This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
{1.3a)
('If;, a¢)= a('¢,¢),
{1.3b)
('¢, ¢) = (¢, '¢)*' ('¢, 'lj;) ~ 0, ('¢, '¢) = 0 if and only if 'lj; is a null vector.
(1.3c) (1.3d)
Two vectors 'If; and ¢ are said to be orthogonal to each other if their corresponding inner product vanishes, namely ('¢, ¢) = 0. For example, let us consider the vectors in enspace 'lj; = (xi, X2, ... , xn) and¢= (y1, y2, ... , Yn) where Xi and Yi are complex numbers. The inner product of 'If; and ¢ written as n
('If;,¢)= L:x;yi = xiy1
+ x2Y2 + .. · + X~Yn·
(1.4)
i=l
Consider the set of continuous function of f (x) where a ::::;; x ::::;; b. An ordered pair of functions f(x) and g(x) define the inner product as (f(x), g(x)) = f(x)*g(x)dx. This vector space is called .C2 (a, b)space
J:
when lf(x)i2 and lg(x)l 2 =finite ..
4
Chapter 1
1.2.1 Schwarz inequality \
We are now in the position to prove the Schwarz inequality. Let, 'If; a~d ¢ be any two vectors. The Schwarz inequality reads as
1('¢, ¢)1 = J('I/J, ¢)(¢, '¢) ~ J('I/J, '1/J)J(¢, ¢).
{1.5)
Proof Since ('lj; +a¢, 'If;+ a¢) ~ 0, where a = ~ + i'fj is a complex number. Regard this inner product ('If; + a¢, 'If; + a¢) = !( ~, 'fJ) as a function of two variables ~ and 'f/· Then
which is positive definite. Let us look for the minimum of !( ~, 'fJ) at ~o, 'fJo by solving 8!(~, 'fJ) 8~
=0,
{1.7)
and we obtain
{1.8)
Therefore
nf•) ('¢,¢)(¢,'¢) :> 0 !( ...(:o, 'f/0 ) = (nf,
(1.9)
that can be cast into the familiar expression of Schwarz inequality.
1.2.2 GramSchmidt orthogonalization process The inner product we have been considering can be applied to the orthogonalization of the basis in the ndimensional vector space. Let
5
1.2 Inner pmduct
{'¢'1, '¢'2, ... , '¢'n} E V be the set of nlinearly independent vectors. Since ('¢'i, '¢'j) f= 0 in general, we can construct a new set of vectors {'¢'L '¢'~, ... , '¢'~} such that ('¢'~, '¢'j) = 0 for all i and j unless i = j, namely '¢'~ and '¢'j are orthogonal to 'each other for i f= j by the following procedure: First take '¢'~ = '¢'1 and construct '¢'~ = '¢'2 + a'¢'~. In order to force '¢'~ to be orthogonal to '¢'~, we solve the a as to meet the condition ('¢'~, '¢'D = 0, i.e. {1.10) and we obtain a
= ( '¢'2, '¢'D* /('¢'~, '¢'D =
('¢'L '¢'2)/('¢'L '¢'0, hence (1.11)
The same procedure can be performed repeatedly to reach '¢'3 '¢'3 +a'¢'~ + (3'¢'~ which guarantees ('¢'3, '¢'~) = ('¢'3, '¢'~) = 0 with a ('¢'~,'¢'3)/('¢'~,'¢'~) and (3 = ('¢'L'¢'3)/('¢'L'¢'D. In general,
= =
(1.12) The set of orthogonal basis {'¢'~, '¢'~, ... , '¢'~} can be normalized immediately by multiplying the inverse square root of the corresponding inner product, i.e.
{1.13)
and {'¢~, '¢2, ... , ~n} becomes the orthonormal set of the basis in the vector space. From now on we shall take the basis to be orthonormal without mentioning it particularly. ·
6
Chapter 1
Example Consider the following set of continuous functions in C( oo, oo)
{1.14)
We construct the new set of orthogonal vectors by applying the GramSchmidt process and obtain:
fMx)
=
fo(x)
=
exp( ~),
{1.15)
(x2) ,
!I) f ( ) ! 1'( X ) = f 1  f~(f~, (!~, ~~) = 1 X = X exp 2
f.2'( X ) =
f!
2
If(!{, h)!~(!~, h) = ( {f{, fi) {!6, f6) X
2 !) exp (x2). 2 2
{1.16)
{1.17)
Similarly we have J3(x) = (x 3  3x/2) exp( x2 /2). The orthonormal functions can be calculated according to
{1.18)
where Hn(x) are called Hermite polynomials. One also recognizes that fn(x) are in fact, the eigenfunctions of the Schrodinger equation for onedimension harmonic oscillation.
1.3 Completeness and Hilbert space Let us introduce some other terminologies in discussing Hilbert space.
1.3.1 Norm A norm on a vector space is a nonnegative real function such that, if'¢,¢ are vectors, the norm of 'If; is written as 11'¢11, satisfying:
7
1..'] Completeness and Hilbert space
11¢11 ;;::: 0, 11¢11 = 0 lla¢11 = lal·ll¢11, 11¢+¢11 ~ 11¢11 + 11¢11.
iff 'ljJ is null vector,
{1.19a) {119b) (1.19c)
Example
If f(x) E C(a, b), namely if f(x) is a continuous function for a variable that lies between a and b, the norm of f (x) can be defined either as llf(x)ll = Max{!f(x)!,a ~ x ~ b} or as the inner product of f(x), i.e.
llf(x)ll 2 = (f(x), f(x))
=
I
1b
lf(x)l 2 dx.
a
1.3.2 Cauchy sequence and convergent sequence Consider an infinite dimensional vector space and denote the basis by {¢1,¢2,¢g, ... }. We construct the partial sum 'l/JN = Li~cPi, where i runs from 1 to N, and obtain ... , 'l/Jj, 'l/Jj+b ... , 'l/Jm, ¢m+l, ... , 'l/Jn, ... for increasing values in N that forms an infinite sequence. The sequence is called a Cauchy sequence if lim 'l/Jn = lim 'l/Jm, ntoo mtoo or more precisely to put in terms of norm, i.e, It is said that a vector
lim 'l/Jm
mtoo
lim ll'l/Jn  'l/Jm II = 0. n,mtoo
'l/Jm converges to 'ljJ if
= ¢,
or
lim ll'l/Jm ¢11 = 0,
mtoo
then { ... , 'l/Jm1, 'l/Jm, ... } is called a convergent sequence. It is easily concluded that every convergent sequence is a Cauchy sequence. Yet it is not necessary true conversely. Namely a Cauchy sequence is not always a convergent sequence.
8
Chapter 1
1.3.3 Complete vector space A vector space, in which every Cauchy sequence of a vector '1/Jm converges to a limiting vector 'lj;, is called a complete vector space.
1.3.4 Hilbert space A Hilbert space is a complete vector space with norm defined as the inner product. A Hilbert space, finite dimensional or infinite dimensional, is separable if its basis is countable.
1.4 Linear operator A linear operator A on a vector space assigns to each vector 'lj; a new vector, i.e. A'lj; = 'lj;' such that A('lj; + ¢)
= A'lj; +A¢,
A(a'lj;)
= aA'lj;.
{1.20)
Two operators A, B are said equal if A'lj; = B'lj; for all 'lj; in the vector space. For convenience in later discussion, we denote • 0: null operator such that O'lj; = 0 for all 'lj;, and 0 is the null vector. • 1: unit operator or identity operator such that I'lj; = 'lj;. The sum of the operators A and B is an operator, such that (A+B)'lj; = A'lj; + B'lj;. The product of operators A and B is again an operator that one writes as A· B or AB such that (AB)'lj; = A(B'lj;). The order of the operators in the product matters greatly. It is generally that AB =/= BA. The associative rule holds for the product of the operators A(BC) = (AB)C.
1.4.1 Bounded operator An operator A is called a bounded operator if there exists a positive number b such that
IIA'l/JII
~
bll'l/JII,
for any vector 'lj; in the vector space.
1..4 Linear operator
9
The least upperbound (supremum) of A, namely the smalleast number of b for a given operator A and for any 'If; in V, is denoted by
IIA'¢11 IIAII =sup { lf¢f'
'If; =/= 0 }
(1.21)
,
then IIA'¢11 ~ IIAIIII'¢11. We are now able to show readily that IIA + Bll ~ IIAII + IIBII.
Proof Let us denote IIA'¢11
IIA + Bll =sup {
~
IIAIIII'¢11 and IIB'¢11
~
IIBIIII'¢11. Then
} {IIA'I/J+B'¢11 } II(A+B)'¢11 II'¢ II , 'If;=/= 0 =sup II'I/JII , 'If;=/= 0
IIA'¢11 ~sup { ~,'¢
IIB'¢11 =/= 0 } +sup { lf¢f''lj; =/= 0 } = IIAII + IIBII.
Similarly, we have IIABII ~ IIAIIIIBII.
1.4.2 Continuous operator Consider the convergent sequence { ... , '1/Jm, '¢m+I, ... , '1/Jn, .. .} such that lim 11'1/Jn  '¢11 = 0. If A is a bounded operator, then { ... , A'¢m, ntoo
A'¢m+l• ... , A'I/Jn, .. .} is also a convergent sequence because lim
ntoo
IIA'I/Jn  A'¢11
~ IIAII lim
ntoo
11'1/Jn '¢11 = 0.
We call operator A the continuous operator.
10
Chapter 1
1.4.3 Inverse operator An operator A has an inverse operator if there exists B R such that AB R = I, then we call operator B R the right inverse of A. Similarly an operator BL such that the product operator BLA =I, then we call operator BL the left inverse of A. In fact, the left inverse operator is always equal to the right inverse operator for a given operator A, because
The inverse operator of a given operator A is also unique. If operators B and C are all inverse operators of A, then C = CI = C( AB) = (CA)B =B. The implication of uniqueness of the inverse operator of operator A allows us to write it in the form A  1 , namely AA 1 = A  1 A = I. It is easily verified that (AB) 1 = B 1 A 1 .
1.4.4 Unitary operator An operator U is unitary if IIU¢11 = 11¢11· A unitary operation preserves the invariant of the inner product of any pair of vectors, i.e. (U'ljJ, U ¢) = ('ljJ, ¢). This can be proved as follows: Let x = 'ljJ + ¢ and we have
(Vx, Vx) = (U('Ijl + ¢), U('ljl + ¢))
+ (U¢, U¢) + (U¢, U'ljl) + (U¢, U¢) 2 2 IIU¢11 + IIU¢11 + 2~{(U¢, U¢)},
= (U¢, U'ljl) =
and on the other hand,
(Vx, Vx) = (¢ + ¢,¢ + ¢)
+ (¢,¢) + (¢,¢) + (¢,¢)
=
(x,x)
=
ll'lfll + 11¢11 + 2~{(¢, ¢)}. 2
= (¢,¢) 2
Since IIU¢11 = 11¢11, IIU¢11 = 11¢11, wehave~{(U'if, U¢)} = ~{(¢,¢)}. Similarly if = 'ljJ + i¢, we obtain (Vx', Vx') = (x', x'), that implies 8'{(U¢, U¢)} = 8'{(¢, ¢)},therefore (U¢, U¢) =('If,¢).
x
1..4 Linear operator
11
1.4,.5 Adjoint operator Consider the inner product of(¢,~¢) where A is a given linear operator of interest. This numerically scalar quantity certainly is a function of operator A and the pair of vectors 'ljJ and ¢, namely ('ljJ, A¢) = F(A, '1/J, ¢) is a scalar quantity. Instead of performing the above inner product straightforwardly, we shall obtain the very same scalar of (¢,A¢) by forming the following inner product (At¢,¢) such that(¢, A¢)= (At¢,¢). The operator At is called the adjoint operator of A. The following relations can be easily established (proofs left to readers): (A+B)t =At +Bt,
(1.23a)
(aA)t = a*At,
(1.23b)
(AB)t = BtAt,
(1.23c)
(At)t =A,
(1.23d)
(At)1 =(A I)t.
(1.23e)
It can also be shown that At is a bounded operator if A is bounded and their norms are equal, i.e. IIAII = IIAtiiTo prove the above equality, let us consider IIAt¢11 2 = (At¢,At'ljJ), namely
therefore IIAt¢11 ~ IIAIIII¢11, and we have IIAtll ~ IIAII· On the other hand, we have IIA¢11 2 = (A'Ij;,A'Ij;) = (AtA¢,'1/J) ~ 11'1/JIIIIAtiiiiA'I/JII, which implies IIAII ~ IIAtll Therefore IIAII = IIAtll is established.
12
Chapter 1
1.4,.6 Hermitian operator When an operator is selfadjoint, namely an adjoint operator At equals to operator A itself, i.e. A = At, then we call A a Hermitian operator.
1.4,. 7 Projection operator Let 1l be a Hilbert space in which we consider a subspace M and its orthogonal complement space M_1_ such that for each vector 'ljJ in 1l = M EB M_1_ that are decomposed into unique vectors '1/JM in M and '1/JM_j_ in M_1_ such that 'ljJ = '1/JM + '1/JM_j_, and ('1/JM, '1/JM_j_) = 0. The projection operator PM when acting upon vector 'ljJ onto a subspace results in PM¢ = ¢M· It is obvious that PM¢ = 'ljJ if 'ljJ E M and PM¢= 0 if 'ljJ E M_1_. One can 11lso be easily convinced that
('1/J,PM¢)
('1/J,c/JM) = ('1/JM +'l/JM_i,¢M) = ('I/JM,¢M) = ('1/JM,¢) = (PM'I/JM,¢) =(PM¢,¢).
=
Therefore PM is also a Hermitian operator, i.e. P1t =PM· Similarly we define P M_j_ such that P M_j_ 'ljJ = '1/JM_j_ and the sum of PM and P M_j_ becomes an identity operator, i.e.
PM +PM_j_ =I. 1.4,.8 Idempotent operator The projection operator is an idempotent operator, namely P3vt = PM because P3vt'I/J = P M'I/JM =PM¢·
1.5 The postulates of quantum mechanics We start to formulate the postulates of quantum mechanics. We shall treat the first three postulates in this chapter, and leave the 4th postulate for the next chapter when we investigate the time evolution of a quantum system.
13
1.5 The postulates of quantum mechanics
1st postulate of quantum mechanics: For every physical system, there exists an abstract entity, called the state (or the state function or wave function that shall be discussed later), which provides the information of the dynamical quantities of the system; such as coordinates, momenta, energy, angular momentum, charge or isospin, etc. All the states for a given physical system are elements of a Hilbert space, i.e. physical system physical state
++ ++
Hilbert space 1l state vector 'ljJ in 1l
Furthermore for each physical observable, such as the 3rd component of the angular momentum or the total energy of the system and so forth, there associates a unique H errn,itian operator in the Hilbert space, i.e. physical (dynamical}
corresponding
observable
hermitean operator
total energy E coordinate
x
angular momentum
f
++ ++ ++
= Ht x = xt
H
The physical quantity measured in the system for the corresponding observable is obtained by taking the inner product of the pair 'ljJ and A'ljJ, i.e.
(A)= ('lj;,A'Ij;),
(1.24)
which is called the expectation value of dynamical quantity A for the system in the state 'lj;, which is normalized, i.e. 11'1/JII = 1. Since the action of operator A upon the vector 'ljJ changes it into another vector ¢, which implies that the action of the measurement of the dynamical quantity in a certain state usually would disturb the physical system and the original state is changed into another state due
14
Chapter 1
to the external disturbance accompanying the measurement. In particular, if an operator A such that A'I/Ja = a'I/Ja, i.e. when A acts upon a particular physical state '1/Ja, the resultant state is the same as the one before, then it is said that the physical state is prepared for the measurement of the dynamical observable associated with the operator A. We shall name:
• '1/Ja : the state particularly prepared in the system for the measurement of the dynamic quantity, called the eigenstate of the operator A. • a : the value of the measurement of the dynamical quantity in the
particular prepared state, called the eigenvalue of the operator A. We shall now explore some properties concerning the eigenvectors and the eigenv~lues through a few propositions.
Proposition 1. The eigenvalues for a Hermitian operator are all real.
Let A'I/Ja = a'I/Ja and At '1/Ja = a'I/Ja, and consider the inner product (A)'I/Ja = ('1/Ja, A'I/Ja) = ('1/Ja, a'I/Ja) = a('I/Ja, '1/Ja) =a. On the other hand, we have (A)'I/Ja = (At'I/Ja,'I/Ja) = (A'I/Ja,'I/Ja) = a*('I/Ja,'I/Ja) =a* which implies a = a* if '1/Ja is not a null vector.
Proposition 2. Two eigenvectors of a Hermitian operator are orthogonal to each other if the corresponding eigenvalues are unequal.
Let A'I/Ja
= a'I/Ja
and A'I/Jb
= b'I/Jb where '1/Ja =/= '1/Jb,
and since
('1/Ja, A'I/Jb) = b('I/Ja, '1/Jb) = (At'I/Ja, '1/Jb) = a*('I/Ja, '1/Jb) = a('I/Ja, '1/Jb),
15
1. 5 The postulates of quantum mechanics
therefore (a b)('I/Ja,'I/Jb) = 0. That implies ('1/Ja,'I/Jb) = 0 if a f= b. It often occurs that there exists more than one eigenvector of an operator with the same eigenvalue. Consider the Hermitian operator C, such that
(1.25) where CI = c2 = . . . = Cm = c, and ('1/Jq, 'I/Jc2 , ••• , '1/Jcm) are linearly independent. The eigenvalue cis called mfold degenerate if there are m linearly independent eigenvectors corresponding to the same eigenvalue c of the operator C.
Proposition 3. If the eigenvalue c of the opemtor C is degenemte, any linear combination of the linearly independent eigenvectors is also an eigenvector.
Due to the linearity of operator C, the linear combination L ai'I/Jci is also the eigenvector of C with the eigenvalue c. By means of the GramSchmidt orthogonalization process, one is able to easily construct a new set of orthonormal vectors {¢c1 , ¢c2 , ••• , ¢em} out of the previous rn linearly independent set {'I/Jc1 , 'I/Jc2 , ••• , '1/Jcm}, such that
(1.26) Following the results of Propositions 2 and 3, we conclude that
(1.27) where a, b refer to the eigenvalues and i, j refer to the index of degeneracy. We shall drop  on top of the orthonormal basis without mentioning it further.
16
Chapter 1
2nd postulate of quantum mechanics: The set of eigenvectors '1/Jai of a given Hermitian operator corresponding to a physical observable form the basis of a Hilbert space. Any state in the physical system can be denoted by a vector in the Hilbert space as a linear combination of 'lj;Ui, i.e. '1/J = L aai '1/Jai, where ('1/Ja., '1/Ja'.) = 8aa'8ij. J
The coefficient aai is obtained by taking the inner product
(1.28) The formulation of the 2nd postulate of quantum mechanics is purely artificial. In fact, it has been proved and well studied that a function space can be spanned by the eigenvectors of a Hermitian SturmLiouville operator. Since the dynamical operators in quantum system are not confined to those of the SturmLiouville form, we would formulate on purpose the second postulate of quantum mechanics in order to build and integrate the whole mathematical structure and the logical development of the quantum theory on a solid and self consistent ground.
1.6 Commutability and compatibility of dynamical observables We shall introduce in the following subsections some terminologies which will help to differentiate various types of the compatible observable.
1. 6.1 Compatible observables If there exist a complete set of linearly independent vectors 'lj;Ui which are eigenstates of both operators Rand S, then the two physical observables corresponding respectively to Hermitian operators R and S are said to be compatible.
17
1.6 Commutability and compatibility of dynamical observables
Proposition
4.
If two observables are compatible, their corresponding operators RandS commute, i.e. [R, S] = 0.
It is obvious because R and S have the following properties:
R'¢'a
= ra'¢'a,
and S'¢'a
= sa'¢'a,
which lead to (RS SR)'¢'a
=
[R, S]'¢'a
=
0,
if we define the commutator of R and S as [R, S] = RS  SR. Therefore [R, S]'¢' = 0 for any'¢', and Proposition 4 is established.
Proposition 5. If R and S are operators corresponding to two compatible observables, and if '¢'r are eigenvectors of R, then ('¢'r, S'¢'r')
= 0,
for
r
f= r'.
(1.29)
The proof of Proposition 5 is straightforward, i.e.
We have (r' r)('¢'r, S'¢'r')
= 0.
Hence Proposition 5 is proved.
Proposition 6. If Pr is the projection operator onto subspace with vectors '¢'r, then
The proof is again straightforward. For
18
Chapter 1
(1.30) we have
('1/Jr', [Pr, S]'I/Jr") = ('1/Jr', (PrS SPr )'1/Jr") = (P r'I/Jr', S'I/Jr")  ('1/Jr', SP r'I/Jr") = (8rr'  8rr") ('1/Jr', S'I/Jr") = 0, (prove it) that leads to ('ljJ, [Pn S] 'ljJ)
= 0 for
all 'ljJ, hence
(1.31)
Proposition 7. If RandS .are two commuting Hermitian operators, there exists a complete set of states which are simultaneously eigenvectors of RandS.
Let us construct a vector ¢~s), projected by P 8 upon the vector '1/Jr which is the eigenvector of the Hermitian operator R, i.e. ,~,(s) '+'r
p S'f'T• "''
(1.32)
It is obvious that ¢~s) is automatically the eigenvector of the Hermitian operator S with eigenvalues, namely
(1.33) On the other hand, Proposition 6 ensures that ¢~ ) is also an eigenvector of the operator R , because
8
(1.34) Hence Proposition 7 is proved.
1.6 Commutability and compatibility of dynamical observables
19
1.6.2 Intrinsic compatibility of the dynamical observables and the direct product space We have seen in the last section that if two operators corresponding respectively to two dynamical observables commute with each other, there always exists a complete set of states which are simultaneously eigenvectors of these two operators. The construction of the simultaneous eigenvectors could be simplified even further in some particular cases in which the compatibility of these two operators is solely based upon the first and the second fundamental commutation relations, i.e. [qi, qj] = [pi,Pj] = 0, without making use of the third fundamental commutation relation. The dynamical observables corresponding to the commuting operators in this particular category are said to be intrinsic compatible. The construction of the simultaneous eigenvectors of the intrinsic compatible observables is formulated in the following proposition.
Proposition 8. Let A and B be two operators corresponding respectively to two intrinsic compatible observables, and let '1/Ja;. and 'Pbj be the eigenvectors of A and B with the eigenvalues ai and bj respectively, then the direct product of '1/Jai and 'Pbj, denoted by '1/Jai ® 'Pbj is the eigenvector of the operator F(A, B) with the eigenvalue F(ai, bj)·
It can be easily shown that '1/Ja. ® 'Pbj is the simultaneous eigenvector of A and B with eigenvalues ai and bj respectively if we make the following identifications: A= F(A, 1),
and
B = F(I, B) .
.Two operators in different Hilbert spaces are always of intrinsic compatibility. Consider the system of a particle with spin, the physical observable associated with the configuration space is compatible with the physical observable in the spin space. Therefore one is able to express the quantum state as the direct product of a vector in the configuration space and another vector in spin space.
20
Chapter 1
We shall leave it for the reader to verify that Propositions 5 through 7 are consistent with the above formulation. We will also elaborate more on the algebra of the direct product space in Section 2.5 in order to provide a rigorous proof for Proposition 8. 1.6.3 3rd postulate of quantum mechanics and commutator algebra
3rd postulate of quantum mechanics:
Every Poisson bracket in classical mechanics for canonical variables (pi, qj) is replaced by the commutator of the corresponding operators with the following relations: Classical mechanics
Quantum mechanics
[qi, qj] = 0 [pi,Pj] = 0 [pi, ll.il = bij where h = 2n/i is Planck's constant.
We discuss the commutator algebra for further applications in later chapters. The commutator of operators A and B, as it is defined previously
[A,B] =ABBA,
(1.35)
[A, B] = [B, A], [A, A]= 0, [A, B + C] = [A, B] +[A, C], [A, [B, C]] + [B, [C, A]] + [C, [A, B]]
(1.36a)
then we have
{1.36b) (1.36c) =
0.
(1.36d)
21
1. 6 Commutability and compatibility of dynamical observables
An operator C is called a constant operator if it commutes with any operator corresponding to the dynamical observables. Obviously any real number times unit operator I is a Hermitian constant operator. When an operator is exponentiated, namely eA, it is defined as the usual sense of the exponential function, i.e.
(1.37) We shall now show a useful identity A
e Be
A_
= B
+ I1.1 [A, B] + 211. [A, [A, B]] + ....
(1.38)
Proof Let f(J..) = e>ABe>A and expand the function f(J..) in terms of power series of ).. at >. = 0, i.e.
Since,
f'(>.) = Af(>.) f(J..)A =[A,/(>.)], we can evaluate each order of the derivatives of f(J..) at )..
f(O) = B, f'(O) = [A, B], f"(O) = [A, [A, B]], and thus
f(J..) = f(O)
+ ~ f'(O) + ~~ J"(O) + ...
= 0,
i.e.
22
Chapter 1
becomes
f(J...)
= e>ABe>.A = B +>..[A, B] +>..~[A, [A, B]] + .... 2.
