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

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108

L. D. Faddeev and 0. A. YakubovskiT

which we also verify in Cartesian coordinates. To this end we compute

the operator - J12, using the notation x, y, z for the projections of x:

a 2

a a z a

a 2

-L = x19

-yax

+ yaz - za+ zax -x192

2 2 192

192

492

192 2

a19z2

= (x +y +z)

+ ay2 +

-x

_y`

2 -z

a ay

192 _ 192

192 d

- 2xy a2yz a az -

2zx

az ax -

19x -

_ 2z

az .

Taking into account the relations

a a

rar -xax+yay +zaz,

2192

19`2

a 19

rar

-x

axe

a 2

+Z2

1922

+xax

+ya

+zaz

,92

192

+ yz

+ 2zx

19X (9y

ay az

192

we get that - T.2 = r2 A - (r)2

i9r 07

-r9 =r 2A

(r2

c0 ).

§ 24. The rotation group

Denote by G the collection of all rotations of the space R3 about

the origin of coordinates. Each rotation g E G generates a linear

transformation

x'=gx

of three-dimensional space, where g now denotes the corresponding

matrix IIgig 1 1

, called a rotation matrix. It is well known that g is an

orthogonal matrix and det g = 1. The converse is

also true: to each

such matrix there corresponds a definite rotation. Therefore, in what

follows we identify rotations with their matrices. The group of real

orthogonal matrices of order three with determinant 1 is denoted by

SO(3) and called the rotation group

There are several ways to parametrize rotations. For example,

in theoretical mechanics one often uses the ruler angles.

For our

purposes it is most convenient to specify a rotation by a vector a = na

(j n J = 1) directed along the axis n of rotation with length a equal to

§ 24. The rotation group

109

the angle of rotation. Here it is assumed that the direction of rotation

and the direction of the vector a form a right-hand screw and that

i he angle a of rotation does not exceed ir (0 < a < ir). If we start the

,vectors a from a single point, then their endpoints fill a ball of radius

Tr. Different rotations correspond to different interior points of this

ball, and diametrically opposite points of the boundary correspond

to the same rotations.

Thus, the rotation group is topologically equivalent to a hall with

opposite points identified.

We shall establish a connection between the rotation group and

the Lie algebra of skew-symmetric matrices of order three.

A matrix A is said to be skew-symmetric if the transposed matrix

A' is = -A. An arbitrary skew-symmetric matrix is given by three

parameters and has the form

a12

a13

--a12

(123

(-a13

-a23 0

The collection of skew-symmetric matrices becomes a Lie algebra

if the commutator [A, 13] is taken as the Lie operation. This assertion

follows from the properties of commu t ators and the equality [A, f3]' =

-[A, B]. The latter can be verified directly:

[A, B]'= (AB-BA)'= 13'A'-A'B' =BA- AB = -[A, B].

It is convenient to choose the matrices

0 0 0 U 0

0--1

A,=

0 0 -1

ill =

0 0

A3 =

1 0

0 \-i 0

U 0

as the generators of the Lie algebra. Any skew-symmetric matrix can

be represented in the form

A(a) = a, Al + a2A2 + a3A:;.

110

L. D. Faddeev and 0. A. YakubovskiT

A3:

We find the comirlutation relations for the matrices A1, A2, and

[A, ,As]= AIA2 - A2A1

0 -1

0 0- U 0 0= 1 0 0= A3,

and the desired relations are

(1)

A direct verification shows that

[A(a), A(b)] = A(a x b).

