Faddeev L.D., Yakubovskii O.A. Lectures on Quantum Mechanics for Mathematics Students

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118

L. D. Faddeev and 0. A. Yakubovski i

with trace zero:

0 1

_ 0 -i

of _

1 0'

3--'

-1

These are called the Pauli matrices. Let us compute the commutation

relations for them:

- 0' O =

2 0

=2i

[a1o2]

-i

-1

that is, [a', o-2] = 2io3; similarly,

[o, (73] = 2icr1, [Q3, (71] = 2io2.

It is not hard to see that the matrices --iaa /2, j = 1. 2, 3, have

the same commutation relations as the infinitesimal generators Aj of

the rotation group

2Q1

192

2Qg

Therefore, we can construct a representation g --+ U(g):

(t)

U(g) = exp [_(o-iai

2 + 2a2 + Q3as)J .

It should be noted that this is not a representation in the usual

sense of the word, since instead of

(2)

we have

(3)

U(91) U(92) = i1(9192)

U(91) U(92) = aU(9192),

where a = ± 1. It is not hard to see this in a simple example, com-

puting the product U (gl) U (g2) when gi = g2 is the rotation through

the angle 7r about the x3-axis:

U(.9,) U(.92)

e e; e

At the same time, U(0, 0, 0) = I corresponds to the identity ele-

ment of the group according to the formula (1). A mapping g - U(g)

satisfying (3) with J

= 1 is called a projective representation with

a multiplier. If we nevertheless want to preserve (2), then we shall

have to consider that to each rotation there correspond two matrices

§ 27. Representation by 2 x 2 unitary matrices

119

U differing in sign.

In physics such representations are said to be

two-to-one. These representations play as important a role in quan-

tum mechanics as the usual representations. In what follows we shall

not stress this, distinction. We remark further that the appearance of

such representations is explained by the fact that the rotation group

is not simply connected.

We consider the properties of the matrices U (g).

It is obvious

that they are unitary, since the of are self-adjoint matrices. It is not

hard to see that they have determinant equal to 1

Indeed, U(g)

has the form eis, where S is a self-adjoint matrix with trace zero.

This matrix can always be reduced to diagonal form by a similarity

transformation, and it takes the form

Correspondingly,

0 _

eza 0

the diagonal form of the matrix will be

_ i.X

The trace

and determinant are invariant under a similarity transformation, so

det U (g) = 1.

Let us determine the general form of a unitary matrix with de-

terminant equal to 1. The condition for unitarity is

b) (a

c + bb a+ bd _

(c d) (b d) -

ca + db

cc + ddJ

From the equality cu + db = 0, we have that d = -cd/b, and from the

conditions det U = I and ad + bb = 1, we get that

ad - be = - ab - be = - b (aa + bb) _ - b = 1;

that is,

c= --b, d =a.

Thus, unitary matrices with determinant 1 have the form

(-6

a bJ,

ja12-FJb12=1.

The group of such matrices is denoted by SU(2).

120 L. D. Faddeev and 0. A. YakubovskiI

§ 28. Representation of the rotation group on a

space of entire analytic functions of two

complex variables

In t his section we construct all irreducible represent ations of the rotes

t ion group. As the space of the representations we choose t he Hilbert

space D2 of functions f E C,'q E C) of the form

ni n2

= C n C

< oc .

1 M2

nin2

nl,n2=() ni! ni.

nl,n2

with the scalar product

(f,.9) =

r1}

r)) e212

dlL(r!)

Just as in 19, the functions f n1 n2 =

,t i ??Tb2 //nj ! n2!

can be

seen to form an orthonormal basis in this space: (f12, fn' n2) _

6nin; Sn2n2. Taking into account the connection between the groups

SO (3) and SU (2), we can construct a representation of the group

SU(2). In what follows, it is convenient to represent f

rz) as f ('),

where

E C2 The representation U -* W(U) is defined by

the formula

(1)

W (U) f ((') = f (U-10-

We. shall denote these operators also by W (a) or W (g), and the op-

erators of rotation about the axes by Wj (a), j = 1, 213.

