
INDEX hiding dimensions 35, 36 Hillion 1 Hussain 3 I
Infeld 28 infinite energies 29 infinite information 29 reformation 29, 265 information theory 29, 30 information transmission 2 reformation, unlimited 266 initial condition 62 interference phenomena 96 invisible dimension 35, 36 iterated Dirac equations 245 K Kepler 27, 35 King 1 KleinGordon matrix 49, 54 L lack of development 11 Lagrange 111 Lagrange function 4, 5, 19, 22, 23, 24, 25, 183, 187, 190 Laplace transform 212 left difference quotient 9 Lehnert 1 Leibniz 26 Lifschitz 78 Lorentz convention 3 Lorentz equation 4, 189 M magnetic charge 2, 19 magnetic dipole current 2 mathematical axiom 30 Maupertuis 19 Meffert 3, 4, 5, 7, 18, 28, 47, 74, 81, 84, 86, 189, 197 Messiah 211 MilneThomson 12, 13, 16, 18, 111,212, 216, 229, 262, 268 Morse 19, 286 multipole current 2 mutually perpendicular vectors 34
319 Pitajeski 78 Plato 27 polynomial, asymptotic 218 principal solution 13 Ptolemy 27 pure mathematics 26, 266 pure number 29
Q quantum gravitation 29 R
random numbers, Compton effect 182 relativistic canonical momentums 190 relativistically variable mass 189 renormalization 28 right difference quotient 8 Roy 1 S scalar potentials 3 Schiff 211 SchrSdinger 30 SchrSdinger approach 83 self energy 29 separation of variables 65 signal solutions 1 singular points 214,217, 227, 245, 247, 257 Smirnov 165 Spiegel 262 spinning bullet 25 standard change of time 28 steady state theory 28 Stegun 241 Stratton 1 summable power series 219 summation sign 13 symmetric difference quotient 8, 10 T table of sums 17 three components Hamilton function 44 topologic group 27 transient theory 28 transition to random numbers 239 U unlimited information 266 unobservable dimensions 35
N Newton 26 noncommutability 196 NSrlund 12, 13, 16, 18, 111,212, 216, 219, 262, 268
V van der Waerden 27 vector potentials 3 vector potential Ae for Ax ~ Ac 148
O onedimensional coordinate system 4042
W Wallenberg 212, 216 Weyl 28
P
Z
physical law 30
Zeno of Elea 26
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ERRATA
Advances in Imaging and Electron Physics Volume 137, Chapter 4, page 170. Equations (44) and (45) are missing on page 170. These equations are as follows.
pc = (1/47r2)(Ac/cT) 2, AI = eCTAmo~/hN,
Ae = 2AcAe(~)/e
G2(~,0,~) = pcN2fi.~(~){(1 + 2iA1)~O,2,0(~,Or,)  [ 4  6 A ~  2i(2A1 + A~)lg'o,1,o(q,0n)+ (1  2iAa)~I'o,z,o(r G2((, 0)= pcN2Ae(r
(44)
+ 2iA1)~I'o,~,1(~, 0n)
 4[6A 2 + 2i(2A1 + h~)]~o,1,l(C,0n) + (6  12A 2  A~)~o,o,1(r 0n)  [4  6 ~  2 ~ ( 2 ~ + ~ ) ] ~ o ,  ~ ,  ~ ( r
+ (1  ~ ) ~ o ,  2 ,  ~ ( r
(45)
On page xxi, "Henning F. Harmuth" should read "Henning F. Harmuth and Beate Meffert."
ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOLUME 137 ISBN: 0120147793
This Page Intentionally Left Blank