Let us consider the matrix eI A3 . To get an explicit expression

for this matrix, we compute the integer powers of the matrix A3:

-1 0

A2=

-1

---13,

A3 =-A3,

A4=I3,

0 0 0

Expanding e'7 A3 in a series, we get that

A -

x arcA;

0O (1)k2k

(-1}k+1c2k-1

E `

= I + I>

(2k)

(2k-i)!

n=O

k=1

k=-1

= I+I3(cosa - 1)+A3sina,

cos a - since 0

e, A3 - = Sin cx

Cos a

0 1

We see that e'A, is a rotation about the third axis through the

angle a; that is, g(0, 0, a) =

It can be verified similarly that

g(a, 0, 0) = e°A' and g(0, a, 0) = C,A2. F o r rotations about the coor-

dinate axes we shall use the abbreviated notation g (a), i = 1, 2, 3.

§ 25. Representations of the rotation group 111

The matrices

8q(a)

are called infinitesimal generators of the

8a;

rotation group. Using the formulas for rotations about the coordinate

axes, we get that

ag(al

, 0,

a=n

sal

ae°`1 A,

a, =0

0a1

a1=0

= Al .

Similarly,

ag(a)

8a2

a=0

Og(a)

= A3-

(9a3

a=o

We see that the skew-symmetric matrices A1, A2, As are infinitesimal

generators of the rotation group.

Now we can easily get a formula for an arbitrary rotation g(na).

It is obvious that the product of the rotations about a single axis n

through angles a and f3 is the rotation through the angle a +,3 about

the same axis:

g(nj3) g(na) = g(n(a +,3)).

Differentiating this identity with respect to /3 and setting /3 = 0, we

get, that,

+ n A+ n A

na) =

g(na).

(niA 1

22 3

3} g(

We have found a differential equation for g(na) which must be solved

under the initial condition g(nO) = I. The unique solution of this

problem is

(2)

.9(na) = eXP[(niAi + n2Aa + nsAs)a].

§ 25. Representations of the rotation group

A representation of the rotation group G on a Hilbert space E is

defined to be a mapping W that carries elements g of G into bounded

linear operators W (g) on E depending continuously on g and such

that

1) W(I) = I,

2) W(9i92) =W(9i)W(9a)

112 L. D. Faddeev and 0. A. Yakubovskii

In the first condition the first I denotes the identity rotation, and the

second denotes the identity operator on C.

It follows at once from

the conditions 1) and 2) that

W(g-1)=W-1(g).

A representation is said to be unitary if the operators W (g) are

unitary; that is, W *(g) = W -'(g).

In group theory it is proved that any representation of the rota-

tion group is equivalent to some unitary representation, and therefore

we can confine ourselves to the study of unitary representations.

We recall the two equivalent definitions of an irreducible repre-

sentation:

1) A representation is said to be irreducible if there are no

operators other than those of the form C1 that commute

with all the operators W (g) :

2) A representation is said to be irreducible if 6 does not have

a proper nontrivial subspace 6 invariant under the action

of the operators W (g).

It is easy to construct a representation of the rotation group on

the state space ' = L2 (W). To do this we define the operators by

the formula

(1)

W(g) OX = 0(g-'X).

The condition 1) of the definition is obvious, and the condition 2) is

verified as follows:

W (9192)

(92 '911 x) = W(91) /' (.9 ' x) = W (91) W (92) zb (x).

The operators W (g) are one--to-one mappings of 1-1 onto itself, and

therefore to verify that they are unitary it suffices to see that they

preserve the norm of a vector ':

IIW(g)II2 =

1,0(g-IX)12dX

= I= IIV9112,

JRJ

where we have used the equality det g = 1. Let us show that

(2)

W (a) = exp[-i(I,i al + Lla2 + L3a3)]-

§ 25. Representations of the rotation group 113

If ere we have denoted W(g(a)) by W (a), and the. projections of the

angular momentum by L1, L2, L3- To prove (2) we first consider a

rotation about the third axis and show that

W3((-t) = e

-iL3CY'

or, equivalently,

(3)

b(93 1(a) X) =

e_iL3a'0 (X)

for any fi(x) E R. We verify the last equality by showing that the

functions on the left- and right-hand sides satisfy one and the same

equation with the same initial conditions.