To get an expression for W (g), we find the infinitesimal operators

of the representation, which we denote by -i Mj, j = 1, 2, 3:

Wai

=dWl(a)

(),a=of

if (()

da8a1

daf (Ui-

(a) 010=0

n=o

(ei)a

a f 4 (a)

a f drl(a)

da 4-0 + a

n=o

Here we have used the definition (1) and have denoted by

(a) and

ii(a) the components of the vector e 2' a (. The last derivatives are

§ 28. Representation of the rotation group

computed as follows:

i l0

doe e

a-0

and therefore

2a, e2°10c

ct=0

77,

dI(a

JQ=0

2 77

2 2(()'

As a result we get that

of of

-iMi f (0 =

2 (T

The operators M2 and M3 are found in exactly the same way.

write out the expressions for these operators:

(2)

(77 d

+ a

}

M=_i

2 2

na + a

M32 77

077)

121

It is easy to verify that the operators Mj have the same com-

mutation relations as the angular momentum operators, and -iMM,

j = 1,2,3, have the same commutation relations as the matrices

A2 ,

A2, A3. For the operators W (a) we get that

(3)

W(a) = exp[-i(Mial + M2a2 + M3a3)]

The basic convenience of the representation space D2 is that it

is very easily decomposed into a direct sum of invariant subspaces

in which the irreducible representations act. Indeed, the invariance

of some subspace with respect to the operators W (a) is equivalent to

the invariance with respect to the action of the operators MI, M2. Ltd3.

From the formulas (2) it is clear that the subspaces of homogeneous

polynomials of degree n = nl +n2 are such invariant subspaces. These

subspaces have dimension n + 1, n = 0, 1, 2, ....

It remains for its

to show that such subspaces do not contain invariant subspaces of

122

L. D. Faddeev and 0. A. Yakubovskil

smaller dimension. To do this we introduce the operators

/AI

M =M

iM2

M- = M -iM =

and consider how they act on the basis vectors fn1

n2 :

a Cnj It2

M+fnin2 =

-nl

o 1f .n21

n lf .n2I .

ns-lnn2tl

\/n- -(n

/(ni - 1)!(n2 -F 1)!

that is,

(5)

M+fnin2 = -

nl(n2 + 1) fni-1,n2+1,

M_ fnin2 =

(n1 + 1)n2 fnl+l,n2-1

It is obvious that

M+fort = 01

M fnu = 0.

From (5) it is clear that the subspaces of homogeneous polynomials

do not contain invariant subspaces of smaller dimension.

Let us show that the basis vectors are eigenvectors of the operator

M3 and the operator MI = Mi + M2 + M32. We have

nt r1712

-1

fin'

-1(1

n -n2)

fnln2

%/-n--j!

. n2, .

Vn n2.

that is,

(6)

-{n1 - n .

1n2

2)fn1n2

For the operator M2 we have the formula

M2=M+M_+M3 -M3.

Indeed,

M+ M- =

(M1+iML)(Ml-iM2) = M12 +M.2 -i(M1M2-M2M1)

=M2-M2

+M3.

§ 29. Uniqueness of the representations D3

123

Next, we have

M 2 f,,1

n2 = (M4-M- +

M3 -

3) frt, rt 2

-- (v"ni + 1)n2

n2 (n, + 1)

(n, -- n2)2 +

2(nI -

n2)fnin21

so that

(7)

M2 f

n1 712

((ni

+ n2)2 -F'

2 (nl + n2) J fni

nz.

It is convenient to rewrite all the relations obtained, replacing

ni, n2 by j, m according to the formulas

ni+n2

m_- (ni -

n=0113...

2 2

m=-3,-j+1,...,j,

n j =j - m,

n2 = j + m.