Therefore we reach the identity by setting >.. = 1 in the function
f(J...), i.e. /(1)
= eA Be A = B +[A, B] + 1! [A, [A, B]] + .... 2
(1.39)
1. 7 Noncommuting operators and the uncertainty principle The uncertainty in the simultaneous measurement of the coordinate and its conjugate momentum is due to the dual nature of a quantum particle, namely the particlewave duality. From the mathematical point of view, the uncertainty originates from the noncommutability of the two operators that correspond respectively to these two physical observables. In fact, uncertainty will arise for any simultaneous measurement of a pair of dynamical quantities if their corresponding quantum operators do not commute. A demonstration is in order to reestablish Heisenberg's famous uncertainty relations. For simplicity, we only consider the onedimensional case and denote b..q and b..p to be the uncertainty in coordinate and momentum simultaneous measurement respectively. b..q and b..p are also called the variance, or the deviation of q and p correspondingly defined as (1.40) (1.41) where '1/J is the quantum state of interest, (Q)I and (P)I are both constant operators obtained by means of multiplying respectively the expectation value (Q) and (P) by a unit operator. Let us introduce deviation operators Qd and P d as follows:
23
I. 7 Noncommuting operators and the uncertainty principle
= Q (Q)I, Pd = P (P)I,
(1.42)
Qd
(1.43)
and denote
(6.q) 2
= (
(1.44)
(6.p) 2
= (
(1.45)
Apply the Schwarz inequality for I and ¢2, i.e.
2 (
(1.46)
Thus we have
Let us introduce an anticommutator of Qd and P d
,
defined as (1.48)
It can be readily shown that the expectation value of an anticommutator with respect to the state 7/J is always real, say a= (7/J, {Qd, P d}'I/J). Furthermore since QdPd = ![Qd,Pd) + !{Qd,Pd}, and its expectation value is calculated as
Hence Heisenberg's uncertainty relation in the one dimensional case is derived, i.e. (1.50) The minimum uncertainty relation holds if the following conditions are fulfilled: (a) P d'l/J
= >..Qd'l/J,
(b) a= 0.
namely P d'l/J and Qd'I/J are linearly dependent.
Chapter 1
24
The coefficient >. can be evaluated by taking the expectation value of the anticommutator HQd, P d} with respect to the state 7/J, and putting it equal to zero. We reach
On the other hand, the expectation value of the commutator 1i 2 1 2 1i [Qd, P d] = : leads to >.(..6.q)  d6.p) = :, hence Z
Z
A
in
(1.51)
>. = 2(..6.q)2.
An explicit expression of the state function in terms of the least uncertainty 6.q shall be postponed until we cover the materials on qrepresentation in quantum mechanics.
1.8 Exercises
Ex 1.8.1
Consider the following set of continuous functions fn(x) = xn, x E [1,1] which spans a £ 2 (1,1)space. Find explicitly the first 3 orthonormal functions by the GramSchmidt process. What are those functions that occur to your mind? Ex 1.8.2
Prove that the Minkowski inequality holds in Hilbert space, i.e. 117/J + <1>11 ~ 117/JII
+ 11<1>11·
(Hint: take the square of either side.)
25
I. 8 Exercises
Ex 1.8.3 Prove the law of parallelogram holds in Hilbert space, i.e.
Ex 1.8.4 Prove that every finite dimensional vector space is complete. (Hint: since the real and complex numbers are complete.) Ex 1.8.5 Show that the
IIABII
~
IIAIIIIBII if A, B E
bounded operator.
Ex 1.8.6 Show that the expectation value of any dynamical observable in the physical system is always real. Ex 1.8.7 Prove that the adjoint conjugate can equivalently be defined as (A¢,¢)= ('lj;,At¢) if (At'lj;,¢) = ('lj;,A¢). Ex 1.8.8 Let {¢ 1, ¢2, ... ,
(Hint: let f(J..) =
e>Ae>Be>(A+B),
by Eq. (1.39). Then integrate it.)
and obtain d~~) =>.[A, B]f(>.)
26
Chapter 1
Ex 1.8.10 Consider a Hilbert space spanned by a Hermitian operator A. (a) Prove that
II(A al) is a null operator if A'I/Ja = a'I/Ja· a
(b) What is the significance of the following operator,
II A'al ? a a
a# a'
Ex 1.8.11
If two observables A1 and A2 are not compatible, but their corresponding operators both commute with the Hamiltonian operator H, i.e.
[A1,H) = [A2,H) = 0. Show that the energy eigenstates are in general degenerate. Ex 1.8.12 Consider a onedimensional Hamiltonian H
= 2~p2 + V(Q)
and use the fact that the commutator of Q and [Q, H] is a constant operator, to show that
which is referred to as the ThomasReicheKuhn sum rule, where Qsk = ('lj;8 , Q'I/Jk) and '1/Js is the eigenstate of H with eigenvalue E 8 , i.e. H'I/Js =
Es'I/Js·
Chapter 2 SpaceTime Translation, Quantum Dynamics and Various Representations in Quantum Mechanics
2.1 Vector space and dual vector space We start from enspace as an example. In matrix notation, a vector x in enspace can be represented by a vertical array of n complex numbers as follows Xl X2
x=
(2.1)
Xi
Xi+l
Xn
which is called a column matrix, where Xi is the ithcomponent of the vector. Each component is labeled by a subscript i. Matrix algebra ensures that the two basic operations of vectors, as we explained at the beginning of the first chapter, can be reproduced. The orthonormal basis of the ndimensional vector space ei is also represented by a column
27
28
Chapter 2
matrix. Take the ithbasis as an example: 0 0 1
(2.2)
0 0 that allows one to express the vector tion
x as the following linear combina
n
X=
~Xiei.
(2.3)
i=l
To perform the inner product of an ordered pair of vectors in en, one needs to introduce another vector space *en, a dual space to enspace. A vector in *en is denoted by a row matrix as follows
which is exactly the adjoint conjugate, also called the Hermitian conjugate of vector x in enspace, namely by taking the complex conjugate of each entry and transposing the column matrix into a row matrix in *enspace. One justifies immediately that *en is also a vector space. Let us see how to perform an inner product by means of matrix algebra. Take the ordered pair of vectors x andy as an example. We construct a dual vector x, which is an adjoint conjugate of the first vector x. Then the inner product of vectors x and fj is obtained from the multiplication of the row matrix x in *en space by the column matrix fj in enspace, i.e.
29
2.1 Vector space and dual vector space
Yl
 ( * x *
xy x 1
2
. •.
*)
Y2
Xn
(2.4)
Yn The linear operator A in enspace is represented by a square matrix with matrix elements Aij for i, j = 1, 2, ... , n, obtained by sandwiching the operator A between the basis row vector ~ and the basis column vector €j, i.e. (2.5) If we take the adjoint conjugate of the above equation, we have (2.6) on the left hand side of Eq. (2.5), while on the right hand side of Eq. (2.5), we have merely a complex number, because its adjoint conjugate is just the complex conjugate. Hence we have the following relation (2.7) A matrix A which equals to its adjoint conjugate matrix, namely A = At is called a Hermitian matrix. Then we have
(2.8) or Aij = Aji• by taking the complex conjugate on both sides. The expectation value of an operator A with respect to the vector x is taken by multiplying the row vector x, then by n matrix A, and the column vector x successively as follows
xAx = expetation value of A with respect to x. If the operator stands for a physical observable, then it is a Hermitian matrix and the expectation value is real because
xAx
= xAtx = (xAx)t = (xAx)*.
(2.9)
30
Chapter 2
The matrix algebra also ensures that n x n square matrices follow the rules of being linear operators. Matrix representation for enspace allows us to construct the projection operator readily. A projection operator that projects a vector x onto a subspace spanned by basis ei is a square matrix formed by putting a column basis before a row basis, i.e. (2.10) or more specifically,
0
0
0
0
1
(o o ...
1
0
...
o ... o)= o
0
0
0
0
00
1
0
...
...
0
0 0
0
0 0
0
The projection operator PM onto a subspace M simply takes the sum of the projection operators L Pi, i.e. PM =
L Pi
for
ei E
(2.11)
M. n
It is obvious that the sum of the projection operators the basis in the entire space is an identity operator, i.e.
Lpi over i=l
(2.12)
It is advantageous in this matrix representation that it allows one to easily construct any projection operator. If one wants to project a
31
2.1 Vector space and dual vector space
vector onto a particular vector can then be expressed as
y, the corresponding projection operator (2.13)
The Hilbert space corresponding to a physical system is often infinite dimensional and also not necessarily separable. It becomes awkward or even impossible to use the matrix formulation in those cases. To overcome the inconvenience and the difficulty, Dirac introduced new notations for the vectors in Hilbert space. A vector in 1l is den9ted by the symbol Ia), called a ket vector, or simply a ket. While a vector in the adjoint conjugate dual space *1l is denoted by the symbol (al, called a bra vector, or simply a bra. Similar to the case in enspace, to construct the inner product of an ordered pair of vectors in Dirac notation, say the vectors Ia) and 1.8), we put the adjoint conjugate of Ia), namely the bra vector (al in front of the ket vector I.B) as follows
(aii.B)
= (a I.B),
(2.14)
which reaffirms the definition of the adjoint conjugates of operator A. The square norm of a vector Ia) is defined as the inner product of a pair of same vectors, i.e. (2.15) A linear operator A which transforms a ket vector Ia) into another ket vector Ia') is written as
Ia') =Ala).
(2.16)
The inner product of the ordered pair of vectors of Ia) and 1.8') is written as follows
(ai.B') = (a I (AI .B))
= (aiAI.B).
(2.17)
Generally we omit writing the parenthesis as it appears in the second term of last equation. On the other hand, if the inner product is constructed for an ordered pair of vectors Ia') and I.B), in which the vector Ia') is a transformed vector of Ia) by operator A, then the corr~sponding inner product is written as
32
Chapter 2
(2.18)
Similar to the case in enspace, we conclude that (2.19)
and the definition of adjoint conjugate of A is regained. Dirac notations in vector space have the advantage of constructing the projection operator readily. Consider a Hermitian operator A, and the eigenvector Ia) with the corresponding eigenvalue a, i.e.
Ala)= ala).
(2.20)
The set of eigenvectors {la1), la2), ... , lai), ... } forms the basis of the Hilbert space 1l. The projection operator onto the subspace Ia) is designed by putting a ket vector ~nd a bra vector together in successive order, i.e. (2.21)
Any vector Ia) in the Hilbert space is expressed in a linear combination of its basis as follows
Ia)
= LaJiaj),
(2.22)
j
where the coefficient aj is obtained through the inner product, i.e. (2.23) j
j
Projection of the vector onto the subspace spanned by the basis lai) is then read as (2.24)
A projection operator onto subspace M can also be constructed by (2.25)
33
2.2 qrepresentation and prepresentation in quantum mechanics
By extending subspace M to cover the whole Hilbert space, we reach the closure relation, (2.26) In the case of a separable Hilbert space with degeneracy, and denoting the basis by ja, i), the closure relation reads as
LPa,i = a,i
L ja,i)(a,ij =I.
(2.27)
a,i
For the case that the eigenvalues become continuous, the eigenvectors I€) are no longer denumerable. The basis vector is characterized by the eigenvalue €, which is a continuum and the inner product is normalized to be a delta function, i.e.
(€1€') = 8(€ €'),
(2.28)
and the closure relation in this case takes the form as
1P~d€ = 1l€)d€(€1
=I.
(2.29)
2.2 qrepresentation and prepresentation in quantum mechanics Consider the Hilbert space spanned by the eigenvectors of the Hermitian operator X corresponding to a dynamical observable in coordinate x. For simplicity, we limit ourselves to the onedimensional case. The eigenvalue equation reads as
Xlx)
= xlx).
(2.30)
H we make a transport of the physical system to the right by a distance €, namely the spatial translation of the system from x to x + €, it is interesting to observe the change of ~he eigenvector in the new system.
34
Chapter 2
Let us first introduce a unitary operator, called the translational operator, defined as
(2.31) where ~~real
parameter, namely the distance of the spatial translation,
= momentum operator,
P
and U(P; ~) is unitary because
ut (P; ~) = e*~P = u 1 (P; ~). Proposition 1. U(P; ~)lx) is an eigenvector of operator X with eigenvalue x+~.
To prove the above statement, let us start with the commutation relation of X and P, i.e. [X, P] = iiLI, that enables us to evaluate the following commutators [G
[X,Pn] = pn 1 [X,P] and
+ [X,Pn 1]P =.i?11WnI,
rv.l '/ { "f' \'" » f
>'?!' 'y \? 
pc,U]=ill[x.~~~ 1
/
(2.32)
Hefpm] (
· )m1
= ~L (m 1)! ~~
pm1 =
~U,
(2.33)
m
or
XU= Construct the new state
U(X+~I).
(2.34)
35
2.2 qrepresentation and prepresentation in quantum mechanics
Ia) = Ulx). Since
Xla) = XUix) = U(X + €I)Ix) = U(x + €)1x) = (x + €)Ia),
(2.35)
it implies that Ia) is also the eigenvector of operator X, with eigenvalue x + €. Therefore we cast vector Ia) as
Ia) = clx+€) = lx+€). The constant c is just a pure phase factor that is chosen to be a unit without losing its generality.
Proposition 2. A state function or wave function as it is generally called, is defined as the inner product of the vector lx) and the state vector 1'¢) of interest, i.e. 'ljl(x) = (xi'¢). Furthermore, sandwiching the momentum operator in between bra vector (xl and state vector 1'¢) results in differentiating the wave function, or more precisely,
no
(xiPI'¢) = :!:l'¢(x). ~
ux
We consider an infinitesimal translation
U(P;€) = 1
and
~€P,
€,
then
36
Chapter 2
= (x!x' + €) =
(x!U(P; €)1x')
8(x x') 8(x x' €) (x!Pix')
(xl ( 1 
~€P) lx'),
= ~€(x!Pix'),
= ~dd 8(x x') = ~8'(x x'), ~
~
X
where 8'(x x') is the derivative of the delta function with respect to its argument, i.e.
d~ 8(x x') =
8'(x x') =
d(x ~ x') 8(x x'),
and it has the following property:
J
(2.36)
f(x)8'(x a)dx =!'(a)
for any function
(x!PI'I/J)
f (x). With this property, we are able to derive that
J
= (x!Pix')dx' (x'I'I/J) =
~~
J
=
J~8'(x
_.!!._8(x' x)'I/J(x')dx' dx'
x')'I/J(x')dx'
= ~_!!_'1/J(x). ~dx
(2.37)
We are now in the position to make a comparison between the abstract state and the corresponding qrepresentation.
2.2 qrepresentation and prepresentation in quantum mechanics
Hilbert space
37
qrepresentation
(xl'l/J) = 'ljJ(x)
1'1/J) Xl'l/J)
(xiXI'l/J) = x'ljJ(x)
Pl'l/J)
(xiPI'l/J) = :d '1/J(x)
li d
fi d Z X F(:d ,x)'ljJ(x)
F(P,X)I'l/J)
~
X
'1/J(x) in the above table is called the wave function or the state function, and the absolute square of '1/J(x), i.e. l'l/J(x)l 2 is commonly referred to as the probability density of the system. We particularly consider the eigenvalue equation for the energy of the system. The Hamiltonian operator reads 1 H= P+ V(X), 2m
(2.38)
and the eigenvalue equation
Hl'l/J) = El'l/J),
(2.39)
when it is expressed in terms of qrepresentation, becomes
(xiHI'l/J) = E(xl'l/J),
(2.40)
or explicitly, the time independent Schrodinger equation as
fi2 ~ ( 2m dx2
+ V(x) )
'1/J(x) = E'ljJ(x).
(2.41)
The prepresentation of the state function can be formulated parallel to that of the qrepresentation with the help of the following proposition.
38
Chapter 2
Proposition 3. Qrepresentation of the eigenstate of the momentum operator is a plane wave, namely 1 i (xlp) = enPx
V21Ji
where IP) is the eigenstate of P, i.e,with the eigenvalue p, i.e. with Pip) =PiP).
Let us consider (x!Pip) and insert the identity operator after the momentum operator P. We have
(x!Pip)
=
J
li i8'(x x')(x'lp)dx'
(x!Pix')dx'(x'lp)
=
J
p(xlp)
=
li d :d (xlp).
=
J lx')dx'(x'l lid i dx (xip),
or
X
~
Solving this differential equation, we obtain i 1 i (xlp) = AenPx = enPx.
V21Ji
The normalization constant can be verified as follows,
(xlx')
That implies
= 8(x x') =
IAI 2 = 1/27rli.
J
(xip)dp(plx')
(2.42)
2.2 qrepresentation and prepresentation in quantum mechanics
39
Similar to the case in the qrepresentation, the wave function in pspace reads as (2.43) In fact '1/J(p) is the inverse Fourier transform of '1/J(x), i.e.
'1/J(p) = (pi'I/J) =
J
(plx')dx' (x'I'I/J) =
~
J
e*px' dx''I/J(x').
(2.44)
The SchrOdinger equation in the prepresentation can be written as
(piHI'I/J) =
(PI 2~ P 2 + V(X)I'I/J).
(2.45)
The first term is trivially obtained,
(2.46)
while the second term can be evaluated by the closure relations,
(PIV(X)I'I/J)
= /(PIV(X)Ip')dp'(p'I'I/J),
where
(PIV(X)Ip') =
jj (plx')dx' (x'IV(X)Ix")dx" (x"lp')
=  1 Jr f e*(p'x"px')V(x')8(x' x")dx'dx" =
2rrli
}
1 rrli
J
2
eni(p' p)xv(x)dx
= V(pp'),
(2.47)
40
Chapter 2
is the Fourier transform of the potential in pspace. Finally we reach the Schrodinger equation in the prepresentation as follows
: : '1/J(p)
+
j V(p p')'I/J(p')dp' = E'I/J(p).
(2.48)
Before leaving this section, let us go back to the qrepresentation of the state with the minimum quantum uncertainty !:l.p!:l.q = n/2 discussed in Section 1.7, where 1'1/J) satisfies the following condition
(2.49)
The qrepresentation of the above equation reads as
na ( i ox 
(p)
) '1/J(x) =
in
2(!:l.x)2 (x  (x) )'1/J(x),
(2.50)
which allows us to obtain the normalized wave function of minimum uncertainty as
'1/J(x)
1
= [27r(!:l.x)2]1/4 exp
[ (x (x) )
4(/::l.x)2
2
i(p)x]
+ n .
(2.51)
2.3 Harmonic oscillator revisited There exists very rich structure in dealing with the harmonic oscillation mathematically on a quantum level. We shall attack the problem by means of the number operator through the creation operator and the annihilation operator rather than solving the Schrodinger equation in the qrepresentation, in which the quantization of the energy levels is derived from the boundary conditions of the wave functions.
41
2.3 Harmonic oscillator revisited
The technique developed in this section can even be applied to the system of linear oscillators in the limit of continuum, which gives rise to the quantization of the field.
2.3.1 Creation and annihilation operators The Hamiltonian operator for the onedimensional harmonic oscillator reads as (2.52)
Let us introduce the new variables defined as follows: 1
I
Pt:.
:' i I ' \ '\
(2.53)
= JmwnP,
The Hamiltonian operator takes simpler form,
,,
(2.54)
with 1l defined as 1l = (p~ + 1;.2 )/2. The nonvanishing canonical quantization relation of P and X, i.e. [P, X]= nlfi can also be simplified as ' . ,'I
\
[p~:., eJ = ii.
(2.55)
I . \
Define the annihilation operator and the creation operator as follows
and
or reversely
(2.56)
42
Chapter 2
(2.57)
Pt:. =
~(at a).
(2.58)
The commutation relations become: (2.59)
Then 1l can be cast into 1l = (at a+ aat)/2 = N + 1/2 where ata, a Hermitian operator called the number operator, which is treated as a new dynamical observable. Consider the Hilbert space spanned by the eigenvectors of the number operator. The properties concerning the eigenvalues as well as the eigenvectors are summarized in the following propositions.
N
=
Proposition 4. If v and lv) are respectively the eigenvalue and eigenvector of number operator N, then v is positive definite.
For the eigenvalue equation of operator N, Nlv) = vlv),
(2.60)
where the eigenvector is labeled by the eigenvalue and denoted it by Iv). Let us define the vector Ia) = alv). The positive definite of the norm of the vector Ia) requires that (ala) Therefore v
~
=
(vlat alv)
= v(vlv)
0, namely vis positive definite.
~ 0.
43
2.3 Harmonic oscillator revisited
Proposition 5. The state with zero eigenvalue i.e. IO), then IO) will be annihilated into the null vector by a, the annihilation operator, namely aiO) = 0.
Let Ia) = aiO), and the norm square of Ia) reads as
(ala)= (OiataiO) = (OINIO) = O(OIO) = 0,
(2.61)
which implies that aiO) is a null vector.
Proposition 6. atlv) is an eigenvector of N with eigenvalue v + 1.
This can be proved with the commutation relation
(2.62) For ~v) =/= 0 and letting Ia) = atlv), then
Nla) = Natlv) = at(N + I)lv) = at(v + l)lv) = (v + l)la). (2.63)
Proposition 7. For lv) =/= 0, then alv) is the eigenvector of N with eigenvalue v1.
Since [N, a] = a, or N a= a(N I), and letting Ia) = alv), then
Nla) = Nalv) = a(N I)lv) = a(v I)lv) = (v l)la).
(2.64)
44
Chapter 2
Proposition 8. The eigenvalues of the number operator N take only nonnegative integers, i.e. v = n = 0, I, 2, ....
The proof can be facilitated with Proposition 7. If v =/= integer, then vectors alv), a 2 1v), a 3 1v), ... , a 8 lv), are all eigenvectors of N respectively with the eigenvalues vI, v 2, ... , v s. Let Ia) = alv s), we calculate the square norm of Ia), i.e. (ala)= (v s)(v slv s). This does not guarantee that (ala) is positive definite if s is chosen larger than v. Positive definite of the norms of all eigenvectors generated by means of applying a 8 on lv) can only be achieved if v takes nonnegative integers, namely v = n = 0, I, 2, .... Therefore the eigenvectors of the number operator N, as well as that of the new Hamiltonian are written as IO), II), 12), ... , In), ... with the normalization (nln') = 8nn'· The eigenvector with zero eigenvalue, i.e. IO) is called the ground state of Nor 1l. The eigenvector II) is called the one particle state for operator N, or the first excited state of the Hamiltonian operator 1l and so forth. This is the reason that at is called the creation operator or the promotion operator, while a is called the annihilation operator or the demotion operator. at acts on state In) to produce the state cln +I), i.e.
so that the action of at will increase the n particle system into the (n+ I)particle system or will promote the nth excited state into (n+ I)th excited state. The constant c is determined to be J n + I when all eigenstates are ·orthonormalized, i.e. (2.65)
2.3 Harmonic oscillator revisited
45
Similarly operator a acts upon state In) and it becomes the state c'ln 1), i.e. aln) = c'ln 1). The action of a decreases the nparticle system into the (n  I)particle system or demotes the nth excited state into the (n 1)th excited state. Again the constant c' can be evaluated to be .Jii, i.e.
ain) = vnln 1).
(2.66)
With the help of the creation operator, one is able to construct the orthonormal basis of the Hilbert space spanned by the eigenvectors of the number operator N. We start from the ground state or zero particle states and build the higher excited states or many particle state by acting the creation operator upon it successively._ Then we have
IO),
(2.67)
Conversely, we are also able to construct the lower excited states by acting the annihilation operator upon thenparticle states successively, i.e.
46
Chapter 2
IO) =all),
In)=
1
Vn+Ialn + 1). n+l
(2.68)
2.4 Nrepresentation and the Rodrigues formula of Hermite polynomials It is interesting to investigate how the eigenstate In) is to be realized in the qrepresentation. Let us consider the eigenvalue equation of the operator i.e.
e,
(2.69)
One can also easily verify that
with the normalization (ele') = 8(e e'). The inner product of Ie) and In) defines the qrepresentation of thenparticle state, namely thenparticle wave function or the wave function of the nth excited state, which reads as
2.4 Nrepresentation and the Rodrigues formula of Hermite polynomials
Since at=
~(e ip~;), the wave function '1/Jn(e)
47
becomes
By Proposition 2, we have
To obtain the 0particle wave function '1/Jo(e), we take the qrepresentation of aiO) = 0, i.e.