The function

'0(X, a) -- e- iL30t v)(X)

obviously satisfies the equation

80(x,a)

- -iL30(x,a),

or, in more detail,

(4)

2 - X1

1P (x, a) -

ax,

aX2

)

The function on the left-hand side of (3), which we denote by

ip (x, a), has the form

01 (x, a) ='(x1 cos a + X2 sin a, -x1 sin a + x2 cos a, x3).

We verify that this function satisfies the equation (4):

O,1 a 1 a

(--xi sin a + x2 cos a) +

(--x1 cos a- x2 sin a),

Ox1 Ox2

x2 =

8x1

cos a - x2

axe

sin a,

1901

= x

sin a+ x

cos a

1Ox2

iax1 iax2

from which (4) follows for the function 01(x, a). Finally, both func-

tions satisfy the same initial condition

0(XI 0) = *I (XI 0) = *(x) -

114

L. D. Faddeev and 0. A. Yakubovskii

Thus, the operators of rotation about the coordinate axes have

the form

(s)

Wj(a) =

e-iLjcx,

j=1, 213-

The infinitesimal operators of the representation are found at once

from (5):

8W (a)

Oaf

aWj(a)

a=O

(=0

= --iL3 .

Repeating the arguments that led us to the formula (24.2), we

now get (2). We note that the operators - i Li have the same com-

mutation relations as the matrices A3 , j = 1,2,3. Indeed, it follows

from

[L1

L2] = iL3

that

[= --iL3.

In group theory it is proved that the commutation relations be-

tween the infinitesimal operators of a representation do not depend on

the choice of the representation. Moreover, if some operators acting

in a space E satisfy the same commutation relations as the infinites-

imal generators of a group, then they are the infinitesimal operators

of some representation acting in E.

In connection with the rotation group this means that if we

find operators M1, M2, M3 satisfying the relations [M1, M2] = M3,

[M2, M3] = M1, [M3, M1] = M2, then we can construct a representar

tion W by the formula

W(a) = exp(a1M1 +a2M2 +a3M3)-

§ 26. Spherically symmetric operators

An operator A is said to be spherically symmetric if it commutes with

all the operators W (g) in a representation of the rotation group.

Obviously, an operator A is spherically symmetric if [A, La] = 0

fori=1,2,3.

Here are some examples of spherically symmetric operators.

§ 26. Spherically symmetric operators

115

1) The operator of multiplication by a function f (r). Indeed, we

have seen that the angular momentum operators act only on the angle

variables. Therefore, L i f (r) * (x) = f (r) L i iI (x) for any '0 E R.

2) The operator L2. The spherical symmetry of this operator

follows from the relations [L2, Li] = 0 obtained earlier for i = 1, 2, 3

3) The kinetic energy operator T = -(1/2m)A. The spherical

symmetry of this operator is clear at once in the momentum repre-

sentation, where it is the operator of multiplication by the function

p'2 / (2m). In the momentum representation the angular momentum

operators Li have exactly the same form as in the coordinate repre-

sentation.

4) The Schrocliuger operator

H_ --

10+ V(r)

for a particle in a central field, as a sum of two spherically symmetric

operators.

We direct attention to the fact that the very existence of spher-

ically symmetric operators different from CI implies the reducibility

of the representation of the rotation group constructed on the state

space H.

For the spectrum of the Schrodinger operator in a central field

we now clarify some features connected with its spherical symmetry.

Let 0 be an eigenvector corresponding to the eigenvalue E:

H = E.

Then

HW(g) ?P = W(g) H = EW(g)1,

from which it is obvious that W (g)

is also an eigenvector of H

corresponding to the same eigenvalue.