Then the formulas (5) -(7) take the form

(8)

M+fjm = -

(j - rn) Cj + m + 1) fj,rn+i

(9) M _ f m = --

(j - m + 1 } (, + m} f j,m-i

(10)

M3,fj9n = mf jrri,

(11

M2fjm = . (j + 1)fjrri

where f j,, denotes f

ra, n2 = f j -rra,j+m.

The new indices j and m are convenient in that to each in-

dex j there corresponds a representation of dimension 2j + 1, j =

0. 1/2,113/21

. This representation is usually denoted by D,, and

j is called the representation index. The formulas (8) (10) make it

possible to easily construct the explicit form of the matrices MI, M2,

M3 for each Dj. Thus, we have constructed the finite-dirnerisional

representations 1). of the rotation group for all dimensions.

§ 29. Uniqueness of the representations Dj

We prove that the irreducible representations Dj constructed are

unique (up to equivalence). In the course of the proof we shall see

124

L. D. Faddeev and 0. A. Yakubovskil

to what extent the spectrum of the angular momentum operators

is determined by their commutation relations, and how an arbitrary

representation of the rotation group is decomposed into irreducible

representations.

Suppose that some irreducible representation is defined on an

n-dimensional space E, and denote by - i Jr . --- i J2 , - i J3 the infinites-

imal operators of this representation. They satisfy the commutation

relations

[Ji, J2] = ieJ31

iJ2, Jai] = iJ1

[J3,Ji] = 2J<2.

The equivalence of this representation to some representation Dj

will be proved if we can prove that for a suitable choice of basis in

E the matrices Jk, k = 1. 2, 3, coincide with the matrices Mk . We

note first of all that a representation of the rotation group is at the

same time a representation of its subgroup of rotations about the x3-

axis through angles a. This subgroup is Abel i an, and therefore all

its irreducible repiesentations are one dimensional and have the form

e-'rn a (we do not assume anything about the numbers mk, allowing

the possibility of "multivalued" representations). This means that

for a suitable choice of basis in E the matrix of a rotation about the

x3-axis has the form

e-iha =

e irnla

0 ... 0

e-im2

.........

..........

... 0

e- im"Q

Thus, E has a basis of eigenvectors of the operator J3:

J3erra = Trtem .

Further. it follows from the irreducibility of the representation that

the operator J2 = '12 + J22+ J,21, which commutes with Jk for all

k = 1.2, 3, must be a multiple of the identity on the space E This is

possible if all the basis vectors e, are eigexivectors of J2 corresponding

to one and the same eigenvalue, which we denote by j (j + 1) (for the

present this is simply a notation). The basis vectors will be denoted

§ 29. Uniqueness of the representations D.

125

by elm

(2)

J2ejm =X(.7 + 1) e)m

We introduce the operators JJ = J1 ± i J2 . It is easy to verify

the formulas

(3)

(4)

(5)

[J2, jam]

-- o,

[J3, J±] = ±J±,

J2=J±J::F +J3

j3.

We find a bound for the possible values of 17,nl for a given eigenvalue

j (j + 1).

To this end we take the scalar product with e j m of the

equality

(J,2 + J2) elm

= (J(1 + 1) -

m2)

and get that

((J? + Jl) ejrr1, ejrn) = j(3 + 1)

-m2.

The left-hand side is nonnegative, therefore

(6)

lmi < /j(j + 1).

From the relations (3) and (4) it follows that

+1)Jfe J3Jte

=(rn±1)JCjj(j

m t m

Therefore, the vectors J± e j,, (if they are nonzero) are eigenvectors of

the operator J2 with eigenvalue j (j + 1) and of the operator .13 with

eigenvalues m ± 1. Thus, for an arbitrary basis element e7 r, we can

construct a chain of eigenvectors with the same eigenvalue of J2 and

with eigenvalues ml, m] + 1, ... , 7,n2 of J3. Here m 1 and m2 denote

the smallest and largest eigenvalues, respectively. The existence of

nil and rn 2 follows from the inequality (6): the chain of eigenvectors

innst break off in both directions.