(eiaiO) =
o,
namely that
or (2.70)
Therefore we can solve the equation and obtain the normalized wave function as
(2.71) and the wave function '1/Jn(e) becomes (2.72)
48
Chapter 2
Applying the following identity
we conclude that
de e_.!e (e ded)n = (1)n1e(d)n e2
2
•
Then
(2.73)
and the Rodrigues formula of Hermite polynomials is identified as
(2.74)
2.5 Two dimensional harmonic oscillation and direct product of vector spaces Let us consider the system of a twodimensional isotropic harmonic oscillator whose Hamiltonian takes the following expression H
= _!__(p2 + p2) + m w(X2 + y2). 2m
x
Y
2
(2.75)
Similar to the case of onedimensional harmonic oscillation, we define the creation operators and the annihilation operators as follows:
2. 5 Twodimensional harmonic oscillation and direct product of vector spaces
1
49
at=
J2(e ip~;),
(2.76a)
a=
~(e+ip~;),
(2.76b)
(2.76c)
(2.76d)
where
e, rJ, Pt; and PrJ are defined as '== ~
'IJ
=
rm:x,
Vh
Pt;
=
1 P Jmfu» x,
Vrmw hy'
(2.77a)
(2.77b)
The fundamental commutation relations can be simplified as
[e, ,.,J = [e,p1]J = o, [p~;,'IJ] = [p~;,p1)] = 0, [p~;, eJ = [p1), ,.,J = ii, which lead to the following commutation relations among the annihilation operators and the creation operators, namely
[a, b] = [a, bt] = [at, b] = [at, bt] = 0, [a, at]= [b, bt] =I. The Hamiltonian can therefore be written as
H=
(ata+~l)fu»+ (btb+~l)fu»,
(2.78) (2.79)
50 or H
Chapter 2
= Ha + Hb
if we put
(ta a+ 211) (
Ha
l1J.JJ
Na
1)
+ 21
11J.J.J,
To solve the Schrodinger equation of the two dimensional harmonic oscillation, we shall make use of Proposition 8 in Section 1.6.2. Evaluating the commutator of the operator Na and the operator Nb, we find that only the first and the second fundamental commutation relations are required to show that [Na,Nb] = 0. Therefore we recognize that Na and Nb are, in fact, the intrinsic compatible observables. The eigenvectors 1'1/J) in the following Schrodinger equation
(2.80)
can be expressed as the direct product of the two eigenvectors in onedimensional case, i.e.
(2.81) with the corresponding eigenvalues given as
E
= (na + nb + 1)11J.JJ = (n + 1)11J.JJ.
(2.82)
The energy of the twodimensional harmonic oscillator is also quantized to be (n + 1)11J.JJ, with degeneracy if na + nb = n > 0. We are now in the position to discuss the algebra of the direct vector space. A direct product space of the Hilbert spaces 1la and 1lb is also a Hilbert space, denoted as follows
(2.83)
2.5 Twodimensional harmonic oscillation and direct product of vector spaces
51
Let 1la be the Hilbert space spanned by lai}, the eigenvectors of the dynamical operator A, then the basis of 1la takes the following set of vectors,
Similarly, if the Hilbert space 1lb is spanned by another set of basis, denoted by
which are the eigenvectors of another dynamical operator B. The product space 1i = 1la ® 1lb is then spanned by the set of direct product of a base vector lai) in 1la and another base vector lbj} in 1lb, denoted by
(2.84) This very direct product vector iai; bj) is also the eigenvector of any linear operator in 1i if the operator is constructed in the form of a function of two operators A and B, denoted by F(A,B), and the eigenvalue equation for the operator F(A,B) reads as follows
(2.85) where F(ai, bj) is the eigenvalue of F(A, B). The dimension of the product space 1i equals to the product of the dimensions of the two individual spaces 1la and 1lb, namely d = dadb with da, db and d standing for the dimensions of 1ia,1ib and 1i respectively. The linear operator in 1i also takes the form of direct product of two linear operators, denoted by RandS, where R is a linear operator in the space 1la and Sis a linear operator in the space 1lb. The direct product operator is commonly denoted by R ® S, whose algebraic operational rule reads as
(2.86) We summarize the properties of the direct product operators as follows:
52
Chapter 2
R®O = 0®8 =0, 1®1 = I,R®I = R,l® 8 = 8, R ® (81 + 82) = R ® 81 + R ® 82, (R1 + R2) ® 8 = R1 ® 8 + R2 ® 8, aR ® (38 = a(3R ® 8, (R ® 8) 1 = R 1 ® 8I, R1R2 ® 8182 = (R1 ® 81)(R2 ® 82),
(2.87a) (2.87b) (2.87c) (2.87d) (2.87e) (2.87f) (2.87g)
where a and (3 are scalars. To demonstrate the difference between the compatible observables and the intrinsic compatible ones, we take the motion of a charged particle in the uniform magnetic field as an example. Consider a uniform magnetic field of Ho in zaxis, the vector potential A(x, y) can be expressed as __. A(x, y)
1 =
Ho( yi + xj). A
A
2
The principle of minimal interaction allows us to cast the Hamiltonian of a particle, with the mass m and the charge q moving in the uniform field Ho, into the following form q H =1 ( PA 2m c
)2
1q ) = 1 { ( Px+HoY
2m
2c
2( +
1q P  HoX Y 2c
)2} .
(2.88)
The sum of the two square terms in the Hamiltonian of last equation ensures that the total energy of the system is always positive definite. Namely when we solve the following Schrodinger equation
HI'¢') = El'¢1), we shall obtain the positive definite eigenvalue E.
(2.89)
2.5 Twodimensional harmonic oscillation and direct product of vector spaces
53
By expanding the quadratic terms of the above Hamiltonian, we obtain the following expression, i.e.
H
= _l_p2
2m x
+
m
2
(Wc)2 x2 2
+
__!_p2 2m Y
+
m
2
(Wc)2y2 !w L (2.90) 2 2 e z,
where We is the cyclotron frequency and Lz stands for the 3rd component of the angular momentum operator which can be calculated explicitly as follows:
qHo
Lz
We=,
me
= XPy YPx.
One recognizes immediately that combining the first two terms of Eq. (2.90) is , in fact, the Hamiltonian of a two dimensional harmonic oscillator with the force constant k
H{2) = __!_(p2 h.o. 2m x
=
+ p2) + Y
m (We)2 2 2 , i.e. m
2
2 (We) (X2 2
+ y2).
(2.91)
It can be easily verified that H~~~. and Lz commute with each other i.e. {2)

[Hh.o,,Lz] 0.
(2.92)
We will run into a dilemma if we apply the method of Proposition 8 to calculate the eigenvalues of the following Hamiltonian (2.93) which is the sum of two compatible observables. The eigenvalue equations for these two observables read as
54
Chapter 2
(2.94) (2.95) where m takes the integers, while n 1 and n2 take only the positive ones. Of course we have made use of the result that the angular momentum Lz is quantized as it is expressed in Eq. (2.95), whiCh we shall investigate in the next chapter. Should we try to evaluate the total energy of the system by applying
H = H~~~ wcLz/2 to act upon the direct product vector ln1; n2) ® lm}, we would end up in the following eigenvalue equation
(2.96)
with E = (n1 + n2 + l)fil»c/2 mfil»c/2, which is no longer a positive definite quantity if m > (n1 + n2 + 1). Namely the total energy of the system E obtained in this way does not guarantee to be positive definite, which contradicts our previous assertion. The contradiction arises from the fact that the observable H~~l. and the observable Lz are not intrinsic compatible even though they commute with each other. One needs, in fact, all the three fundamental commutation relations in order to prove that the commutator of H~~~ and Lz vanishes, i.e. {2) [Hh.o.,
]
Lz = 0.
(2.97)
The dilemma can be resolved by introducing another set of linear operators in terms of the original four dynamical observables with the following relations:
2.5 Twodimensional harmonic oscillation and direct product of vector spaces
55
(2.98a)
(2.98b)
(2.98c)
dt
= ....!.._ (mW; J2 { V W (X + iY) 
iJ
2 ml7wc
(P  iP ) } . x
Y
(2.98d)
It can be verified that the commutation relations among c, ct, d and dt become as follows:
[c, d]
=
[c, dt]
= 0,
(2.99a) (2.99b) (2.99c)
If we construct two Hermitian number operators defined as Nc = c t c and Nd = dtd. It is obvious that Nc and Nd become the intrinsic compatible observables because that [Nc, Nd] = 0 is derived only based upon the first two sets of the commutation relations, i.e. Eqs. (2.99a) and (2.99b). Therefore we can construct the direct product vector Inc; nd) = Inc) ® lnd), where Inc) and lnd) stand for the eigenvectors of the operators Nc and Nd respectively. Using the property that Inc; nd) is automatically the eigenvector of the of the operator F(Nc, Nd) with the eigenvalue F(nc,nd), we are able to calculate the eigenvalue E of the Hamiltonian operator H in the following equation
(2.100)
56
Chapter 2
After some algebra, the two terms H~~~ and Lz in the Hamiltonian above can be expressed in terms of the new dynamical operators Nc and Nd as follows: (2.101)
(2.102)
which allow us to take F(Nc, Nd) as follow
(2.103)
and the eigenvalue E is evaluated as
(2.104)
which is positive definite as we have expected at the beginning of our discussion on the system of a charged particle moving in the uniform magnetic field.
2.6 Elastic medium and the quantization of scalar field Consider the linear chain of N identical particles of mass m, with the springs of force constant a attached to both sides of the nearest mass points. Each mass point is also attached with another spring of force constant (3 to the equilibrium positions which are of equal spacing along the chain, as shown in Figure 2.1. The Hamiltonian of the system reads as
(2.105)
57
2. 6 Elastic medium and the quantization of scalar field
where Qr is the displacement of the rth mass point away from its equilibrium position, and Pr is the corresponding canonical conjugate momentum. The commutation relations are expressed as
The Hamiltonian can be simplified tremendously if we introduce the normal coordinates denoted by Q s through the following Fourier expansion
_[J;L Nm
qr

21rirsQ s· eN
(2.106)
s
Here we assume the total number of mass points to be N, and
8
={
! (
!(
takes from s = N  1) to s = N  1), they are integers if N is odd, and half odd integer if N is even.
The condition that qr is real will lead us to take Q_ 8 = Q!. Before we try to decouple the mixing terms in the Hamiltonian, we establish the following formula:
l
equilibrium point Figure 2.1: Linear chain of identical particles
58
Chapter 2
Proposition 9. For r
= integers, and s  s' = integers, then N
_!__"""' 21rir(ss') N L.....,;e N

8
ss'·
r=l
Denoting z
27ri ( ') = eN ss , we have
1 N z(1  zN) 1 """' r 2 F(z)= NL.....,;z = N(z+z + ... +z )= N( 1 z)' r which implies that for the cases s'
(2.107)
= 0,
F(z) = F(1) = 1. For the case that s  s' = nonzero integers, say s  s' = m then z = exp(27riN/m), which is one of the roots f.or the equation zN 1 = 0. Therefore 1
F(z) = N (1
+ z + ... + z N1 ) =
0,
and one reaches the conclusion
F(z) =
8ss'·
(2.108)
This proposition allows us to invert Qr into Q 8 • We leave it as an exercise to invert Eq. (2.106) and obtain that
(2.109)
Similarly the Fourier expansion for the momentum is expressed as
2. 6 Elastic medium and the quantization of scalar field

Pr
{:;;Ji ~ VN L.....te
27rirs
N
Ps,
59
(2.110)
s
with
and
Ps
=
1
~
vmNn
r
~ L.....t e
_21rirs
N
Pr·
(2.111)
It also leaves you as an exercise to show that each term in the Hamiltonian operator can be calculated as follows,
(2.112)
~~ ~ . 2 ( 27rs) L.....t(qr  Qr+I) 2  an m L.....t 4sm N Q 8 Q 8t , 2 r 2 s
(2.113)
(2.114)
Therefore the Hamiltonian takes the following expression
(2.115)
where w2 ( s) is defined as
4a . 2 (21rs) w2 (s)=sm m
N
(3 +, .m
(2.116)
60
Chapter 2
and w(s) is an even function ins, i.e. w( s)
= w(s).
It can be verified that the commutation relations for the operators expressed in terms of normal coordinate and momentum become (2.117) Let us investigate the system in the Heisenberg picture in which the operators are regarded as time dependent. The equations of motion for Q 8 , Q1, P 8 and P 1 take the identical expressions as the canonical equations of motion in classical mechanics, namely,
or
= _pt
'8H
aQt
or
B'
w 2 (s)QsP• 8t •
Hence we have ..
2
= 0,
(2.118)
(a seiw(s)t +ats eiw(s)t) '
(2.119)
Qs +w (s)Qs
and the solution is cast into Q
s
=
1 J2w(s)
where a 8 / J2W(S) and a~ 8 / J2W(S), playing the roles of the constants of integration, are chosen such as to meet the condition Q 8 = Q~ 8 . Similarly we have the solution for P 8 as
P. s
=
i
fwW (a seiw(s)t +ats eiw(s)t) · V2
(2.120)
2. 6 Elastic medium and the quantization of scalar field
61
Let us define b s =aseiw(s)t and bts =ats eiw(s)t. Then the normal coordinate and the normal momentum become as follows
.zy/w[i) 2( bs + b_t
Ps 
2
8 ),
which enables us to write the commutation relations among the operators as
The Hamiltonian operator can then be expressed as
1 =It Lw(s)(Ns + 21),
(2.121)
s
where N 8 = b!bs is the boson number operator of sth mode. Let us investigate the system of an elastic medium in the limit of continuum. Take the length of the linear chain be l with N mass points equally spaced when they are at equilibrium. Then the position for the rth particle is given as (2.122)
We also define the wave number
ks
27!'
= ls,
(2.123)
62
Chapter 2
and rescale the displacement away from the rth equilibrium position at the time t defined as
=
=
e ;/rs 1 (a eiw(s)t +at eiw(s)t) Vf!i"""' l ~ V2W(S} s s 2
f!i"""'
Vl
~
1
(a ei(ksXrW(s)t) 8
J2w(s)
+ atei(ksXrw(s)t)) 8
'
(2.124) and the corresponding momentum as
II(xr, t) =
~i L
ff (
asei(ksXrw(s)_t)
+ a!ei(ksXrw(s)t)) ,
s
(2.125) where the last term in Eqs. (2.124) and (2.125) are obtained by making use of ks = ks and w(s) = w( s). The commutation relation between ~(xr. t) and II(xr'• t) can be evaluated as
Let us take the limit as N + oo, the positions Xr = rl / N become continuous, and the subscript r can be removed from Xr, namely that
x
=
lim
N+oo
Xr
=
l lim r N.
Ntoo
(2.126)
63
2. 6 Elastic medium and the quantization of scalar field
We also simplify the wave number ks into k, therefore ~(x, t)
=
lim
N+=
~(xr. t)
= ~"" 1 Vl ~ J2w(k)
(2.127)
(a ei(kxw(k)t) +at ei(kxw(k)t)) k
k
'
II(x, t) = lim II(xr. t) NtCXJ
=
~ L i~ ( akei(kxw(k)t) + alei(kxw(k)t)) , k
(2.128) and the commutation relation between [~(x, t), II(x', t)]
= lim
NtCXJ
~(xr. t)
and II(xn t) reduces to
[~(xr, t), II(xr'• t)]
N
=in lim l <5rr'I = iM(x x')I.
(2.129)
NtCXl
It is on the quantum level that the continuous elastic medium is treated as a space being filled by the quantized scalar field or boson field. The linear system can be generalized into the threedimensional case. The field and its conjugate momentum take the expressions as
~(r
(a ,t) = v0_"" l ~ J2w(k)
IJ(r, t)
1
=
ei(krw(k)t) +at ei(krw(k)t)) k
~ L i~ ( akei(krw(k)t) +
k
,
(2. 130)
alei(krw(k)t)),
k
(2.131)
64
Chapter 2
and the commutation rules read as follows:
= [II(r, t), II(r', t)] = 0, [~(r, t), II(r', t)] = iM(r r')I. [~(r, t), ~(r', t)]
(2.132) (2.133)
2. 7 Time evolution operator and the postulate of quantum dynamics The universe evolves in time and so does the physical system. Any quantum state in the quantum mechanical system always changes in time, just like the motion of the particles in classical mechanics changes its configuration according to laws of motion. All the physical states as well as the dynamical observables we have dealt with in the first three postulates of quantum mechanics were expressed in such a way that time is totally absent from the equations. In fact, the physical quantities we had met or dealt with are time dependent implicitly, namely that we set our observation of the system at a convenient time scale, say t = 0. Choosing the time scale is entirely arbitrary . We would like to express the physical state at the particular time t = to to perform the observation or to make the measurement. Then'the state will be denoted by I'¢; to). All the contents in our previous discussion do not change and all the theories and conclusions we have derived so far remain the same if we replace 1'¢) by I'¢; to). 2. 7.1 Time evolution operator and Schrodinger equation What is more interesting and challenging for us is how the state I'¢; to) will change from the moment t 0 to any future time t. Let us denote a time evolution operator by U(t, to). It is a linear operator whose action upon a physical state I'¢; to) will evolve the state into a new state I'¢; t) governed by the law of quantum dynamics. Mathematically it is expressed by I'¢; t)
= V(t, to)l'¢; to).
(2.134)
Conservation of the norm square at any time, or conservation of the probability when they are expressed in the qrepresentation, requires that the time evolution operator be unitary, namely,
2. 7 Time evolution operator and the postulate of quantum dynamics
('¢';tl'¢';t) = ('lf!;toiVt(t,to)V(t,to)l'if!;to) = ('¢';tol'¢';to)
65 (2.135)
or
J
('¢'; tlx}dx(xl'¢', t)
=
J
l'if!(x; t)l 2 dx =
J
l'if!(x; to)l 2 dx.
Therefore
ut (t, to)V(t, to) =I
or
ut (t, to) = u 1 (t, to).
(2.136)
Let us investigate a state at timet, which evolves into state at time t + M, an infinitesimal time interval ~t after t, i.e.
I'¢'; t + M) = V(t + ~t, t)l'¢'; t).
(2.137)
Since U(t, t) =I, the time evolution operator U(t + ~t, t), of course is very close to identity operator, can be approximated up to first order in ~t by the following expression
V(t + ~t, t) =I if'Mt, where the operator
or
n
(2.138)
is Hermitian because
nt = n.
It is the operator n that dictates the change of the state, and this forms the last postulate of quantum mechanics, namely the law of quantum dynamics, in which !lis taken to be H(P, Q; t)jti.
66
Chapter 2
4th Postulate of Quantum Mechanics:
The evolution of the quantum state is governed by the following dynamical law,
I'¢'; t + 8t) = V(t, t + M)l'¢'; t) =
(I ~H(P,
Q; t)8t) I'¢'; t)
(2.139)
or to rewrite it in the familiar expression of the Schrodinger equation, 
1

a

at
_2
1 1
tr ifi lim (l'¢';t+8t) l'if';t)) =lifil'¢';t) ::::eJI(P,Q;t)l'if';t).'l
,__.. atto 8t
. [\
.~
•"\
(2.140)
In other words, the total energy of the system is the generator that drives the quantum state into evolution. Therefore the Schrodinger equation is referred to as the law of quantum dynamics. 2. 7. 2 Time order product and construction of time evolution operator The product of two time evolution operators is again a time evolution operator, given by
. V(t, to) = V(t, t')V(t', to).
(2.141)
Hence we have
V(t + 8t, to)= V(t + 8t, t)V(t, to)=
(I ~H(P,
Q; t)) 8t.
(2.142)
From which the differential equation for U(t, to) is obtained as follows
av(t, to) 1 i a = 8tt0 lim £(U(t + 8t, to) V(t, to))= i"H(P, Q; t)V(t, to). t ut n (2.143)
2. 7 Time evolution operator and the postulate of quantum dynamics
67
The solution for U(t, to) is trivial for the conserved system in which the Hamiltonian operator does not depend upon time explicitly. V(t, to) then takes the following expression
V(t, to) =
e*H(P,Q}(tto).
(2.144)
If the Hamiltonian operator becomes time dependent, we can solve the evolution operator by iteration, i.e.
V(t,t 0 ) =I+ (
~)
1:
or with the abbreviation H(P, Q; t)
V(t, t 0 ) =I+
H(P,Q;t')V(t',to)dt',
= H(t),
(i) 1t dt'H(t')+ (i) li
2
1 t dt'H(t')
li
to
then
to
1t'
dt"H(t")+ ....
to
Let us define the time order product as follows
then t
1 ~
dt'H(t')
1 t
~
dt"H(t") =
~ ~~ { 2.
t
~0
dt' dt"T(H(t')H(t")),
and U(t, to) can be cast into more compact expression as
68
U(t,t0 )
Chapter 2
=I+ ( ~)
+ ~!
1:
(~)
3
dt'H(t')
111:
+
~! ( ~)
2
11:
dt'dt"T(H(t')H(t"))
dt'dt"dt"'T(H(t')H(t")H(t"'))
+ ... (2.145)
2.8 Schrodinger picture vs. Heisenberg picture Measurement of the physical quantities differs from time to time in the dynamical system. The expectation values of the dynamical observables in a quantum system correspond to the data of the measurements, whose values depend upon the time at which the measurements are performed. Therefore, when we take the expectation value of a dynamical observable A with respect to the ~tate 1'¢'; t), i.e. the value (A) = ('¢'; tiAI'¢; t) that we obtain, in fact, is time dependent. From now on we shall denote the implicitly time dependent expectation value of the dynamical operator A with respe.ct to the state 1'¢'; t) by
(A)t = ('¢'; tiAI'¢; t).
(2.146)
· Taking the time derivative of the above equation, we have the following relation
.
=
z t h_('¢'; tolD (HA AH)UI'¢'; to)
.
=
z h_('¢'; ti[H, A]l'¢; t).
(2.147) If we take A to be the position operator Q, the equation becomes
69
2.8 Schrodinger picture vs. Heisenberg picture
(2.148)
The commutator of H and Q can be evaluated according to the fundamental commutation relations among coordinate and momentum, and we obtain
hence
(2.149)
Similarly in the case for the momentum operator P,
\aH) .
d (P)t=dt 8Q
t
(2.150)
It is in this Schrodinger picture that treating the quantum state as time dependent while keeping the dynamical observables as fixed quantum operators, Hamilton's equations of motion are regained as in the following table:
Quantum equations of motion in Schrodinger picture
classical equations of motion
70
Chapter 2
where He= H(pc(t), qc(t)). There also exists another picture to look at the quantum equations of motion, namely the Heisenberg picture. Instead of regarding the state vector as time dependent, one takes the new dynamical observables by the unitary transformation of the old ones, namely taking the new dynamical operator as
.A(t) = ut (t, O)AU(t, O) = ut (t)AU(t),
(2.151)
here for simplicity and clarity, we take to = 0 and put U(t, to) = U(t). Since the commutation relations are invariant under the unitary transformation, the fundamental quantization rules remain the same as before, provided that all new operators are of at equal time, i.e.


~:~
[P(t), Q(t)] = dl,
(2.152)
which allows us to establish the following commutation relations
n aF [Q(t), F(P(t), Q(t))] = :, z aP(t)


[P(t), G(P(t), Q(t))] =
n,ac
i aQ(t).
(2.153)
(2.154)
Then the quantum equations of motion take the following expression
aQ(t) = i ut(t)[H Q]U(t) = ~H at n ' aP(t)'
(2.155)
ai>(t) = iut(t)[H P]U(t) =  ~H at n ' aQ(t)'
(2.156)
and we have the following table:
71
2. 9 Propagator in quantum mechanics
Quantum equations of motion in Heisenberg picture
aQ(t) at a"P(t) at
classical equations of motion
aH ai'(t)' aH  aQ(t),
. () aHc qc t = apc(t)' . () aHc Pc t =  aqc(t).
2.9 Propagator in quantum mechanics Let us consider a onedimensional system in which the Hamiltonian does not depend on time explicitly. The time evolution operator therefore takes the simple form,
U(t, to) = e~H(P,Q)(tto) = U(t to). The wave function in the Schrodinger picture can be expressed as
'1/J(x; t) = (xi'I/J; t) = (xiU(t t')I'I/J; t') =
/(xle~H(tt')lx')dx'(x'I'I/J;t') =
J
K(x,t;x',t')'ljJ(x',t')dx',
and the equality of the last equation defines the propagator
K(x,t;x',t')
= (xle~H(tt')lx').
(2.157)
It is in fact the propagator that is taken as the matrix element of the time evolution operator U(t, t') between the states lx) and lx'). The propagator brings the state function at spacetime point (x', t') to the state function at another spacetime point (x, t). It is obvious that when t = t', the propagator reduces to the delta function, i.e.
K(x, t; x', t) = (xle~H(tt)lx') = (xlx') = 8(x x').
(2.158)
We shall explore the property of the propagator by putting it in a different form. Let us insert an identity projection operator formed by
72
Chapter 2
energy eigenstates in front of state lx'), then we express the propagator in the following form
K(x, t; x', t') = I)xle~H(tt')la)(alx') = I)xle~Ea(tt')la)(alx'). (2.159)
The time derivative of the propagator can be performed· as
in! K(x, t; x', t') = L E0 (xia)e~Ea(tt') (alx') 0!
a X ) K (x, t; X ,'t,). = H ( inax'
(2.160)
If we introduce the step function
TJ(t t')
1
for t > t',
0,
for t < t',
={ '
and make use of the property
!
TJ(t t')
= c5(t t'),
then the time derivative for the product of TJ(t t') and K(x, t; z', t') can be calculated as
in ata (TJ(t t')K(x, t; x', t)) = TJ(t t')HK + iM(t t')K = TJ(t t')HK + iM(t t')c5(x x'), (2.161)
2.10 Newtonian mechanics regained in the classical limit
73
where the last term is obtained by the property that 8(x a)f(x) 8(x' a)f(a). Hence
=
(iii :i H) TJ(t t')K(x, t; x', t') = iM(t t')8(x x').
(2.162)
Therefore the propagator is also referred to as Green's function or kernel. A simple demonstration is in order to evaluate the propagator for a free particle system in which we take Ia) = lp), and summation over a is replaced by integration over the momentum p, then
= 1
2nn
J
en.ip(xx') e _iL(tt')d n2m p
V =
[i
m m(xx') 2nin(t t') exp 2n (t t')
2 ]
.