We see that the eigenspace HF of H corresponding to the eigen-

value E is invariant under rotations (that is, under the action of the

operators W (g)) . The representation W of the rotation group on the

space h induces a representation WE, g --+ WF (g), on the subspace

where WE (g) is the restriction of W (g) to HF (below we use the

flotation W (g) also for the operators W,; (g)). There are two cases:

116

L. D. Faddeev and 0. A. Yakubovskii

either the representation induced on HE is irreducible, or R F, con-

tairis subspaces of smaller diniensiori that are invariant under tiW(g),

and then this representation will be equivalent to a direct suns of

irreducible represenitations.2'

We see also that the multiplicity of our eigenvalue of a spherically

symmetric Schrodinger operator is always no less than the dimen-

sion of some irreducible representation of the rotation group, and it

coincides with this dimension in the first case mentioned.

In physics the appearance of multiple eigenvalues of energy is

called degeneracy, and such energy levels are said to be degenerate.

If on each of the eigenspaces the induced representation is irreducible,

then one says that the operator II does not have accidental degen-

eracies. In this case the multiplicity of the spectrum is completely

explained by the chosen symmetry of the problem. In the presence

of accidental degeneracies there may exist, a richer symmetry group

of the Schrodinger equation. This is the case for the Schrodinger op-

erator for the hydrogen atom, which, as we shall see, has accidental

degeneracies with respect to the rotation group.

We remark that for a spherically symmetric operator H with pure

point spectrum there are eigenvalues with arbitrarily large rrlultiplic-

ity. Indeed, in this case 7-1 can be represented in the form

`l =

"'E1 ®nE2

a...

On the other hand,

fl=Rl.®7-12e ,

where the N,, are subspaces on which irreducible representations of

the rotation group act. In studying such representations in § 30 we

shall see that among the 7-1, there are subspaces of arbitrarily large

dimension. But for any ln, at least one of the intersections 7-1,, n HE,,

25 Here

we use a well-known theorem in group theory

if g - W (g) is a unitary

representation of the rotation group on a Hilbert space E, then there exist finite-

dimensional subspaces El, e2,

that are invariant with respect to W (g) and such

that the representation W is irreducible on each of them. they are pairwise orthogonal,

and they have sum E, that is, E = El ®E2 9

§ 27. Representation by 2 x 2 unitary matrices

117

is nonernpty, and then it contains l-(

entirely, so that the eigenvalue

Ek has multiplicity not less than the dimension of 7n .26

If the system does not have accidental degeneracies, then the

cigenvalues of H can be classified with the help of irreducible rep-

resentations of G in the sense that each eigenspace H E is also an

eigenspace of the corresponding representation. Therefore, the prob-

lent of finding all the irreducible representations of the rotation group

is important. We dwell on this question in the following sections.

In concluding this section we remark that such a relatively simple

quantum mechanics problem as the problem of motion in a central

field could be solved in general without invoking group theory. Our

goal with this example is to show how group theory can be used

in solving quantum mechanics problems. The xnicroworld (atoms,

molecules, crystal lattices) is very rich in diverse forms of symmetry.

The theory of group representations makes it possible to take these

symmetry properties into account, explicitly from the very beginning,

and it is often the case that one can obtain important results for very

complex systems only by using an approach based on group theory.

§ 27. Representation of rotations by 2 x 2

unitary matrices

We shall construct a representation of the rotation group G acting in

the space C2. To this end we introduce three self-adjoint matrices

26A

very Simple example of a spherically symmetric operator with pure point

spec trump is the Sehrodiriger operator

f f =

Pj 4 P2 4- P2

,02

1 ± -2 ±

)

2 2

or, in the coordinate representation,

Fl=_

A+

The problem of separation of variables reduces to the one-dimensional case, and for

the eigenval ues one obtains the formula

En1n2n3 = n1 + n2 + TL3 +

3),

ni, n2. n3 = 0. 1, 2,

To each triple of numbers n1, n2, n3 there corresponds an eigenvector 'Pn 1 n2 n3

It is

clear that the multiplicity of the eigenvalues E grows unboundedly as E ---> oo