Let us compute the norm of J± e j,n, using the facts that ikjm 11 = 1

and J-L = JT.

iiJ±C,m

112

= (JTJ±ejm.

(('12

- 3 T J3) ejm, ejm)

_ j(j + 1) -rn(rat+1)= (.j

rn)(j±m+1).

126 L. D. Faddeev and 0. A. Yakubovskij

Therefore. we can write

(7)

J=e.in = -

(j T m) (j ± m + 1) ej.r1at

This formula enables us for any unit basis vector to construct new

unit basis vectors e j,7 ± 1 satisfying all the requirements The minus

We have not yet explained what values the numbers j and m can

take. We start from the equalities

(8)

J_ ejmi = 0,

J+ejr2 = 0

Multiplying these equalities by J +. and by J_ and using (5), we get

that,

(J2J+J3)jmi=0.

(J2-JJ.3)jm2=0,

J(j + 1) _ M2 + mi = 0,

j(j +1) -rn -m2 =0

From (9) we get at, once that (mi + M2) (MI - m2

- 1) = 0. We take

the solution ml = -m2 of this equation, since m2 > rnl. Further,

rn2 - rn z = 2rn2 is an integer or zero. Therefore, rrm2 can take the

values 0, 1/2. 1, 3/2, .... Finally, we see from (0) that j can be taken

to be m2. This means that the eigenvalues of 'I 2 have the form

j (j + 1), where j = 0, 1/2, 1, 3/2, .

. ,

and for a given j the eigenvalues

m of J3 ruri through the (2j + 1) values -j, + 1, ... , j - 1. j. The

riuinbers j and in are simultaneously either integers or half-integers.

We emphasize once more that these properties of the spectrum of the

operators J2 and J have been obtained using only the commutation

relations.

To finish the proof it remains for us to see that the eigerivectors

constructed form a basis in £. This follows from the irreducibility of

the representation. Indeed, the subspace E` spanned by the vectors

e37 n with m = -j, -j + 1,. .. , j will be invariant with respect to the

operators Jk, k = 1, 2, 3, and therefore must, coincide with e, and the

dimension of the representation is n = 2j + 1. The formulas (1) and

(7) show that for this choice of basis the matrices Jk coincide with

the matrices Mk.

30. Representations of the rotation group on LZ(S2) 127

We note that at the same time we have constructed a way of

decomposing an arbitrary representation into irreducibility represen-

tations. Suppose that a representation of the group of rotations -iJk

and its infinitesimal operators acts in some space e. To distinguish

the invariant subspaces, we must find the general solutions of the

equations

(10) J2e.rn

+ 1) ejm.

J3e3m = mejm.

F o r given j and rn = -j, --3 + 1, ... , j the vectors ejm form a basis

of an irreducible representation of dimension 23 + 1. The problem of

finding the general solutions of (10) is solved most simply as follows.

First a vector e. ,j satisfying the equations

J+ejj = 0,

J3ej.j = jejj

is found, and then the vectors a j m are constructed using the formula

J_ejn = --

(j + m}(j -- rn + 1) ej,m-1,

which makes it possible to find all the vectors e j,,, frorri e j j .

§ 30. Representations of the rotation group on

the space L2(S2). Spherical functions

In § 25 we constructed a representation of the rotation group on the

state space f = L2 (R3) by the operators

W (a) = exp[--i(L1a1 + L2a2 + ,3a3)]

where L1, L2.13 are the angular momentum operators. We recall that

these operators act only on the angle variables of a function (x) E

L2 (R3)

, and

therefore it is convenient to regard the space L2 (R3)

a s 1,2 (R+) ® L2 (S2) . Here L2 (R4) is the space of square-integrable

functions f (r) with weight r2 on R+, and L2(S2) is the space of

square-integrable functions i(n) = 0(0, co) on the unit sphere. The

scalar products in these spaces are given by the formulas

fr2fm(r)Jdr,

f V), (n) GZ (n) dn,

where do = sin 0 dO dcp is the surface element of the unit sphere.