This formula reminds us the familiar form of the delta function
8(x a)= lim E0+0
and lim K(x, t; x', t') t+t'
v~
= 8(x x')
jT exp
2
[ (x a)
] ,
(2.163)
f:
is reproduced.
2.10 Newtonian mechanics regained in the classical limit Modern physics is treated usually as a new branch of physics. New theories are developed to analyze the new discoveries and phenomena in physical systems, particularly in the microscopic systems of atoms or subatoms. Quantum theory rose after the end of the 19th century
74
Chapter 2
and is commonly believed to be a correct theory for investigating the phenomena that were discovered in microscopic quantum systems in the modern era. It is a new theory that replaced Newtonian theory which was successful and widely accepted in the studies of classical dynamics. Yet if we carefully examine the propagator we have treated in the last section, and take the very same theory in the limit that 1i tends to zero, we are able to recover the whole of the theories developed in classical dynamics. In order to treat the spatial coordinates and the temporal ones on equal footing, we introduce a new notation as follows (2.164) The time dependent state function and the Schrodinger equation we had before can be reexpressed respectively as
'1/J(x; t) = (x, ti'I/J) = (xlekHti'I/J),
(2.165)
(2.166)
Furthermore, the propagator from the initial spacetime point (xo, to) to final spacetime point (x, t) takes the following inner product,
(x, tlxo, to)= K(x, t; x 0 , t 0 ) = (xle~H(tto)lxo),
(2.167)
which in fact is the transition probability for a quantum object at spacetime (xo, to) to another spacetime (x, t). Since the closure relation holds at any instant, we have,
J
lx', t')dx' (x', t'l =I,
that can be inserted in between the state (x, tl and the state lxo, to) at the instant t'. If we consider the case that the transition takes place in N steps from the initial spacetime (xo, to) to the final spacetime (x, t), and let
75
2.10 Newtonian mechanics regained in the classical limit
t to = N E,
tn = to
+ nE,
where n = 1, 2, ... , N  1,
with XN = x and tN = t as shown in Figure 2.2, then the propagator can be expressed as in the following form (x, t!xo, to)=
In
dxi(x, t!xN1, tN1) ... (xn, tn!Xn1, tn1) ...
2
(2.168)
A typical term (xn, tn!Xnb tn 1 ) in the above formula shall be explored in detail as follows ( Xn, t n I Xn1, t n1 ) = ( Xn Ie iH"I li Xn1 ) = ( Xn Ie _i(_L)P2+V(X))EOI li 2 m_ Xn1 )
We have made the approximation in the last equality by neglecting the term higher than order E2 . The term (xn Iexp ( 2~~ P 2 E) I() can be
t
(x, t)
Figure 2.2: Spacetime diagram.
76
Chapter 2
calculated explicitly by Fourier transform as in the case of K ( x, t; x', t') for a free particle system and we obtain,
The typical term (xn; tniXn1, tn1) can finally be cast into the expression,
(2.170)
where the discrete form of Lagrangian Ln takes the following form
2
) _ m (Xn Xn1) _ V( L n2 I: Xn1 ' and the propagator of Eq. (2.167) is eventually evaluated as
(x, tlxo, to)
=
(
m )m2/ IT dxn exp (ifiELn ) . 2nin~:
Instead of performing the multiple integration for all steps, we let N tend to infinity, and the summation over the discrete Lagrangian in the exponent becomes integration over time. This new way of performing the integration has a special name called path integration, after Feynman. Therefore we express
77
2.10 Newtonian mechanics regained in the classical limit
= where
E
j Vx(r)exp (~ 1: L(~(r),x(r))dr),
(2.171)
and Xn (or Xn1) are replaced by dr and x(r) respectively, and
L(x(r),x(r)) = limLn =lim {m (Xn Xn 1 )  V(xn1)}. E0+0
E0+0
2
f:
In the classical limit when n + 0, the path integration over the exponential function would oscillate so rapidly and the propagator, the transition probability, would become negligibly small. If we regard the initial spacetime as the location of a particle at the beginning time to, the particle ends up at the final spacetime, the final position x at time t, along the path determined by laws of classical dynamics, then we shall get the maximum transition probability. This can only be achieved if the phase in the exponential appearing in the integrant of the path integration is most stable, namely the variation of the phase be zero. Mathematically it is expressed by
88 = 81t L(x(r), x(r))dr,
(2.172)
to
which is exactly the principle of least action in classical mechanics that leads to Lagrange's equations of motion, namely
d ( 8L ) 8L _ O dr 8x(r)  8x(r)  ' and classical mechanics is regained.
(2.173)
78
Chapter 2
2.11 Exercises
Ex 2.11.1 Show that delta function 8(x) can be expressed as
. 1 1m sin N x (a ) vi:( x ) = 1 7r N+oo
or
X
Ex 2.11.2 Show that
and hence that
(xiF(P)Ix') =
F(~ d~) 8(x x').
Ex 2.11.3 Consider a physical system of one dimension that is translated by a distance (to the right, the wave function then becomes (xiU(P; ()1'1/J). Evaluate the translated wave function explicitly and interpret the result. (Hint: take adjoint conjugate ofUt(P;()Ix) as (xiU(P;().)
79
2.11 Exercises
Ex 2.11.4 Show that the prepresentation of the Schrodinger equation can also be expressed as
p2
2m '1/J(p)
fid) '1/J(p) = E'l/J(p).
+ V ( i dp
Ex 2.11.5 Solve the Schrodinger equation for the potential energy
V(x) = g8(x a). with g positive definite, in qrepresentation and prepresentation.
Ex 2.11.6 Find the one dimensional quantum state of minimum uncertainty in the prepresentation.
Ex 2.11.7 Find the quantum uncertainty
~e ~Pt;,
in the n particle state.
Ex 2.11.8
k;
Show that (OieikxiO) = exp( (OIX2 IO) ), where IO) is the ground state of the one dimensional harmonic oscillator, and X is the position operator.
Ex 2.11.9 Prove that
80
Chapter 2
if we define
(Hint: by means of Proposition 9.) Ex 2.11.10
Prove that 2 2a """"' L..,.(qr qr+l) _ T
ali . 2 L... 4sm m """"' 2 S
(Hint: also by Proposition 9.) Ex 2.11.11
By using Mehler's formula
(
21r N s ) OsQ 8t •
81
2.11 Exercises
prove that the propagator in the case of the one dimensional harmonic oscillator can be expressed as (Hint: use Eq. (2.159) for Ia) = In).),
K(x, t; xo, to)
=
mw 21filisinw(t to)
Ex 2.11.12
For a dynamical observable A(t) which is time dependent explicitly, the time derivative of the expectation value takes the form as
Use the above formula repeatedly and show that
(.6x)F
= (.6.x)5 + ~ { ~(xp + px)o (x)o(P)o} t + ~25t 2 ,
for the case of a free particle in onedimension. Ex 2.11.13
Prove that the one dimensional Schrodinger equation in free space
is invariant under Galilean transformation:
x I = x vot, t' = t.
82
Chapter 2
(Hint: let 'lj;'(x', t')
= f(x, t; vo)'lj;(x, t).)
Ex 2.11.14 Derive explicitly the propagator in the rrrepresentation, i.e.
where
'T"'
'T"' vxup
= 11m . II (dxndPn) E070
with
n
i<
21fn
,
Chapter 3 Symmetry, Transformation and Continuous Groups
3.1 Symmetry and transformation Nature shows symmetries in physics, not only in the geometrical sense; it also contains a wide range of deeper implications among the invariance and the conservation laws in physics. Let us take space as an example. If we assume the homogeneity in spatial property, a translation along any direction in this space will not change the description of the system. The physical quantities involved will be invariant under the spatial transformation, or briefly, it is said to be translational invariant, and hence the total momentum of the system is conserved. So as to the isotropy property of the space, the total angular momentum is conserved due to the rotational symmetry. Transformations in physics often form a group which is characterized by a set of continuous parameters, called the group parameters. Henceforth we call it the continuous group. We shall start with some simple examples in the following section.
3.1.1 Groups and group parameters Let us consider the translation inndimensional space. A vector x in nnspace is transformed into a vector x' by a translation specified with n continuous parameters denoted by the contravariant components ~ = (~ 1 , ~n). We shall write the transformation as
e, ... ,
83
84
Chapter 3
., X '=x+c
or
x'i=xi+ci .,
£or
z· = 12 , , ... , n.
(3.1)
Further translation from the vector x' to the vector x" by another displacement rJ which is characterized by another set of n continuous parameters rJ = ('r/ 1 , '{} 2 , ... , rJn) is written as
x"
= x' + rJ
or
x"i
= x'i + r/ =xi+ (~i + r/).
(3.2)
It is obvious that all the translations form a group, called the ndimensional translational group. If we denote the group elements simply by ~ and rJ for the previous two consecutive translations, and put a dot· in between~ and rJ, then the group operation for the consecutive translation is expressed as
"'= rJ. ~' and the composition rules for the group parameters are just the addition, I.e.
n) 1 2 "f= ( "',"/ , ... ,"f '
(3.3)
Hence the transformation of the space translati9n satisfies the group postulates. (a) "' is again a translation with the group element specified by the set of n continuous parameters, i.e. x" = x + "', "'i = ~i + rJi. (b) There exists the identity element e
=
(0, 0, ... , 0) such that e · ~
=
~e=~.
(c) The inverse element of~' denoted by ~ 1 exists, such that ~1 . ~
= ~ . ~1 = e,
that implies (3.4)
Instead of performing the displacement upon the vectors in n nspace, if we consider the transformation of the function F(x) by making the displacement in the coordinates system, then the expression for
85
3.1 Symmetry and transformation
the group elements will be drastically changed. When the function F(x) = F(xl, x 2 , ... , xn) is transformed by a displacement into F(x') = F(x + ~) = F(x 1 + x 2 + xn + ~), we can expand the function in terms of a Taylor series and put it in a more compact expression by summing over the repeated indices as
e,
e, ... ,
F(x') = F(x + ~) . a 1 .. a a ~i...Q.,. =F(x)+f:'a.F(X)+1 f:f1 a.a.F(x)+ ... =e ax•F(x). xz 2. xz x:J (3.5) One notices that the group elements for translation take a completely different form. The group element of the displacement is written as exp (f!alaxi) instead of~ Therefore the group operation is the usual multiplication, namely the successive displacements of~ and rJ is written as the product of two group elements exp (rJia1axi) and exp (~ia1axi), and reaching to another element as
a ) . exp (~z· axi a) exp ( rJz.axi
a] = exp (''/ ·a = exp [ (~z.+. rJz) axi axi) ,.
and the addition of the group parameters 'Yi = ~i + rJi is regained in the exponent but in totally different context. Namely the fact that the partial derivatives with respect to two different coordinates commute with each other ensure us to have 'Yi = ~i + rJi. The parameters in the translational group form an ndimensional Euclidean space itself. This nnspace is called the group parameter space, or group m~nifold in short. Let us consider the linear transformation on ndimensional vector space as another example. Vector x when expressed by a column matrix, is changed into vector x' by ann x n matrix A, i.e.
x' Ax ' or if we express it in terms of the matrix element as
x'i
= A~xj =
(8~
+ a~)xi,
where . A~= 8~ +a~.
86
Chapter 3
The linear transformations form a group provided that all the transformation matrices are of nonvanishing determinants. Consider two successive transformations and x' =Ax, x" =Ex', then we conclude: (a) x" =Ex'= EAx = Cx and C = EA with property det C because det A f= 0 and det E f= 0.
f=
0
(b) The identity matrix exists by choosing a~ = 0. (c) Since det A
f= 0, of course A l
exists.
The group element A of t4is linear transformation consists of n 2 parameters, namely a~ for i, j = 1, 2, ... , n. Therefore its group param2 eter space or the group manifold is a nn space, a Euclidean space of n 2dimension, and this group is called the GL(n, R) group. The composition rules for the group parameters in GL(n, R) are more complicated than in the case of spatial translation. The group parameters of the group element C = EA obtained from the successive transformations A and E take the following composition rules (3.6)
or cJi · = 8Ji· + bJi · + a·Ji
il· + bla J"
(3.7)
Different groups dictate different sets of the composition rules for the group parameters. We shall consider the general case of rparameter group and denote the element simply by the set of r parameters. Let us denote the group elements by a,b
for
a=(al,a 2 , ... ,ar), b=(b1 ,b2 , ... ,br).
For the product of a and b, i.e. c = ba = f(ai, bi) = f(a, b), we can express the group parameters of the resultant element cl = Jl(ai, bi) with the composition functions Jl(ai, bi) being infinitively differentiable with respect to ai and bi. Denote e as the identity element and identify its position at the origin in the parameter space, i.e. e = (0, 0, ... , 0), then we have
87
3.2 Lie groups and Lie algebras
f(a,e)
= f(e,a) =a.
(3.8)
Also if we denote the inverse of element a by a 1 , then the following relations hold
Continuous groups were systematically investigated by Lie who particularly concentrated on the behavior of function f (a, b) when both elements a and b are at the vicinity of the identity transformation, namely putting elements a and b close to identity element e, and looking into the functional behavior of fl(ai,lJi) for infinitesimal of ai and bi. We shall summarize the important results in the next section.
3.2 Lie groups and Lie algebras Let us start with the composition rules of the group parameters
Since the elements a = (a 1 , a 2 , ... , ar) and b = (b 1 , b2 , ... , br) are close to the identity element e = (0, 0, ... , 0), we perform Taylor's expansion of the functions Jl(ai, bi) at the origin of the group parameter space in power series of ai and bi up to second order, namely,
l c
= fl(
a
z.i)
i
= fl(
'if
e, e
) 8fl(e,e) i 8fl(e,e)bi azjl(e,e) iz.i m(3) + 8ai a + 8bi + 8ai8bi a i f +v .
This equation can be further simplified by the following relations:
l(e,e) = 0,
fl(a,e) = al,
fl(e,b) = bl,
and hence
with
l.. = az Jl(e, e) = constant coeffi cient. 8ai8bi
f z:J
(3.10)
88
Chapter 3
The above formula enables us to calculate the group parameters of the inverse element a 1 = ((a 1 ) 1 , ( a 1 ) 2 , ... , ( a 1 )n). When a rv e, we evaluate f(a, a 1 ) as
0 = cl = l(a,a1) = al
+ (a1)l + J}jai(a1)j + 0(3),
therefore, up to 0(3) we have, (a1)l = al
+ ffjaiaj + 0(3).
Let us consider the transformation of a vector in ndimensional space, as shown in the Figure 3.1, such that the transformation forms a group G(a), i.e. x f
= h(x,a),
x'
x
~~x'+da! a+da
·
Figure 3.1: Transformation of a vector inndimensional space. where G(a) is the rparameter group, and a is the group element. Let us perform an infinitesimal transformation from x 1 to x" by 8a, an element close to identity element e, right after the first transformation by a, as follows x" = x 1 + dx 1 = h(x 1 , 8a). The changes in the vector can be expressed as 1
dx
1
8h(x ,e)s:
( 1 )s:
= a( 8a) ua = u x ua,
where u(x1) stands for an x r matrix. The combined transformation can also be obtained through a single transformation by the group element a+ da with the following relations:
89
3.2 Lie groups and Lie algebras
x"
= h(x, a+ da),
a+ da
=
f(a, 8a).
Therefore da can also be expressed as the first order approximation in 8a as follows _ 8f(a,e) 1: d aab ua,
or
dal
=
8fl(a, e) 1: m abm ua
= flm (a )1:uam ,
(3.11)
where 8f(a, e)/8b is a r x r square matrix evaluated at b = e or if we express 8a in terms of da by inverting the matrix 8f(a, e)/8b, namely
(3.12)
where 'lj;(a) again is a rxr square matrix, the inverse matrix of 8f(a, e)j8b. Therefore dx'
= u(x')'lj;(a)da,
or when it is written in terms of the components as follows dx'm
= u'f1};]daj.
Let us consider the transformation for an arbitrary function F(x) = F (x 1 , x 2 , ... , xn) in the vector space by a group element at the vicinity of identity e. The change in F(x) is calculated as
We define the differential operator (3.13) which is called the generator of the transformation, or the infinitesimal generator of the group, or simply group generator.
90
Chapter 3
Let us go back to the space translation discussed in previous chapter. A vector X in n 3 space is displaced by a group element~= (e,e,~3 ), namely T(~)
x~x'=x+~,
or expressed in components
The generators of the 3dimensional translational group can be calculated as follows  m a  ax'm a  ~m a  a X iU·         v     ~
axm
a~i
axm
~
axm
axi,
which is related to the ith component of the momentum operators in qrepresentation in quantum mechanics, i.e. (3.14) eM·P as the group which allows us to express U(P, ~) = e~i a~• element for a coordinate transformation to the right by xi + xi + which corresponds to the transformation of the physical state to the left by ~i. We take the rotation about the azimuthal axis as another example. It is a oneparameter group defined by
r,
(3.15) (3.16) that allows us to calculate
91
3.2 Lie groups and Lie algebras
1 ul
=
8x'aell
0=0
2 = x,
(3.17)
(3.18)
and we obtain the generator of the rotational group about zaxis as follows
(3.19)
which is a familiar expression related to the qrepresentation of the 3rd component of the angular momentum operator. The angle() of the rotation in fact is just the group parameter. The parameter takes the value 0::;; () < 27r. We call it the 0(2) group, an orthogonal transformation in R.2space. A continuous group with a finite number of parameters that takes a bounded domain in the group parameter space is called compact Lie group. Otherwise we call them noncompact ones. The translational group in the previous example is a noncompact 3parameter group, while the 0(2) group is a compact 1parameter group. Lie algebra is defined as the commutator of the group generators. Let us consider the commutator of Xi and Xj, i.e.
(3.20)
Yet the last parenthesis can be simplified as
92
Chapter 3
(3.21)
where cfj = cji is a constant coefficient and antisymmetric with respect to i and j. The proof, though a little tedious, goes as follows: Since dxm
= u'k'I/J~dan,
we have
~:: = u'k'I/J~.
With the property
of infinite differentiability of the transformation functions h(x, a) with respect to the group parameters, namely that
82xm 8alaan
a2xm aanaal,
(3.22)
we reach at
(3.23)
aum
Putting aa~
aum
= ax1 u~ '1/J~, the above equation takes the expression
as
or cast into the following expression
(3.24)
By inverting 'ljJ into
f, the above equation can be written as
93
3.2 Lie groups and Lie algebras
(3.25)
The term on the right hand side of the last equation depends upon the coordinate only, while the left hand side is product of a coordinate dependent function uk and a group parameter dependent factor
(~~t  ~y) f} fj. The condition can only be satisfied if, 8 V;f ( aan
_ 88al VJ~) ~~~~J = c"f.ZJ = c~.JZ = Z
constant coefficient.
Therefore we have,
(3.26)
and hence the commutator of Xi and Xj can then be expressed as
(3.27)
and they are called Lie algebras. stants.
ctj are called the structure con
A few terminologies concerning Lie groups are introduced as follows: (a) A Lie group is said to be Abelian if all the structure constants are zero, i.e.
ctj = 0,
i,j,k ~ r,
which imply that the generators of the group commute with each other.
(b) A subset of the elements of a group G is called a subgroup of G if those elements satisfy the group postulates. The generators of the subgroup, X 1 ,X2 , ... ,Xi, ... ,Xp for p
(c) It is called an invariant subgroup H of G, if the generators in H commute with the generators outside H, namely that
(d) A group is called a simple group if it contains no invariant subgroup. It is called a semisimple if it contains no invariant, Abelian subgroup. The generators of the group also satisfy the Jacobi identity, i.e. (3.28)
which leads to the following relation on the structu:r:e constants,
cLcik + skcu + dkiclj = o.
(3.29)
3.3 More on semisimple group We shall confine ourselves to semisimple Lie group from now on for further investigation. A symmetric g tensor is introduced for the discussion on Cartan's criteria of the semisimple group, and also the inverse of the g tensor, i.e. g 1 , another symmetric one will be constructed for the discussion on the Casimir operator.
3.3.1 Cartan's criteria of the semisimple group With the structure constants we construct a symmetric tensor g defined as (3.30)
95
3.3 More on semisimple group
which is called the metric tensor or g tensor or sometimes by the name of Killing form. We formulate the criteria given by Cartan in characterizing the semisimple group in the following proposition.
Proposition 1.
A group G is semisimple if and only if detlgl
i 0.
The condition detlgl =/= 0 for a semisimple group is necessary and sufficient. Let us assume a Lie group contains an Abelian invariant subgroup. We denote the index of the generators in the Abelian invariant subalgebra by putting a bar on the top, i.e. and k. It can be easily demonstrated that detlgl of the group vanishes, i.e.
z, ]
detlgl = 0,
(3.31)
if the group contains an Abelian invariant subgroup. Let us calculate the element 9{j of the g tensor as follows (3.32) which implies that the elements of the whole column of the g tensor equal to zero. Hence the determinant of the g tensor is zero, i.e. detlgl = 0, and the sufficient condition of Cartan's criteria is reached. The condition of zero in determinant of the g tensor is also necessary. Consider a system of r simultaneous homogeneous equations as follows
g ··x· j  0• ~J
(3.33)
The condition detlgl = 0 also ensures the existence of the nontrivial solutions of xi. Let us construct a generator X, with the coefficients taken from the nontrivial solutions xj. The generators then form an invariant subgroup as demonstrated in the following proposition.
96
Chapter 3
Proposition 2. If xi are the nontrivial solutions of the following simultaneous homogeneous equations, gkiXi
=0
and Xi is the generators of the group G, then the generators xi Xi form an invariant subalgebra of G.
The proof is quite straightforward by considering the following commutator (3.34)
If we contract gik with yk, we obtain the following result

k  XlCzji  CjignlX n l_o gikY k_  gikX lClj  ,
(3.35)
which implies that yk is also the solution of Eq. (3.33), and hence the generators xi Xi form an invariant subalgebra. While in reaching Eq. (3.35), we have made use of the cyclic properties of the indices in Czji = Cjil which will be shown later in Proposition 3.
3.3.2 Casimir operator The nonzero determinant of g, i.e. detjgj f= 0, enables us to construct the inverse of g, a tensor denoted by g 1 , with elements gii such that il i g glj = ~jl
then g 1 will allow us to form a quadratic of the generators of the Lie group, called Casimir operator defined as (3.36)
97
3.3 More on semisimple group
which shall commute with any generator of the group as can be verified by taking the commutator of C with, say, the generator Xk in the following expression (3.37) Let us define a 3rd rank tensor through the following relation l
Cik
l.
= 9 J Cjik·
We shall prove the antisymmetric property of Cijk in the following proposition.
Proposition 3. The 3rd rank tensor Cijk is antisymmetric with respect to the interchange of any pair of the indices.
The proof goes as follows Cijk
l l n m = Cij9lk = Ci_jClmckn
(3.38) By summing over the repeated indices, the last line of the above equation becomes invariant under the cyclic permutation of i, j and k. Therefore the conclusion of Proposition 3 is reached. Eqation (3.37) can then be expressed as
[C, Xk] = gii imcmik(XlXj + XjXl) (3.39) Since the last term in the last equation is symmetric with respect to l and j, so is the upper indices m and i~ yet Cmik is antisymmetric with respect to the lower indices m and i, the summation leads to [C, Xk] = 0, which concludes that the Casimir operator of the group commutes with all the group generators. We shall make use of this property to investigate the theory of angular momentum in the following chapter.
98
Chapter 3
3.4 Standard form of the semisimple Lie algebras Following the approach of Cartan and Weyl, let us consider the following equation
[A,X] =pX,
(3.40)
where A is an arbitrary linear combination of the generators Xi with the coefficient ai, i.e. (3.41) and X is another linear combination of the generators Xi with the coefficients xi, namely X xix.
(3.42)
~,
such that Eq. (3.40) can be met. We shall give a special name to X of the eigenvalue equation in Eq. (3.40) and call X the eigengenerator, while pis also called the eigenvalue for this particular form of the eigenvalue equation. If we express Eq. (3.40) explicitly in terms of the structure constants then we have the following simultaneous homogeneous equations, i.e.
ct,
(3.43) with the secular equation given by detla
i k cij 
k
P~i I = 0.
(3.44)
For the rparameter Lie group, there are r roots of the solution of p in Eq. (3.44). We shall state the conclusions from Cartan's without
providing the argument in details that if ai is chosen such that the secular equation has the maximum numbers of distinct roots, then only the eigenvalue p = 0 becomes degenerate. Let l be the multiplicity of the degenerate roots of the secular equation, then l is said to be the rank of the semisimple Lie algebra. The eigengenerator corresponding to the degenerate eigenvalue p = 0 is denoted by Hi (i = 1, 2, ... , l), which commutes with each other, i.e. (3.45)
99
3.4 Standard form of the semisimple Lie algebras
ct
with the structure constant = 0 where i, j ~ l and J.l ~ r. The eigenvalue equations with nonzero eigenvalues read as (3.46) where we have used the eigenvalue a to label the eigengenerator as En. Since A commutes with Hi, it allows us to express A as the linear combination of Hi, namely (3.47) Let us investigate the commutator of A and [Hi, En] as follows
[A, [Hi, En]] = [A, HiEn]  [A, EnHi] = a(HiEn  EnHi) = a[Hi, En],
(3.48)
which implies that there exist l eigengenerators corresponding to the same eigenvalue a in the eigenvalue equation of Eq. (3.40). It is theresult contradictory to the assumption that a is not degenerate. Therefore we conclude that [Hi, En] must be proportional to En, i.e. (3.49) where the structure constant cfn = aic)~, with a and (3 taking all the distinct nonzero eigenvalues. One also proves readily that (3.50) by making use of Eqs. (3.46), (3.47) and (3.49). To further our investigation, let us make use of the Jacobi identity and we find that
[A, [En, Ef3]]
[En, [E13, A]]  [Ef3, [A, En]] = (a+ (3)[En, E13],
=
(3.51)
which means that [En, Ef3] is the eigengenerators of A with eigenvalue a + (3 if and only if a + (3 f= 0. Hence we can express the commutator of En and E13 as another eigengenerator as follows (3.52)
100
Chapter 3
with the structure constant c~f3 = Naf30~+f3" For the case that a + f3 = 0, the commutator [Ea, Ea] shall be proportional to the linear combination of Hi. Therefore we reach the following conclusions: [Ea, Ea] [Ea, Ef3]
= C~_ 0 Hi, = Naf3Ea+f3·
(3.53) (3.54)
Let us evaluate the elements of the g tensor in the standard form and divide them into three categories, namely 9ij, 9ia and 9af3, where the indices in the subscripts take the following values: i,j ~ l, {
a, f3 takes the values, r  l in total numbers, corresponding to
the roots of nonzero eigenvalues.
The evaluation of 9ij is the simplest as follows, for 9ij
= cff3Sa = aiaj.
(3.55)
We shall leave them as an exercise to show that 9ia
= 0.
(3.56)
Finally let us calculate the element 9af3 as follows _ v p, _ a ':Y 9a{3  C0 !1C{3v  Ca:yC{3a
+ Cp,
0 _ 0
a C{3p,
+ """' L....J
a+:r ':Y Ca'Y Cf3(a+'Y),
(3.57)
Tf.a
where J.L, v takes the values from 1 to las well as all those corresponding to the nonzero eigenvalues. With the properties of Eqs. (3.49), (3.53) and Eq. (3.54), we reach the conclusion that 9af3 is 0 unless f3 = a, namely 9a{3
=0
if a
+ f3 f. 0.
(3.58)
Suppose that a is the eigenvalue of Eq. (3.40), but a is not, then the whole column in the g tensor becomes 0 because 9af3 = 0. The determinant of g will vanish in this case, and Cartan's criteria for semisimple
101
3. 5 Root vector and its properties
group will be violated. Therefore the roots a and a of the secular equation Eq. (3.44) must appear in pair and 9aa will be different from 0. We shall choose the nonzero 9aa = 1, due to the fact that the eigengenerator Ea is linear on both sides of the eigenvalue equation of Eq. (3.40). This degree offreedom in the normalization of 9ij allows us to evaluate c~a as follows i
Caa
B Yf3 _ gijcaa3. _ gijc·3aa _ 9ij Cja a
(3.59) Hence we can rewrite Eq. (3.53) as follows (3.60)
[Ea, Ea] = aiHi.
Here we summarize the algebra of the semisimple group in the following standard form [Hi,Hj] = 0,
(3.45)
= aiEa,
(3.49)
[Ea, E_a] = ai Hi,
(3.60)
[Hi, Ea]
[Ea, Ef3]
=
Naf3Ea+f3, if
a+ {3# 0.
(3.54)
3.5 Root vector and its properties Let us regard the root a as an element in {dimensional vector space with its covariant components ai, i.e. (3.61) where a bar is placed under the root, written as _g to indicate it as a covariant vector. While the contravariant vector, denoted by a, with the component ai, which has been derived previously, can be expressed as a= ( a,a, 1 2
l) ... ,a.
(3.62)
If we denote the scalar product of two roots a and {3 as follows (a,{3) = aif3i = aif3i,, (3.63)
102
Chapter 3
then we obtain the following proposition.
Proposition 4· If a and {3 are two roots, then (a,{3)
(a, a) =
1 .
2 zntegers.
(3.64)
We start the proof by taking the commutator of the eigengenerators
Ea and B.P namely [Ea, E.,]. Since we enjoy the degree of freedom to rescale the eigengenerators, the structure constants can be chosen to be one, i.e. (3.65)
Let us assume that 'Y is a root but not 'Y +a. Then we can construct a series of roots by performing the commutator successively of Ea and a general eigenvector E.,ja with different j, i.e. (3.66)
The series of roots will start from 'Y and terminate at 'Y  ga, or explicitly as follows ,"f,"f a,"f 2a, ... ,{3, ... ,"f ga,
(3.67)
which is called astring, in which the root {3 stands for any root in the string. The finite length of the astring comes from the fact that it is a rparameter group, the finite number of roots prohibit us to perform the commutator indefinitely. Let us assume the string stops at the root 'Y ga, then the commutator vanishes, i.e. (3.68)
Now let us take the commutator of eigengenerators Ea and any generator, say E.,(Hl)m then we obtain as follows (3.69)
103
3. 5 Root vector and its properties
where the structure constant Jl.j+l can not be normalized further for the reason that all eigengenerators corresponding to the roots in astring have been rescaled already. Applying the Jacobi identity for the generators Ea, Ea and Eyja, which leads to the following relation Jl.j+lEyja
=
[Ea, [Ea, Eyja]]
= [Ea, [Eyja, Ea]]=
[Eyja, ai Hi]
[Eyja, [Ea, EaJ1
+ Jl.j [Ea, Ey(jl)a]
= ai[Hi, Eyja] + J.LjEyja, and we arrive at a recursion formula for the structure constant Jl.j+l as follows J.l.HI
= ('y,a) j(a,a) + Jl.j,
(3.70)
where j takes the integer with J.Lo = 0. The structure constant Jl.j then can be evaluated as follows
Jl.i
= j('Y, a)
1
;;JU 1)(a, a).
Since j terminates at g, therefore Jl.g+l
= 0.
(3.71)
It follows that
1
('Y,a) = 2"g(a,a).
(3.72)
Let us now take {3 = 'Y j a as any root in the astring, and eliminate 'Y in Eq. (3.72), then we reach
(a,{3) =
which proves Proposition 4.
1
.
(9 2J)(a,a).
2
(3.73)
104
Chapter 3
3.6 Vector diagrams We are now in the position to investigate the graphic representations of the root vectors. As we have shown in the construction of astring that
(a,{3)
=
1
(3.74)
2m(a,a),
where m takes the integer. Similarly, it follows that
(a, {3) =
1
2
(3.75)
n({3, {3),
if the {3string is taken into consideration. Therefore we obtain the following condition
(a, !3)({3, a) 1 2 cos cp= (a,a)({3,{3) =4mn,
(3.76)
and cos2 cp takes only the value 0, 1/4, 1/2, and 1. As we have demonstrated that the roots a and a come in pair, we need only to consider the positive values of cos cp, which lead us to take the following possible angles for cp, i.e. 7r
7r
7r
7r
cp=0,6'4'3'2" The length of the root vector a and {3 can then be calculated according to the various combinations in m and n as follows:
=
V 2 (~{3),
1!31 = J(f3,{3) =
V 2 (~{3).
lal =
J(a,a)
3.6 l!ector
105
diagra~s
With the method introduced by Vander Waerden, we summarize the results in 5 cases as follows: (1) for cp = 0, thus we have the trivial result that two vectors a and f3 are identical. (2) for cp = 1r /6, m = 1 or 3 and n = 3 or 1 respectively, and the ratio of the length square, i.e. ((3, f3)J(a, a) = 1/3 or 3. (3) for cp = 1r /4, m = 1 or 2 and n = 2 or 1 respectively, and the ratio of the length square, i.e. ((3,(3)/(a,a) = 1/2 or 2. (4) for cp = 1r /3, m = 1 and n = 1 respectively, and the ratio of the length square, i.e. ((3, (3)/(a, a) = 1. (5) for cp = 1r /2, the scalar product vanishes and the ratio of the length is undetermined. It is easy for us to demonstrate the graphic representation in two dimensional case, namely for rank two Lie groups. The root vectors can be drawn on a two dimensional space. For the above case 2 to case 4, the vector diagrams are: A. cp
= 7r/6
(a) ({3, {3)/(a, a)
=3
(b) ({3,{3)/(a,a)
= 1/3
The group is called G2 group after Cartan, an exceptional group containing 2 null root vectors and another 12 root vectors.
106
Chapter 3
B. cp=7r/4
(c) ((3,(3)/(a,a) = 2
(e) ((3, (3)/(a, a)
=
1
(d) ((3,(3)/(a,a) = 1/2
(f) ((3,(3)/(a,a)
=1
Figure 3.2: Vector diagram of G2, B2, C2, A2 and D2 group. They are called B 2 group and C2 group, which has 2 null root vectors and 8 root vectors and are associated with 80(5) group and Sp(4) group, the symplectic group in 4dimension, respectively.
c. cp = 7r/3 It is called A2 group, a group associated with SU(3), which contains 6 root vectors and 2 null root vectors.
3. 7 PCT: discrete symmetry, discrete groups
D. cp
107
= 7r/2
We shall consider another one called D2 group, with two pairs of mutually orthogonal root vectors. D2 group is commonly referred to as 80(4), which is isomorphic to the direct product of two 80(3) groups. The groups of general rank l take the names Al, Bl, Cl and Dl corresponding to 8U(l + 1), 80(2l + 1), 8p(2l) and 80(2l) respectively. Further investigation in those groups mentioned above as well as the exceptional Lie groups G2, F4, E6, E7 andEs shall be left to the readers to consult with more advanced books and journals.
3. 7 PCT: discrete symmetry, discrete groups Besides the continuous groups we have investigated in the past sections, physics involves another category of symmetry, called the discrete symmetry. The groups associated with discrete transformations are called the discrete groups. Parity, charge conjugation and time reversal transformation are among the most frequently discussed subjects in quantum mechanics, particularly in particle physics. We shall devote ourselves to study of PCT in this section.
3. 7.1 Parity transformation Let us start with parity transformation. To investigate the phenomenon of any physical system, it is imperative to use a coordinate system. Yet as far as the motion of the particle is concerned, physicists do not care if a righthanded coordinate system is used or if a lefthanded one is adopted. The reason is simple. For the motion of a system of particles in the righthanded coordinate space, there always exists an identical motion in the space of its mirror image (3dimensional one, of course). In other words, the motion of the particles is invariant under the mirror reflections. It is said that the equation of motion in classical mechanics is invariant under the parity transformation. It can be visualized as the symmetry between a physical phenomenon in space and its image phenomenon in mirror space. This is contrary to our previous understanding of the symmetry discussed in the past few sections, which are restricted solely to continuous transformations such
108
Chapter 3
as the spatial translation or rotation. The symmetry arising from the parity transformation on the other hand, is discrete. For this very reason, we coin a term, discrete symmetry. The immediate consequence of this discrete symmetry is that we, are unable to reorient continuously a righthanded coordinate system into a lefthanded one and vice versa. Therefore parity transformation falls into the category of discrete transformation. Quantum mechanically, the parity transformation can be represented by an operator 'P, which changes the position x into x, i.e. ,..,
_, P.T. _..., t+ X"'
r :x
_, = x,
(3.77)
or in general,
P: J(X)
tis, JU!P) = J( x).
(3. 78)
It is obvious that
p2 =I,
(3.79)
because that two successive parity transformations leave the position vector unchanged. The parity transformation of a dynamical observable V is expressed as follows (3.80)
We shall introduce two terms which are relevant to parity transformation. An operator V is called polar vector operator, or simply vector operator if it anticommutes with the parity operator, i.e.
{'P, V} = 0,
(3.81)
pvpl = VP = v.
(3.82)
or more explicitly that
The position operator and the momentum operator are of this kind, i.e.
109
3. 7 PCT: discrete symmetry, discrete groups
pxpl = x,
{3.83a)
pppI = P.
(3.83b)
On the other hand, an operator A which commutes with the parity, is called the axial vector operator, namely ['P,A]
= 0,
{3.84)
or (3.85)
One can easily prove that the angular momentum L is an axial vector operator. It is of interest to investigate a quantum system with symmetry under parity transformation. We shall summarize a few remarkable results in the following proposition.
Proposition 5. If a dynamical operator S is invariance under the parity transformation, then there exists a pair of vectors which are the simultaneous eigenvectors of S and 'P with the eigenvalues s and ±1 respectively.
Let us consider the following eigenvalue equation Sis)= sis). By applying the parity operator on both sides of the last equation, we can easily show that 'PSis)= 'PS'P 1 'Pis) = SPis)P = Sls)P = sis)P, where SP =Sand 'Pis)= ls)P.
{3.86)
110
Chapter 3
This implies that is)P is also the eigenvector of S with the eigenvalue s, namely that the eigenvectors are of 2fold degeneracy. Therefore it allows one to construct a pair of vectors (3.87)
with the following properties {3.88) which proves Proposition 5. If we take S to be the Hamiltonian operator of a central forces system, i.e.
s = H = 2~P2 + V(IXI).
(3.89)
It is obvious that 1'H1' 1 =H. Hence we have
in!
(1') = ([1', H]) =
o.
(3.90)
Therefore the parity is conserved, or in another way of saying, symmetry is preserved under parity transformation.
3. 7.2 Charge conjugation and time reversal transformation There exists another discrete transformation called the charge conjugation. A charge conjugation is denoted by C, which takes a particle into its antiparticle, namely C changes the charge of a particle q into the charge of its antiparticle q. We construct a onedimensional charge space in which all the particles take their positions on a line or an axis, the coordinate system of the charge space, according to the amount of the electric charge they bear. Particles with positive charge and particles with the negative one are respectively located on the right hand side and the left hand side of the origin of this onedimensional frame of
3. 7 PCT: discrete symmetry, discrete groups
111
reference. The operation of the charge conjugation can then be regarded mathematically as mapping a point q into the mirror image point q, a reflection with respect to the origin, i.e. C.T
C : q 17 q.
(3.91)
Hence the charge conjugation operator C is isomorphic onto the parity operator 'P of the space reflection in 1dimensional space. If we denote the charge operator Q associated with the physical observable of the electric charge of a particle, then the charge conjugation C takes Q into Q in the following similarity transformation
c: cQc 1 = Q, or
{C,Q} = 0.
{3.92) {3.93)
It implies that the charge conjugation operator C always anticommutes with the charge operator Q. The onetoone correspondence between the charge conjugation and the parity transformation in one dimension allows one to apply the results derived from the parity operation to the case of the charge conjugation. Let us now proceed to explore the third discrete transformation 7, called the time reversal transformation. We start with the following time dependent Schrodinger equation
ingt'I/J(x,t) = H
(~v,x) '1/J(x,t).
{3.94)
If we take the complex conjugate on both sides of the equation above, and replace t by t, then
(3.95)
It occurs often that the Hamiltonian H( (n/i)\7, x) = H((n/i)\7, x) if H contains only the quadratic term in P. Then the Schrodinger equa
112
Chapter 3
tion becomes
:t
in '1/J*(x, t) = H ( ~v, x) '1/J*(x, t).
(3.96)
This implies that both '1/J(x, t) and '1/J*(x, t) are solutions of the Schrodinger equation. The appearance of the complex conjugate as well as the replacement of t by t in the wave function of the last equation allows one to construct the time reversal operator which bears some sort of property in antilinearity. If we decompose T into the product of KandT, respectively standing for an antilinear operator and a temporal reflection operator, then we reach the conclusions in the following proposition.
Proposition 6.
If the time reversal operator 7 takes as product of K and T, i.e. T=KT,
w h ere
K = antilinear operator, { T
= temporal reflection operator,
then both '1/J(x, t) and K '1/J(x, t) are solutions of the time dependent Schrodinger equation as long as KHK 1 =H. Moreover, K 2 '1/J = rJ'I/J for any wave function '1/J and unimodulus factor 'TJ·
The proof goes as follows. Let us consider the transformation of H and '1/J, 1 T HT = KH
(!!.v ' x' t) 't.
T'I/J(x, t) = K'I/J(x, t).
K 1
'
3. 7 PCT: discrete symmetry, discrete groups
113
Then the Schrodinger equation takes the expression as
{3.97)
For the stationary system with Hamiltonian containing the quadratic term in linear momentum P, then KH(n\lji,X)K 1 = H(n\lji,x). Therefore we have
in%tK'I/J(x,t) = H
(~\l,x) K'I/J(x,t).
(3.98)
This proves the first part of the proposition. If we take the inner product of any pair of vectors 'ljJ and <.p, then by the antiunitary property of the operator K, we have (3.99)
which implies that rJ is unimodulus, i.e. ITJI = 1. Hence we complete the proof of Proposition 6. An immediate consequence of the second part of Proposition 6 is the existence of Kramers degeneracy if the quantum system is invariant under time reversal transformation. We derive the result as an example.
Example
Since 'ljJ and K '1/J are solutions of the Schrodinger equation, and are also orthogonal to each other as can be proved in the following line
('1/J, K'ljJ) by taking 'TJ established.
=
=
(K 2 '1jJ, K'ljJ) = TJ('I/J, K'ljJ) = ('1/J, K'ljJ) = 0,
{3.100)
1. Therefore the existence of Kramers degeneracy is
114
Chapter 3
3.8 Exercises
Ex 3.8.1
Consider the following transformation,
(a) What are the conditions if the transformation forms a 2 parameters group? (b) Denote the group element by a= (a1 ,a2 ). Find the inverse element of a, namely, a 1 = ((a 1)1, (a1)2). (c) What are the composition rules of the group parameters, i.e. try to find c = ba for c1 = c1(ai,!Ji) = Jl(ai,!Ji)? (d) Can you construct the 2 x 2 matrix representation of the group to justify your answer in (a), (b) and (c)? Ex 3.8.2
Consider the rotational transformation about the azimuthal axis with an angle (), namely,
then the function F(x)
= F(xl,x 2 ,x3 )
will be changed into
Prove explicitly that the transformed function can be obtained by
115
3.8 Exercises
(Hint: use exp(OX3)
F(x)
= exp (08j8cp)
and write
= F(rcosc.p,rsinc.p,x 3).)
Ex 3.8.3
A conjugate subgroup is defined as a 1 H a if H is a subgroup, where a E G. Show that the selfconjugate subgroup can alternatively be defined as the invariant subgroup. Ex 3.8.4
Verify that the Jacobi identity leads to following relation among the structure i.e.
Ex 3.8.5
Show that the elements Lie algebra vanish.
9ia
of the g tensor in the standard form of
Ex 3.8.6
Give the argument to verify the elements 9a{3 of the g tensor in the standard form of Lie algebra also vanish if o: + (3 =1 0. Ex 3.8.7
Let the parity operator be defined as (Hi;nt: by using Eq. (1.39).)
116
Chapter 3
Show that
pxp1=X, ppp1 =
P.
Chapter 4 Angular Momentum
4.1 0(3) group, SU(2) group and angular momentum Rotational transformation is one of the most commonly studied subjects in physics. Rotational transformation is an orthogonal transformation, that leaves the norm of the vector invariant. Orthogonal transformations of the coordinate system of a threedimensional vector space form a group of 0 (3). Yet as far as the invariance of the norm is concerned, there exists another transformation, namely a unitary transformation, the SU(2) group, which is homomorphic onto 80(3) group, and transformation also leaves the norm of a vector in C2space invariant. We shall review the orthogonal transformation in threedimensional vector space and investigate the group structure and properties of the 0(3) group, as well as the SU(2) group which are utilized as the tools for the discussion of the theory of angular momentum in the following section.
4.1.1 0{3} group Let us perform the rotational transformation in R 3space, i.e. the threedimensional vector space, about a fixed direction denoted by a unit vector n as the axis of rotation with an infinitesimal angle d(). It can also be written as a vector representing the direction of the rotational axis as well as the magnitude of the rotation as expressed in the following form (4.1)
117
118
Chapter 4
where we decompose the infinitesimal rotation dif into its components in the last term of the above equation. For a finite rotation with an angle if about the fixed axis fi can be also written as
(4.2) Henceforth, each rotation will be represented by a point at (01 , 02 , 03 ) in the threedimensional parameter space. The infinitesimal rotation will change the vector into expressed in the usual vectorial algebraic notation in the following form as
x
_,
X X
= dx = dO """"*

X X,
x',
(4.3)
where dif = (d0 1 , d0 2 , d0 3 ) is an infinitesimal vector in the parameter space of the rotational group at the vicinity of the origin. The parameter space is also a threedimensional Euclidean space that is isomorphic onto the previous vector space described by the coordinate system. To express the change in the vector due to the infinitesimal rotation in terms of the vector components, one has 
that allows us to calculate the matrix elements u~ in the rotational group 0(3), i.e.
Since Xi =
u~ {)~l , we obtain that (4.4)
and the Lie algebra of the 0(3) group is given as follows
4.1 0{3} group, SU(2} group and angular momentum
119
(4.5) where the fourth term of the above equation is obtained by the Jacobi identity,
(4.6) Hence we have the following Lie algebra of the rotational transformation in 3dimensional vector space,
(4.7) that reproduce the commutation relations for the quantum operators of the angular momentum, i.e.
(4.8) .
i
If we put xi= Li n . There are various representations for the generators of the symmetry transformation as well as for the Lie algebras. The representations of the group generators and the Lie algebras we adopted so far are expressed with the purpose to investigate the properties in the symmetry transformation of the functions. We refer this representation as the canonical formulation. If we focus our attention on the transformation of the vectors directly, we attain completely different representations. Take the rotational transformation of a vector in matrix notation as follows
x' = Rx or
(:~) = x'3
R (::) , x3
(4.9)
120
Chapter 4
where R is a 3 x 3 square matrix, which is called an orthogonal matrix because the rotational transformation leaves the norm of the vector invariant, i.e.
(4.10) Orthogonality imposes the following 6 conditions on the matrix elements of R, i.e.
R ijRjk
J:.k
u~
•
(4.11)
The above orthogonality conditions, asides from having the following relation
will also reduce the independent parameters in the matrix R from 9 to 3. It is obvious that the orthogonal matrices R form a group and that is the very reason to name it the 0(3) group, the orthogonal group in 3dimension. To investigate the matrix representations of the group generators, let us construct the orthogonal matrix R = R(0\ 02 , 03 ), where 0\ 02 and 03 stand for the group parameters. If we take the parameters small enough and keep only the first order in Oi, the matrix can be expressed as follows
where the matrix A(O\ 02 , 03 ) is close to a null matrix, namely that
A(O, 0, 0)
= 0.
(4.12)
Since the orthogonality condition of the matrix R leads, to first order approximation, the following relation
which implies that matrix A is antisymmetric The generators of the rotational group 0(3) are obtained by taking the derivative of matrix A(0\ 02 , 03 ) with respect to the corresponding group parameter, i.e.
121
4.1 0(3) group, SU(2} group and angular momentum
a 1 ,02 ,()) 31 . Xi= aOiA(O
e•=o
,
(4.13)
which again is an antisymmetric matrix as well. Let us consider the rotational transformation about the 3rd axis with an angle cp, the orthogonal matrix R(O, 0, cp) can, up to first order in cp, be approximated as  sincp 0
0 which allows us to obtain the generator x3 of 0(3) group by taking the derivative of matrix A respect to cp, namely
(4.14)
The reason that X 3 is called the generator of the group is due to the retrievability of the rotational matrix R(O, 0, cp), even with the finite angle cp, by exponentiating the matrix cpX3 and obtaining the following result coscp  sincp 3
R(O, 0, cp) = ecpX =
(
s~cp
coscp
(4.15)
0
Similar to the derivation of the group generator X 3 , the other two generators can also be calculated and given as
122
Chapter 4
x2 =
(
0 0 0) (0 1 0
~ ~ ~),
x1 = o o
1 0 0
1
,
(4.16)
by performing the infinitesimal rotations about the axis accordingly. These three matrices are referred to as the generators of 0(3) group in Cartesian bases. Let us construct the orthogonal matrix R in terms of the Euler angles cp, () and X· It is obtained by performing a rotation about the 3rd axis with an angle cp followed by another rotation about the 2nd axis with an angle () and then making a third rotation about the 3rd axis with an angle x to achieve the matrix R as follows R(cp, (), x)
= eXX3e')X2e'PX3.
(4.17)
It can be easily verified that R(cp,O,x+2m7r) = R(cp,O,x),
(4.18)
for m = integers. That implies only a finite domain in the parameter space is enough to exhaust all the elements of the 0(3) group. Therefore the parameters take the following ranges 0
~
cp
~
21f,
0
~
()
~ 1r
and
0
~
x
~ 1r.
(4.19)
Furthermore the orthogonality condition is automatically builtin because of the antisymmetric property in the group generators, namely that (4.20) It is interesting to observe that
det R= 1, which takes only the positive root in the equation (det R) 2  1 = 0. This can be understood that the matrix R(cp, (), x) is reached from the identity element I = R(O, 0, 0) with the determinant being equal to one, by varying the group parameters continuously away from the origin of
4.1 0{3) group, SU{2) group and angular momentum
123
Figure 4.1: Unit vector n(O, cp). the group parameter space. The elements of this particular orthogonal matrices form a group which is called the group of 80(3). Since the Euler angles take a finite range of the values, therefore the group parameters of S0(3) are confined within the domain of a sphere with the radius 1r in the parameter space. If we assign x as the angle of rotation about a unit vector n = (sin(} cos cp, sin(} sin cp, cos 0) which serves as the axis of rotation as shown in Figure 4.1, all points within the sphere of radius 1r in the parameter space, representing all the elements of the rotational group, can be reached by choosing suitable n and x, only if the two diametrically opposite points, namely points of antipode on the surface of the sphere in the group manifold, are identified. Let us abbreviate the unit vector as n(O, cp), and denote the group generators as X = (X1 , X 2 , X 3 ), then the group elements can be expressed as follows R(cp,O,x)
= exn(O,cp)·X.
(4.21)
The elements of the 0(3) group consist ofR(cp, (}, x) and R(cp, 8, X) and R(cp,O,x), the mirror images of the elements R(cp,O,x), namely taking spatial reflection followed by a rotation. In the other word,
124
Chapter 4
0(3) group contains the elements R(rp,O,x) as well as the elements
R(rp, e, x). 4.1.2 U(2) group and SU(2) group Let us now consider the complex vector space of 2 dimensions, i.e. C  space, in which a vector is represented by a column matrix with two components of complex numbers as follows
e
2
(4.22)
A linear transformation on the vector by a matrix M is expressed as
e' =Me, or
(4.23)
where the matrix elements of M are all of complex numbers. If we restrict the transformation to be unitary, which leaves the norm of the vector invariant, then (4.24)
and we reach that (4.25)
The transformation stated above is called the U(2) group. If we take det M = 1 as an extra restriction on matrix M, then matrix M is not only unitary but also unimodular, and the transformations form the group of SU(2). From now on we shall confine ourselves in discussing SU(2), in which the condition Mt = M 1 can be expressed explicitly as follows
125
4.1 0(3) group, SU(2} group and angulnr momentum
a*
(
1*
{3*) 8*
1 ( 8  det M  1
!) . ....
(4.26)
Therefore we have the following relations (4.27) and the matrix M takes the expression as
M(a  {3*
a*{3) ,
which contains only 3 independent parameters due to the condition det M = lal 2 + 1/31 2 = 1. Let us investigate the group property of the matrix M when it is close to the identity element I, and if we put a = 1 + then the matrix M can be rewritten as
e,
M =I+ (
e e{3)
!3*
=I+ D.
The unitarity condition of matrix M, at the vicinity of the identity element, becomes MtM = (I+Dt)(I+ D)~ I+Dt +D =I, which implies Dt +D = 0, or more specifically that e+e* = 0. Therefore we are able to rewrite the matrix M in the following form
M=I+ (·
~c
~(a+
ib)
~(a:ib)), ~c
126
Chapter 4
where a, band c, all real, are the three parameters of the SU(2) group. The generators of the group are obtained as follows
(4.28)
We can verify immediately that the commutators among the generators take the general expression (4.29)
which is exactly the same as the Lie algebras in the case of the 0(3) group. The generators of the SU(2) group are related to Pauli matrices as follows (4.30)
The identical Lie algebras for both groups 0(3) and SU(2) do not necessary have the same domain in the group manifold. Let us construct the group elements, i.e. the matrix M by exponentiating the group generators as follows (4.31)
where n = n(O, cp) has the same definition as that of 0(3) group, yet the parameter X takes different domain in the group manifold for the reason that both matrix M and matrix  M fulfill the unitarity condition of the transformation. Therefore the group elements of SU(2) will fill the sphere of radius 27r, namely it takes all the points in a domain of the sphere with radius 27r in the parameter space to exhaust the group elements. The parameters cp, (}and X range as follows
4.1 0{3) group, SU{2) group and angular momentum
127 (4.32)
with again the points of antipode on the surface of the sphere are identified. It will be left to you as an exercise to show that
M( xn~ · Y) = cos X
.~
2 + zn · O" sm 2 ,  • X
(4.33)
which enables us to conclude that (4.34) While in the case of the 0(3) group, in which R(rp,O,x + 2m1r) = R(rp, 0, x), one concludes that elements M andMin SU(2) correspond to an element R in 0(3). It is said that the SU(2) group is homomorphic onto the 0(3) group. To obtain a deeper understanding of the homeomorphism of SU(2) onto 0(3), let us construct a 2 x 2 matrix X defined as follows
(4.35)
A transformation in the coordinate system performed by the rotational matrix R(rp, 0, x) can equivalently be achieved by the unitary transformation of the matrix X which is sandwiched in between the SU(2) matrix M = exp(xn · Y) and its adjoint conjugate Mt = exp( xn · Y) as follows
(4.36) The above transformation leaves the norm of the vector invariant and is justified by taking the determinant on both sides of the above equation, i.e. x'ix'i = det X'= (det Mt)(det X)(det M) = xixildet Ml 2 = xixi. (4.37)
128
Chapter 4
Let us take the rotation about the 3rd axis with angle cp as an example and evaluate the matrix M = e'PY3 explicitly as follows
(4.38)
Equating the matrix elements on both sides, we obtain
(4.39) (4.40) (4.41) and the coordinate transformation by rotation is reproduced.
4.2 0{3)/SU{2) algebras and angular momentum Let us consider the Lie algebras of the 0(3) group or the SU(2) group, in which we take the generators as follows
(4.42)
in order to be in conformity with the notations used in quantum mechanics. The algebra then has the following usual commutation relations
(4.43)
129
4.2 0(3)/SU(2) algebras and angular momentum
Let us now construct Cartan's standard form of the algebra by solving the eigenvalue equation of the following type [A, X]= pX,
(4.44)
where A is a linear combination of the generators with arbitrary choice of the coefficients ai, and X which is also a linear combination of the generators with the unknown coefficients xi to be determined in order to satisfy the above eigenvalue equation. Denote A and X as follows (4.45) and the eigenvalue equation becomes as
[A , X]
•f<_i j k J X Eij k
= 'lnu
=
s:k j T puj X vk·
(4.46)
Therefore we reach the following set of 3 homogeneous equations as (4.47) For the existence of the nontrivial solutions in xj, one imposes that (4.48) or more explicitly that p
ina3 ina2
ina3 p
ina 2 ina1
ina 1
p
=0.
(4.49)
Hence, we obtain the 3 eigenvalues given as JLo
= 0,
(4.50a) (4.50b) (4.50c)
130
Chapter 4
It is the very example of 3parameter Lie group of rank equaling to one. If we denote Jo, J+ and J_, the eigengenerators corresponding respectively to the eigenvalues flo, 1l+ and /l, we find that J0 is proportional to A. By setting
A
(4.51)
we obtain the following standard form for the commutator of Jo and J± in Lie algebras of the 0(3)/SU(2) group, [J0 , J±] = ±nJ±. Let us compute the commutator of the generators J+ and J_, by applying the Jacobi identity, one finds that
[[J+, J_], Jo] = 0,
(4.52)
which implies that the commutator of J+ and J_ is proportional to J 0 , i.e. (4.53)
where we take a to be one due to the fact that the eigenvalue equation is linear in X on both sides of the equation, that allows the eigengenerators J+ and J_ to absorb any arbitrary factor. Henceforth the Lie algebra of the 0(3)/SU(2) group is summarized as follows
[Jo, J±] = ±nJ±, [J+, J_] = nJo,
(4.54) (4.55)
with the hermiticity property given as
Jo
= Jci'
Jt
= J'
tJ J +·
(4.56)
The calculation of the Casimir operator of the group, we shall leave as an exercise, and it is found that (4.57)
The factor 2 is introduced with the purpose to identify C with the total angular momentum operator J 2 , i.e.
131
4.2 0{3}/SU{2} algebras ancl ang11lar moment?tm
(4.58) Since C commutes with the generators of the group, it is then invariant under the transformation. Furthermore the eigenvalues of the Casimir operator will be used to label the dimension of the irreducible representation of the group. Let us construct the eigenvectors of both the operators C and Jo, and denote it as lc, v), referred to commonly as the irreducible representations of the 0(3) /SU(2) group, hence we have Clc, v) = cic, v), Jolc, v)
= vnlc, v).
(4.59a) (4.59b)
With the commutation relations,[J0 , J±] = ±nJ±, one is now ready to show the following proposition.
Proposition 1. If lc, v) is the eigenvector of the Casimir operator C and the operator J0 with the eigenvalues c and vn respectively, then J±lc, v) are also the eigenvectors of operators C and Jo with the eigenvalues c and (v ± 1)n respectively.
The vectors J ±I c, v) can be easily proved to be also the eigenvectors of operator C because [C, J±] = 0, which leads to the following relation (4.60) To show that J±lc, v) are the eigenvectors of the operator Jo with the eigenvalues (v ± 1)n, let us apply the operator Jo uponJ±Ic, v) and make use of the commutation relation of Eq. (4.53), then we find that
JoJ±Ic, v) = (J±Jo ± nJ±)Ic, v) = (v ± 1)nJ±Ic, v), which imply that J±lc, v) are the eigenvectors of Jo corresponding to the eigenvalues (v ± 1)n respectively. Proposition 1 is then established. Let us now apply Proposition 1 to generate the eigenvectors lc, v) as follows: J+lc, v) =ode, v + 1), J_lc, v) =T ,Bic, v 1).
132
Chapter 4
Therefore we are able to build a sequence of vectors by applying J + and J_ on jc, v) consecutively, and obtain
jc, v_), ... , jc, vq), ... , jc, v1), jc, v), jc, v+1), ... , jc, v+p), ... , jc, v+)· The sequence must terminate somewhere on both ends, otherwise we will encounter the dilemma of having the negative norm of the eigenvector. To see this, let us take the highest positive value v of the eigenvector, such that the norm of the vector J+lc, v+) vanishes, namely
or (4.61) where we have made use of the relation J_J+ = C J6 fiJo. Therefore one proves that the eigenvector jc, a) with a> v+, will lead to negative norm. Similarly, J_jc, v_) = 0, if we denote v_ to be the lowest value of v, and we have the relation (4.62) Solving the above two quadratic equations to obtain values of v+ and v_, we take the difference of v+ and v_, which must be an integer, because it is the total number of steps in applying J+ or J_ to form the nonvanishing eigenveCtors in the sequence. Therefore we have
v+ v_
=
2v~4 + nc2 1 = 2J.'
(4.63)
where j is taken as half integers. It also allows one to express the eigenvalue of the Casimir operator as (4.64) which helps one to calculate that v+
= v_
=J.
(4.65)
4.2 0{3)/SU{2} algebras and angular moment11m
133
It is the construction of the irreducible representation of the 0(3)/SU(2) group that enables us to quantize the dynamical observables C and Jo. In order to express the eigenvectors of J 2 and Jo in the usual form appearing in quantum mechanics, we shall replace vector lc, v) with the familiar form of the vector li, m) from now on, and it is normalized as
(j, mlj', m') = 8jj'8mm', with  j ~ m ~ j,  j' ~ m' ~ j'.
One also has the following relations
n
J+lj, m) =
V'iJ(j m)(j + m + l)lj, m + 1),
(4.66a)
J_lj, m) =
n J(j + m)(j m + l)lj, m 1).
(4.66b)
10 v2
·
Different values of j form different irreducible representations of the group. The !dimensional representation is a trivial one by taking j = 0. For the case that j = 1/2, it forms a twodimensional representation, called the spinor representation. If we identify the vectors ll/2, 1/2) and ll/2, 1/2) with the 2component bases column matrices respectively as follows
then the matrix representation of the generators can be constructed with the matrix element J:!;,f:,) = (1/2, miJil/2, m'). We shall leave them as an exercise again to show that
134
Chapter 4
J.<~) 0
n
(1 o)
=2 0
1
'
(4.67) which allows us to cast the spin matrices as the following expression
(4.68a)
8 = y
_1_(J(~) _ J<~)) = ~ J2i
+

2
(oi i) , 0
(4.68b)
(4.68c)
"
In the case of the 3dimensional representation, of which j generators can be calculated as follows:
J(l) X
~ !:__ J2 (~
y
~ !:__ J2 ( ~
D·
0
(1 0
=n o o
i ~).
(4.69a)
i
0
J~ 1 )
1, the
1
0 1
J.(l)
=
0
~)'
(4.69b)
(4.69c)
0 0 1
which take completely different expression from the 3 generators of the
135
4.3 Irred?tcible representations of 0{3) grmtp and spherical harmonics
rotational group of the Cartesian form obtained in previous Subsection 4.1. In fact the generators of 3dimensional irreducible representation are related to those of the Cartesian form through an unitary transformation which need to be elaborated more as an exercise by the readers.
4.3 Irreducible representations of0(3) group and spherical harmonics ·Let us consider the irreducible representations of the 0(3) group Jl, m), in which l takes only the positive integers land m lies in between l and l, i.e. l = 1, 2, ... and l ~ m ~ l. Then the qrepresentation of the vector Jl, m) in the spherical coordinate system can be expressed as (!9,
Proposition 2.
rzm
Let and YzmI be two spherical harmonics of the same degree l with the order m and m1 respectively, then they are connected by the following relation
Yzm(O,

.
J2
1 y(l+m)(lm1)
eicp
(a + . aa
z cot un
l
u,
(4.71)
The proof is straightforward. Take the qrepresentation of the fol
136
Chapter 4
lowing equation
(0 rnll m)
•r'
=
J2
(0 rniL+Il m 1) J(l+m)(lml)n •r ' '
where if the qrepresentation of operator L+ is expressed by
L + _ eicp(a {)O
+ z. cot oa)"" ocp n,
(4.72)
then we arrive at the conclusion of Proposition 2.
Proposition 3. All the spherical harmonics takes the form of the product of eimcp, an exponential function of
The proof goes by taking the qrepresentation of the equation as follows
(0, cpiL3Il, m) = mn(O, cpll, m). Express the qrepresentation of the operator £3 by (nfi)ofocp, then we have the following differential equation
and the solution of the above equation leads to Proposition 3, i.e.
}[m(o, cp)
= eimcp f(O).
The property of the function f(O) can be analyzed in the following proposition.
137
4.3 lrred?tcible representations of 0{3) grmtp and spherical harmonics
Proposition 4. Function f(O) satisfies the associated Legendre's differential equation, i.e.
d: {(1,i) :J.lf(!l)} + {lu + 1) 1: :
2}
f(!l) = o, (4.74)
or more explicitly by writing f(O) in terms of the associated Legendre polynomial of degree l and order m, i.e. f(O) ex: Pt(cosO) = Pt(J.L),
(4.75)
where we put fl = cos (}, and identify f ((}) = f (1l) .
We start the proof by taking the following equation in qrepresentation, namely
Denote the qrepresentation of the total angular momentum operator by 2
L =
2{
n
a( . aoa) +
1 sin(} ao
sm (}
2 1 a } sin2 (} a2cp
which can be evaluated by means of coordinate transformation from the Cartesian system to the spherical one. Then we reach the following equation
Putting 1l =cos 0, the above equation can be reduced to the following associated Legendre's differential equation as
d: { 2)
(1  !l :1l f(!l)}
+ { l(l + 1) 
1
::2}
f(!l) = 0,
138
Chapter 4
and the solution of it is referred to as the associated Legendre polynomials. Therefore Proposition 4 is proved, i.e.
Proposition 2 can be facilitated to construct the spherical harmonics by avoiding the painstaking methods of solving the associated Legendre's differential equation. rzm(o, rp) can be formulated easily by successive applications of Proposition 2 from the spherical harmonics of lowest order, namely from Yzl(O, rp). Let us first evaluate Yzl(O, rp) by taking the qrepresentation of the following equation
(0, rpiLIl, l) =
o,
or
_ z· cot un~) {)0 orp '17l(nu, rp) _ (~ L
l
(~ _ z· cot un~) {)O orp e ilcpf(n) u
= eilcp (
:0 l 0) f(O) = cot
0,
which leads to the solution of f (0) as
f(O)
=
Asinl 0.
The normalization constant A can be calculated by taking the inner product as follows
Hence A reads as
A=
(2l + 1)! 1 411" 2ll!"
Therefore we are able to express ~l ( 0, rp) as follows
43 Irrecl?tcible representations of 0{3) grmtp ancl spherical harmonics
yzl ((), cp) =
139
(2Z + 1)! 1 ilcp . l () 4n 2lZ! e sm .
Yr ((),
The spherical harmonics cp) can then be constructed by applying the L+ operator l + m times upon 1[l(O, cp), namely
y; m(o ) l ,cp
(l m)! { icp ( a . () a ) (2l)!(l +m)! e ao +~cot acp
=
}l+m y;l(o ) l ,cp
(2Z + 1)(l m)! { icp (a . tO a) 4n(l + m)! e ao +~co acp
1
2ll!
}l+m eilcp sm. l () .
To simplify the calculation of the last term in the previous equation, we apply the identity given in Proposition 5.
Proposition 5.
g
Acting the operator ei'P ( 0 + i cot () ~) n times repeatedly upon the function eipcp f(O) will result in another function of the following form, i.e.
{ . (a + e~'P
ao
a) }n .
i cot() ocp
e~P'P f(O)
= (1tei(p+n)cp
(sinp+n 0d sinP o) f(O). dcos () (4.76)
It is just a pure algebraic manipulation to show the above identity. Let us take the first step to calculate the following equation
140
Chapter 4
(aae
ezcp
a) .
+icot8 ezpcpf(8)
acp
= (1)ei(p+l)cp
If we regard (sinP+l
(sinP+l 8d sinP dcose
e)
f(O).
(4.77)
8dc~O sinP 8) f(O) as a new function of g(B),
i.e.
and take the second step as follows
eicp
(!!_ae + i cote~) ei(p+l)cp g(B) acp = (1 )ei(p+2)cp (sinP+2 8d sin pl dcosB
Repeating n times of the same operation, we have
{ _(aae + ezcp
i cot
_f (8) eacp )}n ezpcp
a
e) g(
B)
44 0(4) group, dynamical symmetry and the hydrogen atom
141
which proves Proposition 5. With all the preparations in the last few propositions, we are now in the position to evaluate Y!m(e, rp) by letting n = l + m,p = l in Proposition 5, then
Yl m((},tf"l)= r
(2l+1)(lm)! 1 { irp(a . t() a )}l+m ilrp. I() 47r(l+m)! 21[! e 8()+~co 8rp e Slll
=
(1)1+m
(2l + 1)(l m)! eimrp . m dl+m . 21 47r(l + m)! 21l! sm () dl+m cos() sm ()
=
(1)1+m
(2l + 1)(l m)! eimrp ( _ 2)W' dl+m ( _ 2)1 1 1 47r(l + m)! 21[! Jl dl+mJl Jl (2l + 1)(l m)! eimrp p,m( ) 47r(l + m)! 21l! 1 Jl '
where
and Rodrigues' formulas for Legendre polynomials Il(JL) as well as for the associated Legendre functions ~m(JL) are reproduced.
4.4 0(4) group, dynamical symmetry and the hydrogen atom Let us investigate the orthogonal group in 4dimensional Euclidean space, denoted by the 0(4) group, in which the norm of the vector,
142
Chapter 4
xixi
=
invariant
fori sums from 1 to 4,
under the orthogonal transformation. The generators of the group in canonical form can be easily constructed by means of rotation on the (xi, :d)plane similar to the case of the 0(3) group as we have done before, namely, they are the 6 generators given as
.a
x~a.
xJ
.a x3a., x~
for i,j
= 1,2,3,4.
If we denote 3
a
1
a
4
a
M2 =X ax1 x ax3'
M3
1
a
2
a
3
a
4
a
= X ax2 X axl '
and 1
a
4
a
N1 = X ax4 X ax1 ,
2
a
N2 =X ax4 x ax2'
N3 =X ax4 x ax3'
then the Lie algebra of the 0(4) group reads as [Mi,Mj]
= tfjMk,
[Mi,Ni]
=
tfjNk,
[Ni,Nj]
= tfjMk.
(4.78)
The above relations enable us to redefine the generators by the following combinations: B·~
=
1 (M·N.·) 2 ~ z '
(4.79)
and the algebra for these redefined generators become
[Ai, Ai]
= tfjAk,
[Ai, Bj]
= 0,
[Bi, Bj]
= tfjBk.
(4.80)
Among the Lie algebra of the 0(4) group, we find that there exist two sets of generators Ai and Bi whose commutators respectively form
44 0(4) group, dynamical symmetry and the hydrogen atom
143
a subalgebra of the 0(3) group. The 0(4) algebra becomes the direct sum of two 0(3) algebras and the 0(4) group is then locally isomorphic onto 0(3) ® 0(3) group. We are now in position to discuss the hydrogenlike atoms from the dynamical symmetry properties. The Hamiltonian operator for the hydrogenlike atoms takes the following expression H
= _!_p22m
ze2
r
= _!_p2 '5__ 2m
r'
(4.81)
where we have fixed the center of the Coulomb potential at the origin. The space described by this coordinate system is isotropic with respect to the rotation about any axis through the origin. It implies that the angular momentum operators are conserved because they commute with the Hamiltonian operator. Furthermore, the attractive Coulomb potential enables us to introduce another three conserved operators, called Lenz operators. Classically, it is the vector, called the Lenz vector or the RungeLenz vector, that originates from the center of the force and points to the aphelion, as shown in Figure 4.2. It is a particular feature that the orbit is closed and fixed in space for the case of the Coulomb potential. If we construct a plane perpendicular to the constant angular momentum through the origin, the closed orbit lies on the plane without
Figure 4.2: Hydrogenlike orbit.
144
Chapter 4
precession of the major axis. Then the Lenz vector can be constructed by means of the following elliptical orbit equation, i.e. mk l
1
 = (1EcosO) 2 r
(4.82) '
where l is the angular momentum of the system and E is the eccentricity of the ellipse given as 2Ei2 1 + mk 2 ,
E=
E
with
< 0.
Let us assume the Lentz vector
R = ar + f3[ X jJ. By taking
r · R, we reach the following relations
 r·R=rRcosO=r
m
k( l
2
2
l } =ar 2 f3l, 2 ) 1 { 1klEE mr
or
l2 ) r  l 2 ( f3 l2 )  0 · ( ar  mkE(l E) m2k2E(l E)  ' that allows us to obtain
1 l2 a   ,: r mkE(l E)'
l2
/3 = m,2 k:2:E(1E:)'
which leads to the classical Lenz vector as _ R=
2
l mkE(l E)
{r +  l xjJ 1 
r
mk
}
.
(4.83)
44 0(4) group, dynamical symmetry and the hydrogen atom
145
To obtain the quantum Lenz operators, we replace the classical dynamical variables by the equivalent quantum operators in the above equation. Then we have
If we take the commutator of computation, we find that
!4 and Rj, though it is a tedious
By rescaling the operators in the following expression
the Lie algebra of the 0(4) group is regained, i.e.
[Li, Lj] = inEijkLk,
(4.85a) (4.85b) (4.85c)
The dynamical symmetry of the Coulomb potential can also be applied to calculate the eigenenergy of the hydrogen like atoms. Let us denote 1 J·z = (L· 2 z +D.) 1~ '
(4.86)
and these operators are the. generators of 0(3) ® 0(3) group, namely,
146
Chapter 4
[Ji, Jj] = inEijkJk, [Ji,Kj] = 0, [Ki, Kj] = inEijkKk,
(4.87a) (4.87b) (4.87c)
with the property J 2 = K 2 because of the relation LJli = 0. Let us calculate J 2 + K 2 , which is related to the Hamiltonian as follows
(4.88)
or
mk2 {
H=2
1
2(J2+K2)+fi2
}
.
(4.89)
Since the hydrogen atom is subject to the condition ;]2 = K 2 , the eigenstates of it are then taken as the direct product of the irreducible representations of 0(3) ® 0(3) as follows
lj, m) ® ik, m')j=k
=IU, m); (k, m'))j=k = l(j, m); (j, m')).
(4.90)
They are also the eigenstates of Hamiltonian (why?), i.e.
Hl(j, m); (j, m')) = El(j, m); (j, m')),
(4.91)
with the eigenvalues
(4.92)
or in terms of the principal quantum number n well known eigenenergy of the hydrogen atom,
= 2j + 1, we reach the
147
4.5 Exercises
where a is the fine structure constant.
4.5 Exercises Ex 4.5.1 Show that SU(2) matrix M(xn · Y) = exp(x~n · iJ) corresponding to a rotation about axis n with an angle x takes the following form,
M( xn~ · Y) =cos X + zn ·~ ·a + • sm X .
2
2
Ex 4.5.2 Taking matrices,
ai
=
O"i
and making use of the following properties of Pauli
(a) hermiticity a it = (b) traceless Tr
ai
= 0,
(c) {ai, ai}
= 28iii,
(d) Tr
=
(J"i(J"j
(e) det
ai
ai,
28ij,
= 1,
show that the rotational matrix element
R)
is given by
(Hint: by Eq. (4.36).) Ex 4.5.3 Construct explicitly the matrix representations of the generators in two dimensions.
148
Chapter 4
Ex 4.5.4
Denote the 2 x 2 traceless matrices A and B by arbitrary parameters as A = a· if, B = b·if. Show that AB = a· bl + i(a x b) ·if. Ex 4.5.5
Show that U( cp)ifUt (rp} where U(r,O)
= (n · if)n + n X if sin rp n X (n
X
if) COS rp,
= exp(i~n ·if).
Ex 4.5.6
Show that the Casimir operator of the rotational group takes the expression
Jdi J+J
in the Cartesian
+ J_J+ + JJ
in the standard form.
Ex 4.5.7
Find the unitary matrix U which diagonalizes
Jic)
01 0) (
=~ 1
0
0
0
0 0
and hence verify explicitly that
J(l) X
010) (
= !!___ v'2 1 0 1
0 1 0
l
b~sis,
J y(l) = !!___ In v2
(~· 't
0
149
45 Exercises
Ex 4.5.8
Show that [H, R?,]
=0
for the hydrogen atom.
Ex 4.5.9
Prove that
Ex 4.5.10
Prove that
(Hint: use the fact that [H, Ji]
=
[H, Ki]
= 0.)
Chapter 5 Lorentz Transformation, 0(3,1)/SL(2,C) Group and the Dirac Equation
5.1 Spacetime structure and Minkowski space Space and time had been treated separately since the birth of physics as an experimental science in mediaeval time. As Newton stated in his work, The Principia, absolute space, in its own n~ture, without relation to anything external, remains always similar and immovable. Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external. Physicists had been confined by the concept of the absoluteness of space and time such that the coordinate transformations from one inertial frame to another, always kept time as an invariant parameter. All this concept of absolute space and time resulted in the formulation of Galilean relativity which has not been challenged until the end of 19th century and the beginning of the 20th century. It was the triumphal results that related the wave phenomena of light to the electromagnetic theory in Maxwell equations, that space and time were interwoven into the formulation of the theory and were treated as equals. The development in special relativity afterwards led naturally to generalize the 3dimensional coordinate space to the 4dimensional one by including the timeaxis, an extra dimension. And so it is no longer the Euclidean vector space, but a fourdimensional Minkowski space with metric tensor defined as
151
152
Chapter 5 9p,v
=0
if fl =/=
V ;

900
= 911 = 922 = 933 =
1.
(5.1)
The zeroth axis is defined as the time axis, and x 0 = ct. A vector in Minkowski space, also called a 4vector, is denoted by x = (x 0 , xl, x 2 , x 3) to specify the time and position. In order to avoid using the term "norm", we define the length square of the vector as
This is a postulate in special relativity that the laws of physics are invariant under a homogeneous linear transformation on the spacetime 4vector if the transformation leaves the length square of a 4vector invariant. This chapter is devoted to the exploration of the properties of these transformations as well as their applications in relativistic quantum wave equations.
5.1.1 Homogeneous Lorentz transformation and 80(3,1} group Let us consider the following linear transformation in the matrix form x
1+
x'
= Ax,
(5.3)
where x and x' are column matrices, and A is a 4 x 4 square matrix. If we denote the Minkowski metric tensor also in matrix form as
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
(5.4)
the length square of the 4vector x becomes xT gx
= (xo)2 + (x1)2 + (x2)2 + (x3)2.
(5.5)
The invariance in the square length of x due to the isotropy property of spacetime restrains the Lorentz transformation matrix A to the following conditions x'T gx1 = XT AT gAx = XT gx, (5.6)
153
5.1 Spacetime structure and Minkowski space
namely that (5.7) This is the crucial difference between the Euclidean space and the Minkowski space because the g matrix is involved in the latter case. If det A =1 0, then the transformation forms a group, called the homogeneous Lorentz group. H we take the determinant on both sides of Eq. (5.7), we obtain (det A) 2 = 1. We shall restrict ourselves to the case det A = 1 with which the group elements can be reached continuously from the identity element. The length invariance will provide 10 equations of constraints among the matrix elements of A, namely
(5.8) Therefore there remain 6 independent parameters in matrix A. Furthermore for the case a = (3 = 0, the constraint reads as
(5.9)
A8
A8 : : ;
which provides two choices, i.e. ~ 1 and 1. If all the group elements are continuously connected to the identity ~ 1 should be chosen, and the group is called proper element, Lorentz group, or simply the Lorentz group, or the 80(3,1) group, to which we shall confine ourselves for further investigation. To construct the generators of the group, let us consider the boost Lorentz transformation, in which one spatial coordinate system O' is travelling with velocity in a particular direction, say along the x 11axis, with respect to another spatial coordinate system 0 along the direction of x 1axis, of which both origins of the coordinate frames coincide at x 0 = x'0 = 0. Then the boost matrix takes the following familiar expression:
A8
(5.10a) (5.10b)
154
Chapter 5
where (3 = vjc and ry = 1/ )1 (32. It is interesting to visualize the above boost transformation as a rotation on the (x 0 , x 1 ) plane with an imaginary angle, or hyperbolic angle defined by
e
(5.11) Then the boost matrix becomes
A=
coshe  sinhe 0 0 sinhe coshe 0 0 1 0 0 0 0 0 1 0
(5.12)
The matrix representation of the generator for the boost transformation can then be easily obtained by taking the derivath;e of the above matrix with respect to the group parameter at the origin of the parameter space, i.e.
e
B1=
0 1 1 0 0 0 0 0
0 0 0 0
0 0 0 0
(5.13a)
The other two generators of the boost transformation in x 2axis and x 3 axis can also be constructed with similar procedures. They read as:
B2
=
0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0
B3=
0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0
(5.13b,c)
155
5.1 Spacetime structun; and Minkowski space
The remaining three generators of the 80(3,1) group are merely those of the spatial rotational ones, expressed as:
Al=
0 0 0
and A2 =
0 0
0
0
0 0
0
0
0 0
0
1
0 0 1
0
0
0 0 0 1 0 0 0
0
0 1 0
0
A3=
(5.14a)
0
0
0 0
0
0
1 0
0 1
0 0
0
0 0
0
(5.14b,c)
This allows us to obtain the Lie algebra of the 80(3,1) group as follows: (5.15) but the commutators among the boost generators Bi take the different sign to close the algebra, i.e. (5.16) The algebra of the generators Ai closes by themselves, they form a sub algebra of 80(3,1), which implies that 80(3,1) contains 80(3), the rotational transformation, as its subgroup. As in the case of rotational transformation, the Lorentz boost matrix A of the relative motion along the x 1 axis can also be regained by exponentiating the matrix generator B1, namely
ef.B1
=
cosh~
sinh~
0 0
sinh~
cosh~
0 0
0
0
1 0
0
0
0 1
(5.17)
156
Chapter 5
Therefore the group elements of the 80(3,1) can be expressed in the general forms as (5.18)
e)
Since the group parameters ~ = (~1, ~2 , are unbounded, hence the Lorentz group, or the 80(3,1) group, is a noncompact Lie group. The algebra of the 80(3,1) group can also be constructed by means of the usual canonical formulation, in which we denote the contravariant tensor generators as
(5.19)
and one will immediately confirm that
It is convenient to cast the algebra in terms of~ and Bi, which will reproduce the 80(3,1) algebra by the following identification o·'1 B2g··M 2'J '
(5.21)
5.2 Irreducible representation of 80{3,1) and Lorentz spinors As in the case of the 80(4) group, let us take the linear combinations of the generators of the 80(3,1) group and denote that
L· = 1 2 2
(Ai +B) _2
2
'
and
R
z
= 21
(Ai  B) . ~
2
(5.22a,b)
Then the Lie algebra of the 80(3,1) group takes the expressions as those of the 80(3) ® 80(3) group, namely, i.e.
5.2 Irreducible representation of 80{3,1} and Lorentz spinors
157
(5.23a) (5.23b) (5.23c) Yet they are not exactly parallel to the case of the 80(4) group, because the generators Li and Ri of the noncompact Lorentz group are not Hermitian. The immediate consequence of these properties gives rise to two options we have to face. Either we choose the finite dimensional representations, in which case the unitarity condition shall be abandoned or we choose to preserve the unitarity of the representation by accepting the infinite dimensional representations. Let us consider the finite dimensional irreducible representation, and take the direct product Jl, m)1 ® Jr, n)r = Jl, m; r, n) as the bases, where Jl, m)1 is the eigenvector of the commuting operators £ 2 and £3 with the eigenvalues l(l + 1) and m respectively, while Jr, n)r is the eigenvector of another pair of commuting operators R 2 and R3 with the eigenvalues r(r + 1) and n respectively, such that
l ::::; m ::::; l,
 r ::::; n ::::; r,
and
l, r = half integers.
(5.24)
Consider the simplest case of one dimension in which we take l = r = 0, and label the representation by the symbol (0, 0). The matrix representations of the generators are of one dimension, namely a 1 x 1 matrix with element as
(0, 0; 0, OILiJO, 0; 0, 0)
= (0, 0; 0, OIRiJO, 0; 0, 0) = 0,
(5.25)
and we have the trivial representation of the group, identity. There are two distinct 2dimensional representations, namely (1/2, D)representation and (0, 1/2)representation. We shall leave them as an· exercise for the readers to calculate that the generators shall take the following forms:
158
Chapter 5
R~~,O) z
= 0,
(5.26a)
(5.26b) which allow us to obtain
Bi(~,O)
~
.
20"./,)
(5.27a)
(5.27b) and obtain finally the 2dimensional irreducible representation of the Lorentz group as follows l o(8, ) oE)  = exp D ( 2•
(i25 · (8 ioE)) ,
(5.28a)
(5.28b)
As a quick check out of curiosity, one finds that
or
namely that unitarity is lost in the finite dimensional irreducible representation.
5.2 Irreducible representation of 80(3,1} and Lorentz spinors
159
With the aim of investigating the relativistic wave equation of spin ~ system, we shall limit ourselves by not going into further exploration of the higher dimensional representations. Let us perform the following identifications:
(5.29a,b)
as the two bases of the twodimensional complex manifold, or
V(~,or
space, and construct the lefthanded Lorentz spinor, or simply the lefthanded spinor as follows
(5.30)
where 'lfJ[{x) or '1/J[(x) are all complex functions of spacetime coordinate x = (x 0, xl, x 2 , x 3). Under the Lorentz transformation, the lefthanded spinor is transformed by applying the (1/2, 0) matrix representation of the 80(3,1) group, i.e. Eq. (5.28a) upon the spinor as follows (5.31) where x' = A  l x implies that active transformation of the quantum states equivalent to a passive Lorentz transformation of the spacetime vector. It should be emphasized that not only the components of the spinor are transformed by the (1/2, 0) matrix, but that the spacetime coordinates in the argument of the complex functions '1/Jl (x) are also simultaneously transformed accordingly. As for the Lorentz transformation of the righthanded spinor, one finds that it can be expressed as
(5.32)
160
Chapter 5
where 'lj!~(x)(a = 1, 2) are the complex functions of the spacetime coordinates, and fa( a = 1, 2) are the bases column matrix of the righthanded spinor space. Similar to the transformation in the lefthanded spinor space, the Lorentz transformation for 'lj!r(x) can then be written as
(5.33)
5.3 SL(2,C) group and the Lorentz transformation
It is interesting to observe that both matrices
n< ~ ,o) ( 0, €)
and
n(o, ~) (if, €) play more roles than just performing the Lorentz transformation upon the spinor wave functions. Let us consider the linear transformation on a twodimensional complex vector space, in which a vector € is acted upon by a 2 x 2 matrix L with complex matrix elements as follows
(5.34)
Let us impose a condition such that
det L
=
a
b
c d
= ab be = 1,
(5.35)
which reduces the 8 parameters to 6 independent ones. Matrices with the properties of such specification form a group of SL(2,C). If we express the SL(2,C) matrix by exponentiating an arbitrary 2 x 2 matrix A, namely
(5.36) then we have the following proposition
161
5. 9 SL(2, C) group and the Lorentz transformation
Proposition 1. If a matrix L can be expressed as L = eA, then
det L
= eTr
(5.37)
A_
Proof of the proposition is straightforward: we find the eigenvectors of the matrix A as the new bases that form the matrix S, which diagonalizes the matrix A, i.e.
s 1 AS. (5.38) Hwe insert the matrix L between s 1 and S, we obtain the following A'=
relation
(5.39)
Therefore we conclude that det L
= det cs1 LS) = ea1+a2 = eTr
A'= eTr s1AS
= eTr
A_ (5.40)
Let us go back to the two dimensional irreducible representations of the Lorentz group, the matrix n< ~ ,o) (if, () and matrix D(O, ~) (if, (). They are in fact the SL(2,C) matrices because det D(~,o)(iJ,()
= det e~u(Bi() = eTr ~u{8i()
1,
{5.41)
~u·(B+iO = 1.
(5.42)
=
and similarly det n
= det
e~u(B+i()
= eTr
The 6 group parameters are designated as jj = (01 , 02 , (]3) and ( = (€1 , €3), which enable us to formulate the generators of the SL (2, C) group in the following matrix representations, i.e.
e,
162
Chapter 5
(5.43a,b)
with the same Lie algebra of the S0(3,1) group, i.e.
[~, Aj] = EtAk>
(5.44a) (5.44b) (5.44c)
We conclude therefore that SL(2,C) is an isomorphism onto the S0(3,1) group. To demonstrate the relation between the two groups with respect to Lorentz transformation, we introduce the spacetime matrix
(5.45)
where u 0 = u0 = I, a unit matrix, and ui are the Pauli matrices. The length of the spacetime vector is related the determinant of the matrix, i.e.
(5.46) Therefore the transformation of X into X' by multiplying SL(2,C) matrices, D
(5.47) will leave the length of the vector x
= (x 0 , x 1 , x 2 , x 3 )
det X'= det X.
invariant because
(5.48)
5.4 Chiral transformation and spinor algebra
163
Take a specific example of a Lorentz boost transformation along the x 3axis, in which = (0, 0, 0), and ( = (0, 0, €), then we obtain
e
(5.49)
Therefore we recover the usual Lorentz transformation by boosting the coordinate frame along the x 3 axis if we compare the matrix elements on both sides of the last equation and obtain the following relations: (5.50a) (5.50b) (5.50c) (5.50d)
5.4 Chiral transformation and spinor algebra As we have mentioned before, physics involves two distinct types of transformations, the continuous transformations and the discrete ones. Here in this section, we shall introduce another discrete symmetry transformation. In addition to parity, charge conjugation or time reversal transformations in quantum mechanics, the chiral transformation is among the least discussed topics. The word "chirality" was derived from the Greek xr::tp, which means hand. Henceforth chirality is generally recognized as synonym to handedness in the geometric sense. It is a discrete transformation from the righthanded irreducible representation to the lefthanded one of the Lorentz group, and vice versa. In contrast to the usual parity transformation, an antilinear and antiunitary operator is required in the chiral transformation in order to obtain the self consistent theory, such that the Lorentz transformation can equivalently be performed in the lefthanded frame of reference as well as the righthanded one. It also provides the interrelation between the lefthanded spinors and the righthanded s,pinors.
164
Chapter 5
Let us denote the chiral operator IC, an antilinear operator and for each vector'¢, there exists a vector a'¢+ bc.p such that IC(a'¢ + bc.p)
= a*IC'¢ + b*ICc.p,
(5.51)
where a and b are complex numbers. The antiunitarity of 1C is defined for each pair of vectors '1/J and c.p, such that the inner product takes the following relation
(1C'¢,1Cc.p) = (c.p,'¢) = ('1/J,c.p)*.
(5.52)
The chiral transformation is neither the parity transformation on the spatial coordinate system nor is it the time reversal transformation on the temporal axis in Minkowski space. The six generators of the Lorentz group therefore are invariant under the chiral transformation, namely the rotational generators as well as the Lorentz boost generators remain unchanged under IC, i.e. (5.53) Yet the new set of the group generators Li and Ri, are transformed according to the following relations
because of the antilinearity property of operator JC. Since chiral transformation on L 2 becomes R 2 and reciprocally it holds for R 2 , namely the chiral transformation will interchange operator L 2 and operator R 2 , therefore we have the following proposition:
Proposition 2. The vector ICLjm is the eigenvector of R 2 and R3 with the eigenvalues j(j + 1) and m respectively. While the vector JCRkn is the eigenvector of L 2 and L3 with the eigenvalues k(k + 1) and n respectively.
165
5.4 Chiral transformation and spinor algebra
The proof goes as follows: let us use the brief notations Ljm, Rkn instead of li, m)l and lk, n)r respectively, and consider R 2 1CLjm = 1CL2 Ljm = j(j + l)ICLjm, and R31CLjm = 1CL3Ljm = miCLjm which implies that ICLjm = y(m)Rj,m· Similarly that ICRkn = 8(n)x
Lk,n· The antiunitarity of the operator 1C allows us to find the coefficients y(m) and 8(n) as follows
(ICLjm,ICLjm')
= y*(m)'Y(m')(Rj,m,Rj,m') = (Ljm',Ljm),
(5.55)
or y*(m)'Y(m')Lm,m' = 8m'm, that leads to h'(m)l 2 = 1. The same argument can be applied to the righthanded case to obtain 18(n)l 2 = 1. It is of interest to observe that the chiral transformation will also make the connection for the irreducible representation of the lefthanded ones to the righthanded ones. Let us take the spinor representation as an example by considering the matrix of the lefthanded Lorentz transformation,
n<~,O) mm'
elementD~~}
= (L l m,eB·A+[BL l m' ) = (veif.A+[BL VL ) '" l2, m'''" l2' m 2' 2'
or it can be cast into the matrix form as follows
(5.56)
Similarly the righthanded Lorentz matrix is related to the lefthanded one by
(5.57)
166
Chapter 5
By choosing the phase factors!'(~)= 8(~) = ')'(~) = 8(~) = 1, and defining a 2 x 2 matrix i = 1, 2 and 3 that
E
as (
0 1 ) , one can easily prove for 1 0 
01) U~ (0 1)
EU~E1 = ( 1 2
0
2
1
0
=
Ui
'
(5.58)
which allows one to reciprocate the following relation: n
= cDa.o)*E 1,
(5.59a) (5.59b)
We are n_ow in the position to investigate further the connection between the SL(2,C) transformation and the Lorentz transformation with spinor algebra. Let us construct a dual space V(~,o) to the spinor space V(~,O)• then V(~,O) is called the colefthanded spinor space, a 2dimensional complex vector space with the basis ea(a = 1, 2) which can be identified with the row matrix as follows
e1 = (1, O), e2 = (O, 1),
(5.60a,b)
where a new set of notations are introduced in the subscript indices in order to differentiate the colefthanded spinor to the righthanded one. A spinor in v(~,O) is then expressed by
'¢ = J;aea = (J;1,J;2),
(5.61)
where J; is related to the components of the lefthanded spinor 'If; ('lj;1, '1j; 2 )T by the following equation
=
(5.62) or in matrix notation as (5.63)
167
5.4 Chiral transformation and spinor algebra
The Lorentz transformation on the colefthanded spinor can be found to be
ni, L.T. nil ntlT 1 'f'l7'1'='f'·E
nt,TDT(l 0) 1 2'€
='f'
Similarly if we construct a corighthanded spinor
dual vector to the righthanded spinor
cp = (cp1, cp2), the
~ (::) is defined as fullows (5.64)
Then the Lorentz transformation on
cp = (cp1, cp2) can be expressed as
. L.T. ·I cpl7cp =cpIT ET =cp·n(l2' O)t •
(5.65)
Let us proceed to consider the irreducible representation denoted by
(~,
!). It is a 4dimensional vector space spanned by the direct product
of two spinor spaces, namely the lefthanded spinor spaces and the corighthanded spinor space, with the bases given by eajb. A general
!
element in ( ~, )space can be expressed as matrix formulation, it can be written as
uteajb,
or if cast into the
(5.66)
A similar argument leads to express the
(!, ~) representation as follows (5.67)
!
It is obvious that Lorentz transformation on vectors in ( ~, )space and in
(!, ~ ) space, takes respectively the following expressions:
168
Chapter 5
L.T. U( l2'2i) 1t
u'
(5.68a)
L.T. U( i2'2l) 1t
u'
n
(5.68b)
=
This transformation property immediately convinces us that the spacetime matrix
(5.69)
which we have constructed in Section 5.3 is an operator belonging to ( ~, ~)representation.
5.5 Lorentz spinors and the Dirac equation As we have discussed in Section 5.1, time and space are regarded as a four vector in the Minkowski space. In fact, the energy and the momentum of a mass point, can also be treated as a Minkowski four vector in relativistic dynamics. Let us define the world line as the trajectory of a mass point in the fourdimensional Minkowski space in which the spacetime coordinates are functions of a parameter r, called the proper time, a Lorentz invariant scalar whose differential is defined as follows (5.70) Consider the Lorentz frame which is boosted instantaneously with the same velocity as that of the moving mass point. Then the time measured in this frame where the mass point is instantaneously at rest, namely that dx' = 0, is in fact the proper time because of the invariant property, 1
2 1 dr = dr' = (  c 2 9ttvdx'11dx'v) = dt'.
(5.71)
169
5. 5 Lorentz spinors and the Dirac equation
Yet this invariant quantity when measured in the initial reference frame, namely 0system, is given by
dr
1 = ( g dxlldxv c2ttv
)1 = g2 2
1 dt c2.
(5.72)
We will then introduce a velocity 4vector utt(r) defined as utt(r) dxtt / dr which is related to the energymomentum 4vector ptt by the following equation (5.73) The energymomentum 4vector for a mass point moving with velocity v can be expressed in the reference frame as pi" = (p0 , j1), which when evaluated in the usual relativistic form becomes:
(5.74a)
_
dx
p=modr
=
moc/J
J1 (32 .
(5.74b)
One can easily verify that the scalar product, or the length of the 4vector ptt is characterized by the rest mass of the moving particle, an invariant under the Lorentz transformation, namely (5.75) Let us construct a 2 x 2 energymomentum matrix as follows
(5.76)
It transforms as the (~, ~) irreducible representation of the Lorentz group, i.e.
170
Chapter 5
(5.77) As a particular case, let us consider a mass point at rest at the origin of a frame 0, and a frame O' which is travelling along x 3 axis with the relative velocity v. Then the 4momentum in the O' system can be expressed as
(5.78) or in matrix notation,
(5.79)
We expect that the energy and the momentum of the mass point, when measured in the O'system, will be respectively given as follows: 10 moe p = moccoshe = , (5.80a) (32
Jl
(5.80b)
Let us consider the chiral conjugate (abbreviated as c.c.) of the spacetime matrix X defined as follows
c.c. X c=E X* E1 . X !+
(5.81)
The Lorentz transformation on Xc can be evaluated by the transformation property of X, namely that x~ = EX'*E 1 = E(DC~,o)xnc~,o)t)*E 1 = nco,~)xcnco,~)t.
One recognizes that X and Xc belong respectively to the (!,
(5.82)
!)and the
171
5. 5 Lorentz spinors and the Dirac equation
(!, ! ) irreducible representations. Yet as far as Lorentz transformation on the spacetime coordinate is concerned, both matrices nC~,o) and nco,~) play equivalent roles but result in the very same Lorentz transformation, namely we have two different ways to perform the same Lorentz transformation by means of employing two different SL(2,C) matrices, i.e. D(~,o) and n(o,~). The spacetime operator X is transformed as (!, !) representation under the Lorentz group. If this tensor operator acts upon a vector in the righthanded spinor space, i.e. Xcp, it turns out that the new transformed vector is a lefthanded spinor , because the contraction of the second indices of the (!,!)tensor with the indices of the (0, !)tensor results in the (!, 0)tensor, namely, a lefthanded spinor. We shall summarize these properties in the following proposition:
Proposition 3.
!
An operator of the (!, )representation that acts upon a vector of the (0, !)representation yields a vector of(!, D)representation. )representation that acts upon Conversely an operator of the a vector of the (!,D)representation will yield a vector of the (0, !)representation.
(!, !
Let A be an operator of the (!, ! )representation, and (0, !)representation, then if we denote 'TJ as follows 'TJ=Ae,
ea vector of (5.83)
then the Lorentz transformation on 'TJ becomes 'TJ
!::'£, 'TJ' = A'e = D(~,o)AD(~,o)tD(O,~)e = D(~,o)'TJ,
(5.84)
which implies that 'TJ is a vector of the (!,G)representation. Similarly if we denote Ac, an operator of the(!, !)representation, then e = Ac'TJ has the following transformation property,
172
Chapter 5
(5.85)
e
which states that is the vector of the (0, ~)representation. The above proposition allows us to conclude that Pc'l/Jl and P'l/Jr are the righthanded spinor and the lefthanded spinor respectively. If we represent the quantum state of a free mass point with spin equal to 1/2 by the spinors, then we have Pc'l/Jl
= moc'l/Jn
P'l/Jr
= moc'l/Jl,
(5.86a) (5.86b)
where we introduce the proportional constant m 0 c in order to obtain the same physical dimension on both sides of the equation. If we take the direct sum of the righthanded spinor and the lefthanded spinor to form a 4component spinor, called Dirac spinor, the above two equations can be simplified in the following expression
(5.87)
or
(5.88)
where '1/Jd(x) = '1/Jr(x) EB '1/Jl(x) is the Dirac spinor. The Dirac equation is usually expressed in the qrepresentation by · identifying that
(5.89)
The Dirac equation then reads as
5.5 Lorentz spinors and the Dirac equation
173
(5.90)
or often expressed in covariant form as (5.91)
where the 4 x 4 matrices 'YM are called Dirac matrices with the explicit expression given as follows:
'Y
M=(Oulk
ut) 0
(5.92)
By making use of the following property of the anticommutator in "'lk, namely that (5.93)
we are able to convert the Dirac equation which is of the first order derivative in space and time, into the second order differential equation of the KleinGordon one. Applying the factor i"jvov once more upon the Dirac equation, we then obtain the following relation
or
(5.94)
which implies that each component in the Dirac spinor satisfies the KleinGordon equation.
174
Chapter 5
The covariant formulation of the Dirac equation does not imply that 111f)J.L is Lorentz invariant, namely that i'YJ.Lf)J.L in the 0Lorentz frame will not take the expression iy'J.Lf)~ in the 0'Lorentz frame. Once the Dirac matrices are chosen in one Lorentz frame, they will in fact be valid in all of the Lorentz frames. The universality of the 111 matrices is the unique feature of the Dirac equation. What has to be modified in the Dirac equation under Lorentz transformation are the spacetime coordinates as well as the Dirac spinor components, namely the Lorentz transformed Dirac equation in the O'system will take the following form (5.95)
with exactly the same 111 matrices appearing as in the 0system. We shall prove the statement in the following proposition.
Proposition 4. The gamma matrices 111 are universal in all Lorentz frame, namely the Dirac equation in another Lorentz frame, i.e. the O' system always takes the same gamma matrices 111 used in 0system. The equation in O' system is expressed as
(iyll8~ + m~c) V;~(x') = 0.
The proof of the statement goes as follows: let us perform the Lorentz transformation of the Dirac equation in 0system by multiplying the following 4 x 4 matrix D( 0, (), the direct sum of the irreducible representation nco,~)(e,() and D(~,o)(e,(), i.e. D(ff, ()
= D(O,~)(ff, () E9 D(~,o)(ff, ()
175
5.5 Lorentz spinors and the Dime equation
If we insert an identity I = above equation reads as
n 1 D
right after the 111 matrix, then the
(5.96)
We shall leave it as an exercise for the readers to show that (5.97) which allows us to express the Dirac equation in 0'system as
(ifll8~ + m~c) V;~(x') = 0,
The representations of the Dirac matrices 111 are not unique. During the development of quantum theory, the formulation of 111 matrices differed from one school to another. The 111 matrices we adopt in Eq. (5.92) are called the Weyl representation. Let us define a new set of gamma matrices by means of similarity transformation with the matrix S given as follows
1 ( I s vf2 I and denote the transformed matrices by matrices reads as follows
I) I
(5.98)
'
iw Then the new set of Dirac (5.99)
or explicitly:
o = (I 0) i = (0 ui)
I
0 I
'
I
ui
0
'
(5.100)
176
Chapter 5
which are commonly referred to as the standard representation of the gamma matrices or the DiracPauli representation as it is often called. The similarity transformation of the gamma matrices reshuffles the components of a Dirac spinor in order to maintain exactly the same form of the wave equation. Let us multiply the whole equation of Eq. (5.91) by the matrix S, i.e.
If we insert an identity I = sIs right after 'YJ.l in the last equation, and denote the new Dirac spinor by ;j;d(x) and ;y11 = S"fJ.LSI, we regain the Dirac equation, namely
where ;J;d(x) = S'l/Jd(x). As the new set of gamma matrices are obtained from the original set by a similarity transformation, therefore the anticommutation relation of Eq. (5.93) is preserved, namely (5.101) this enables us to show that each component in Dirac spinor ;j;d also satisfies the KleinGordon equation. Let us return to the prepresentation of the Dirac equation expressed in Eq. (5.88), i.e.
(5.102)
In the limit of zero mass, the equation reduces to:
Pc'l/Jl = 0,
P'I/Jr = 0.
(o.to3)
5.6 Electromagnetic interaction and gyromagnetic mtio of the t!kctnm
177
Since in the zero mass limit, p 0 = po = IP! and the above equations can be expressed in terms of the helicity operator as follows: (5.104) where a· p =a· pflPI is the helicity operator. The equations imply that the lefthanded spinor and the righthanded spinor are of opposite helicities.
5.6 Electromagnetic interaction and gyromagnetic ratio of the electron Let us investigate a system of a spin 1/2 charged particle moving in the electromagnetic field of the 4vector potential AM(x) = (cp(x), A(x)). The principle of minimal interaction allows us to express the Dirac equation in the following form { 'YM
(PM ~AJ.k(x)) moe} '1/Jd(x) = 0.
(5.105)
We shall adopt the Weyl representation of the gamma matrices and abbreviate P qAjc into the operator 7r. Then the above equation can be separated into the following two coupled equations:
{(Po+ ~cp) a· 7r} '1/Jz = moc'I/Jn
(5.106a)
{ (Po + ~cp) +a · 7r} '1/Jr = moc'I/Jz.
(5.106b)
Let us apply the operator {(Po+ qcpfc) a· 7r} upon both sides of the last equation and make use of the Eq.(5.106a). We then obtain the equation of the righthanded spinor as follows
{(Po+ q:) a· 7r} {(Po+ q:) +a· 7r} '1/Jr = (moc) 2 '1/Jr· (5.107) Consider for the case of pure magnetic field, namely AM = (0, A), then the stationary state solution of the last equation takes a much
178
Chapter 5
simpler expression, i.e.
(5.108)
where we replace Po =  P 0 =  E /c. Eq. (5.108) can be calculated as follows
(a.
7r)
2
=
7r
=
7r
and the second term (
(
~)
2
+ ia · (P ~A) qn c
2 
~)
0".
(moc)
2
'::::'.
(\7
X
x
(P ~A)
A),
(5.109)
2 
(moc) 2 can be approximated as
2 
The first term (cr · 7r ) 2 in
2m0 c
(~moe) = 2moE',
(5.110)
in the nonrelativistic limit, where E' represents the total energy minus the rest mass energy of the particle. Finally Eq. (5.106a) reduces to the familiar expression of the Schrodinger equation for a spin ~ charged particle coupled with the magnetic field H, expressed as follows 1 ( q { 2m PA c 0
)2   qna · H } '1/Jr=E'l/Jr· , 2moc
(5.111)
The second term of the last equation stands for the energy of the intrinsic magnetic moment of the charged spin 1/2 particle J.te coupled to the magnetic field H, namely qn q    a · H =   S · H = J.te ·H. 2moc moe
(5.112)
5. 7 Gamma matrix algebra and PC'l' in Dime: spinor
.~ystem
179
For the case of an electron, the gyromagnetic ratio /e, the ratio of the magnetic moment J.te to the intrinsic spin S, is related to the gfactor in the following relation (5.113)
Comparing Eq. (5.113) with Eq. (5.112), we reach a remarkable conclusion that, to the lowest order approximation, an electron has the gfactor equaling to 2, namely (5.114) The precision in the measurement of the electron magnetic moment had challenged theorists to higher order calculations in terms of fine structure constant, namely the radiative corrections, which are beyond the scope of our investigation.
5. 7 Gamma matrix algebra and PCT in Dirac spinor system Before entering into the discussion on discrete symmetry transformation of the Dirac spinor, we shall explore the properties of gamma matrices. As we have shown that the condition
is valid for any representation of the Dirac matrix. Out of these 4 Dirac matrices 111, let us construct 16 matrices of
4 x 4 and divide them into 5 classes as follows:
(1) I,
(2) 'YJ.l' (3) 1 5
= 1 o1 11 21 3,
(4) /5/J.L'
(5)
O"J.LV
= ! [!mu, 1 v].
180
Chapter 5
Class 1 It contains only the identity matrix, obtained, up to a sign, by making the square of any of 'YM matrix. It commutes with all the gamma matrices in 5 classes.
Class 2 It contains 4 traceless matrices with the square equaling to ±1.
Class 3 It is a traceless, diagonal matrix obtained by successive multiplication of the 4 Dirac matrices. The matrix is denoted by "/5 with the following properties,
Class
4
It contains 4 traceless matrices with the following properties,
{"15"/M, "/5"/v} = 2gl1vl.
Class 5 It contains 6 matrices, also traceless, by taking different values on J.L and v. The square of the matrix is, up to a sign, a unit matrix.
With all the preparation on the properties of the gamma matrices, we are now in position to discuss the parity transformation of the Dirac spinor. Firstly let us consider the parity transformation of a 4vector RM in the Minkowski space expressed as follows
P:
RM =
(R0 ,R) ~RPM= (R0 ,R), (5.115)
5. 7 Gamma matrix algebra and PCT in Dirac spinor system
181
If RM is an operator such that R?M = ( .R?,  R), then we call RM a 4vector in Minkowski space. In fact, there exists another vector SM called the axial 4vector in Minkowski space if the transformation property takes the following relation
P:
or
PSMp 1
= sPM = (SO,s),
(5.116)
when SM is an operator. It is of the outmost interest to look into the parity transformation of a physical quantity S which is invariant under Lorentz transformation, namely a Lorentz scalar. We name S a scalar if it is also invariant under parity transformation, i.e.
or if it is an operator, then
p:
P sp 1 =
sv = s.
(5.117)
In contrast to the last expression, if a physical quantity P is transformed by operator P as to meet the following relation, i.e. ppp1
= P,
(5.118)
then we call P a pseudoscalar. Let us start to investigate the parity transformation property of the Dirac spinor. For the sake of simplicity, we shall express the Dirac equation in momentum operator as follows (5.119a) or
('l Po+ "Y • P moc)'I/Jd
= 0.
(5.119b)
Applying the parity operator P upon the Dirac equation above, and inserting the identity I= pIp right after the momentum operator PM, we obtain
182
Chapter 5
(5.120a) or where 'lj;nx)
('l Po')'· P m0 c)'lj;~(x) = 0,
(5.120b)
= 1''1/Jd(x).
Therefore we have the following proposition:
Proposition 5. Let '1/Jd(x) be the solution of the Dirac equation, and 1''1/Jd(x) be the parity transformed Dirac spinor, i.e. 'lj;~(x) = 1''1/Jd(x) = 'lj;(x 0, x), then yD'Ij;P(x) is the solution of the same Dirac equation.
The above conclusion is obvious if the Eq. (5.120b) is multiplied by 'Yo from the right, and making use of the relation 'Yo"Y = ')''Yo, then we arrive at (5.121) Let us return to the 5 classes of the gamma matrices given at the beginning of this section. With the definition if;(x) = 'lj;t(x)yD, we shall investigate the parity transformation properties of the bilinear spinor combinations given as follows:
To simplify the demonstration, let us take the solution to be nondegenerate, namely
Therefore one can easily verify that
Pif;(x)'lj;(x)P 1 = ijjP(x)'lj;P(x) = eia'lj;t(x)"!0eia'lj;(x) = if;(x)'lj;(x). As for the transformation property of if;(x)'Y5 '1j;(x), one finds that
5. 7 Gamma matrix algebra and PCT in Dirac spinor system
183
It is because of these peculiar properties given above that we call ijj(x)'lf;(x) a scalar and ijj(x)'·/'1/J(x) a pseudoscalar. We shall summarize the results of the transformation properties for the rest of three bilinear spinor combinations as follows:
= (ijj(x)·l'I/J(x), ijj(x)'Y'I/J(x)),
As one can also prove, it is of course much more complicated than in the case of parity that under the Lorentz transformation, the bilinear spinor combinations in class 1 or in class 3 behave as a Lorentz scalar, while class 2 or class 4 behave as a Lorentz vector and class 5 behave as a second rank antisymmetric tensor. We shall leave the proof to students of inquiring minds.
The charge conjugation property can only be analyzed in the system with the electromagnetic interaction. Let us express the Dirac equation for a particle with charge q as follows
(5.122)
Denoting the charge conjugation operator by C, and defining 'lj;c (x) C'lj;(x), then the last equation can be cast into
=
184
Chapter 5
(5.123)
where CqC 1
= q is employed in obtaining the second term.
Taking the complex conjugate of Eq. (5.123), the equation becomes
which can be converted back into the Dirac equation
where the solution is denoted by C'¢c*(x) if the following condition is met, namely,
with eif3 standing for a phase factor and C a 4 x 4 matrix. The solution of matrix C is strictly representation dependent. If we adopt 'YM in Weyl representation, we have the solution C = "/ 5 "/2 , because the Dirac matrices are of real matrix elements except "12 . The charge conjugate spinor takes the following expression (5.124) As for the time reversal transformation of the following Dirac equation, one considers
185
5.8 Exercises
(5.125)
where one takes the timet into tin the equation, T'lf;(x)
= '1/Jt(x), i.e.
r :8o ~ a~ = ao. Equation (5.125) can easily be converted back into the Dirac equation of the following expression
{ (i,l8o + iy · 8)  m~c}T'I/Jt(x) = 0, by multiplying a matrix T, if the following conditions are met, i.e. T/'oT1 = _ 7 o,
We shall leave it to the readers to verify that T ei"IJ/' 0 /' 5'1/J(x) with 'TJ being a phase factor.
= 1'0 1'5 , and '1/Jt(x) =
5.8 Exercises Ex 5.8.1 Show that a spacelike vector is orthogonal to a timelike vector. (Hint: if x 0 > lxl, then y 0 < IYI for x · y = 0.)
Ex 5.8.2 Show that the sum of two timelike vectors in the same light cone is also a timelike vector. (Hint: make use of the Schwarz inequality.)
186
Chapter 5
Ex 5.8.3 Show that two consecutive Lorentz boost transformations in the same direction with velocities {31 and {32 are equivalent to a boost with velocity [3 given by
Ex 5.8.4 Verify that D~::_} = form as follows
'Y(m)D~j2:n,'Y*(m')
can be cast in the matrix
Ex 5.8.5 Verify that
Ex 5.8.6 Let 'If; and cp be the lefthandedspinor and corighthanded spinor respectively. Prove then that cp'lj; is an invariant scalar under Lorentz transformation.
187
5.8 ExercUJes
Ex 5.8.7 Show that the Lorentz transformation matrix element A~ can be expressed as follows (Hint: by Eqs. (5.77) and (5.82).)
AI~= !Tr V 2
{uM D(~,O)u C
V
D(~,O)t} = !Tr 2
{uM D(o,~)u
CV
D(o,~)t} •
Ex 5.8.8 Show that D"/M n 1
= A~'Yv
where
and
"'
v
=
(
0 uv
u~) . 0
(Hint: since X'= DXD 1 by definition.)
Ex 5.8.9 Define a new Dirac spinor as
'1/J~(x) = effca(x)'I/Jd(x). Show that the Dirac equation for a charge particle with EM interaction is invariant under such gauge transformation. (Hint: with a new vector potential A~(x) = AM(x) 811a(x).)
Bibliography
Dirac, P. A.M., The Principles of Quantum Mechanics, 4th Edition, (Oxford University Press, London, 1958). Feynman, R. P. and A. P. Hibbs, Quantum Mechanics and Path Integrals, (McGrawHill, Inc. 1965). Giir:sey, Feza, Editor; Group Theoretical Concepts and Methods in Elementary Particle Physics, (Gordon and Breach, Science Publishers, Inc., New York, 1964). Jordan, Thomas F., Linear Operators for Quantum Mechanics, 2nd Edition, (John Wiley & Sons, 1982). Messiah, Albert, Quantum Mechanics, Vol. I & Vol. II, (NorthHolland Publishing Co., Amsterdam; Interscience Publishers Inc., New York, 1962). Riesz, F. and B. SzNagy, Functional Analysis, translated by L. F. Boron, (F. Ungar Publishing Co., New York, 1955). Tung, WuKi, Group Theory in Physics, (World Scientific Publishing Co Pte Ltd., Singapore, 1985). von Neumann, John, Mathematical Foundations of Quantum Mechanics, (Princeton University Press, Princeton, New Jersey, 1955). Wigner, Eugene P., Group Theory, (Academic Press, New York, 1959). Wybourne, Brian G., Classical Groups for Physicists, ( John Wiley & Sons, 1974).
189
Index
Cartan's criteria for semisimple group, 94 Casimir operator, 96 Cauchy sequence, 7 charge conjugate, 110 charge conjugation, 110 chiral transformation, 163 closure relation, 33 commutability, 16 commutator, 17, 20 compact group compact Lie group, 91 compatibility, 16 compatible observables, 16 completeness, 6 complete vector space, 8 conjugate adjoint conjugate, 11, 25 charge conjugate, 110 contravariant vector, 101 convergent sequence, 7 covariant vector, 101 cyclic permutation, 97 cyclotron frequency, 53
astring, 102 4vector, 152 Abelian group, 94 active transformation, 159 adjoint conjugate, 11, 25, 29 adjoint conjugate matrix, 29 algebra Lie· algebra, 87, 93 0(3) algebra, 88, 128, 143 0(4) algebra, 143 80(3) algebra, 117 80(3,1) algebra, 156 8U(2) algebra, 128 angular momentum, 54, 109, 117, 128 angular momentum operator, 53, 91, 130, 135, 137 orbital angular momentum , 135 annihilation operator, 40, 41, 44 anticommutator, 23 antilinear operator, 112 antiparticle, 110 antiunitary operator, 163 axial vector, 109 axial vector operator, 109
degeneracy, 15, 33, 98 Kramers degeneracy, 113 mfold degenerate, 15 dimension, 3 of the irreducible representation, 131 of vector space, 3 Dirac equation, 168
base vector, 33, 51 basis, 3 boost Lorentz transformation, 153 bra vector, 31 canonical commutation relations, 20, 41, 57
191
192 Dirac matrices, 173 Dirac spinor, 172 DiracPauli representation of gamma matrices, 176 direct product of Hilbert spaces, 50 of operators, 51 of spaces, 19, 50 of states, 19 of vectors, 51, 54 direct product vector, 51, 54 direct sum, 172 discrete discrete group, 107 discrete symmetry, 107 discrete transformation, 107 dual space, 28 dual vector space, 27 dynamical observable, 12, 14, 19 eigengenerator, 98 eigenstate, 14 eigenvalue, 14 eigenvector, 14 electromagnetic interaction, 177 principle of minimal interaction, 177 vector potenti~il, 177 equation Dirac equation, 168 KleinGordon equation, 173, 176 Schrodinger equation, 37, 64, 79 Euclidean space, 2 Euler angles, 122 expectation value, 13, 22, 68 Fourier Fourier expansion, 58 Fourier transform, 39 4vector, 152 function 8function, 33, 36
Index
state function, 13, 35, 37 wave function, 13, 35, 37, 39, 46, 71 fundamental commutation relation, 19 gfactor of magnetic moment, 179 gmetric tensor of Minkowski space, 152 gtensor of Lie group, 95 generator matrix representation, 120, 133, 147, 154, 157, 161 GramSchmidt orthogonalization, 4 group Abelian group, 94 continuous group, 83 discrete group, 107 exceptional group, 105 group generator, 89 homogeneous Lorentz group, 153 infinitesimal generator, 89 invariant subgroup, 94, 115 Lie group, 87, 93 0(3) group, 117 0(3,1) group, 151 0(4) group, 141 proper Lorentz group, 153 rparameter group, 86 rotational group, 91, 118 semisimple group, 94 semisimple Lie group, 94 SL(2,C) group, 151 80(3) group, 117 80(3,1) group, 156 80(4) group, 156 SU(2) group, 117, 124 symmetry group, 83 translational group, 84, 90 U(2) group, 124 group composition rules, 84
193
Index
group generator, 89 group manifold, 85 group parameter, 83 group parameter space, 85 gyromagnetic ratio, 179 Heisenberg Heisenberg picture, 68 Heisenberg's uncertainty relation, 23 helicity, 177 Hermitian Hermitian conjugate, 28 Hermitian operator, 1214 Hilbert SP.ace, 6, 8, 12, 32, 33 homogeneity, 83 homogeneous Lorentz group, 153 homogeneous Lorentz transformation, 152 idempotent operator , 12 identity identity element of a group, 84, 86 identity operator, 8 identity transformation, 87 Jacobi identity, 94, 115 infinitesimal generator, 89 inner product, 3 intrinsic intrinsic compatibility, 19 intrinsic compatible observables, 19 intrinsic spin, 179 invariance invariant subgroup, 94, 115 Jacobi identity, 94 ket vector, 31 KleinGordon equation, 173, 176 Kramers degeneracy, 113
Legendre polynomial, 137 Associated Legendre poly., 137 Lenz operator, 143 Lenz vector, 143 Lie algebra, 87, 93 rank of, 98 standard form of, 98 Lie group, 93 compact Lie group, 91 linear dependent, 2 linear independent, 2 Lorentz boost transformation, 153 Lorentz group proper Lorentz group, 153 Lorentz spinor, 156, 168 mfold degenerate, 15 matrix A matrix of Lorentz transformation, 152 Dirac matrices, 173 orthogonal matrix, 120 Pauli matrices, 126, 147 metric tensor, 152 Minkowski metric tensor, 151 Minkowski inequality, 24 space, 151 mirror image, 107 momentum operator, 34 norm, 6 of a vector, 31, 124 normal coordinate, 57 null vector, 3 number operator, 42 operator adjoint operator, 11 angular momentum operator, 53, 91, 130, 135, 137 annihilation operator, 40, 41, 44 a;ntilinear operator, 112
194 antiunitary operator, 163 axial vector operator, 109 bounded operator, 8 Casimir operator, 96 constant operator, 21 continuous operator, 9 creation operator, 40, 41, 44 demotion operator, 44 deviation operator, 22 helicity operator, 177 Hermitian operator, 1214 idempotent operator, 12 identity operator, 8 inverse operator, 10 Lenz operator, 143 linear operator, 8 momentum operator, 34 null operator, 8 number operator, 42 projection operator, 12, 3032, 71 promotion operator, 44 time evolution operator, 64 translational operator, 34 unit operator, 8 unitary operator, 10, 34 vector operator, 108 orthogonal matrix, 120 orthogonal vectors, 6 orthonormal, 5 particlewave duality, 22 path integration, 76 physical observable, 13 postulate of quantum dynamics, 64 postulates of quantum mechanics, 12 1st postulate of, 13 2nd postulate of, 16 3rd postulate of, 20 4th postulate of, 66 principle of minimum interaction, 52 propagator, 71
Index
proper Lorentz group, 153 proper time, 168 pseudoscalar, 181 recursion formula, 103 relativistic wave equation, 152, 159, 176 representation DiracPauli representation of gamma matrices, 176 irreducible representation, 133, 135, 156 irreducible representation of 80(3,1), 156 nrepresentation, 46 prepresentation, 33 qrepresentation, 33 standard representation, 176 Weyl representation of gamma matrices, 175 Rodrigues formula for associated Legendre polynomials, 141 for Legendre polynomials, 141 of Hermite polynomials, 46, 48 root vector, 101, 104 scalar product, 3 Schrodinger equation, 37, 64, 79 Schrodinger picture, 68 Schwarz inequality, 4 semisimple Lie group, 94 space enspace, 2 £ 2 (a, b)space, 2, 3 nnspace, 2 *enspace, 28 complete vector space, 8 complex vector space, 2, see also enspace dimension of vector space, 3 direct product space, 19, 50
Index
dual vector space, 28 Euclidean space, 2, see also nn_ space Hilbert space, 6, 8, 12, 32, 33 orthogonal complement space, 12 subspace, 12, 17 vector space, 1, 27 spatial translation, 33, 90 spherical harmonics, 135 spin 1/2 particle, 178 spinor colefthanded spinor, 166 corighthanded spinor, 167 Dirac spinor, 172 lefthanded spinor, 159 Lorentz spinor, 156, 168 right handed spinor, 159 standard representation, 176 state, 13 physical state, 13 structure constant, 93 subgroup invariant subgroup, 94, 115 semisimple Lie group, 94 subspace, 12, 17 symmetry, 83 discrete symmetry, 107 dynamical symmetry, 141 rotational symmetry, 83 transformation, 83 translational, 83 temporal reflection, 112 tensor antisymmetric tensor, 120, 183 metric tensor, 152 time evolution, 12, 64 time reversal transformation, 107, 110 transformation, 111 active transformation, 159
195 boost Lorentz transformation, 153 charge conjugate, 110 charge conjugation, 110 chiral transformation, 163 continuous transformation, 107, 163 discrete transformation, 107 homogeneous Lorentz transformation, 152 identity transformation, 87 Lorentz transformation, 151 orthogonal transformation, 91, see also 0(3), 0(3,1), 0(4) group parity transformation, 107 time reversal transformation, 107, 110 unitary transformation, 70, 117, see also U(2), SU(2) group translation, 27, 84 uncertainty deviation, variance, 6.p, 6.q, 22 Heisenberg's uncertainty, 23 quantum uncertainty, 40, 79 uncertainty principle, 22 uncertainty relation, 22 unitarity, 125, 126, 157, 158 unitary unitary group, see U(2),SU(2) group unitary operator, 10, 34 unitary transformation, 70, 117 vector base vector, 33, 51 basis of, 3 bra vector, 31 contravariant vector, 101 direct product vector, 51, 54 4vector, 152
196 ket vector, 31 Lenz vector, 143 linear dependent, 2 linear independent, 2 null vector, 3 orthogonal vectors, 6 orthonormal set of, 5 root vector, 101, 104 state vector, 13 vector space, 1, 27 vector diagram, 104 wave relativistic wave equation, 152, 159, 176 wave function, 13, 35, 37, 39, 46, 71 wave number, 61, 63
Index