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

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198

L. D. Faddeev and 0. A. Yakubovskii

Then

Am iLn cprnn and H AcpM,}, = Am/1n'. From the

completeness of the collection it follows that AB;p = BA'p for any

ypEH.

The assertions proved are valid for an arbitrary number of pair-

wise commuting self-adjoint operators A1,

. . ,

An, ... with pure point

spectrum and with certain stipulations for operators with a continu-

ous spectrum.

We also have the following assertion. If the self-adjoint operators

A1,.

.. ,

An commute, then there is a self-adjoint operator R such that

all the operators Ai, i = 1. 2, ... , n, are

functions Ai = FZ (R) of it.

It is quite easy to construct such an operator R for operators

with pure point spectrum. Let us consider the case of two commuting

operators A and B. They have a common complete system

I of

eigenvectors which satisfy the equations (2). We define the operator

R by the equalities

(In)n

-ryn i mn

where the r(

are distinct real numbers. Obviously, R is a self-adjoint

operator with a simple pure point spectrum. We now introduce real

functions F W and G (x) satisfying the conditions Am = F (r ,,7) and

/-In = G (rmn ). For x different from a all the numbers

the values

of F (x) and G (x) are not important.

It follows at once from the

definition of a function of an operator that A = F (R) and B =

G(R). The general theorem for commuting operators A,

.....

An with

arbitrary spectrum was proved by von Neumann.

We now discuss the physical consequences of the assertions for-

mulated for commuting operators. We recall that the Heisenberg

uncertainty relation does not impose any restrictions on the variance

of the commuting observables, and in this sense we have called them

simultaneously measurable observables. The concept of simultane-

ous measurability can now be made more precise. The last assertion

shows that to measure the numerical values of commuting observables

it suffices to measure the single observable R; that is, it is possible

in principle to find out the numerical values of all the observables

A1, ... ,

An with a single measurement. The existence of a system of

common eigenvectors implies the existence of an infinite set of states

in which all these observables have definite numerical values. Finally,

§ 44. Properties of commuting operators 199

it will follow from results in the next section that for any state it is

possible to construct a common distribution function for the numer-

ical values of simultaneously measurable observables.

We shall introduce the concept of a function of commuting oper-

ators A and B. For a real function f (x, y) it is possible to construct

a self-adjoint operator f (A, B) by setting

f(A'B) Vmn -

where the vectors prn n satisfy (2). This definition agrees with the

definition given earlier for a function of simultaneously measurable

observables.

We have the following assertion. If the operator D commutes

with any operator C commuting with all the operators A1, .. , Art

and [Ai, Aj] = 0, then D is a function of these operators.

In the

proof we again confine ourselves to the case of two operators with

pure point spectrum.

Suppose that to each pair Am, An of eigenvalues there corresponds

a single eigenvector:

ASpmn = Am Pmn,

Bcomn = /1 n(pmrzn

In this case it suffices that D commute with just the operators A and

B. Indeed, if [D, A] = 0 and SO E fm, then also Dcp E Wm, since

ADcp = DAcp = Am Dep.

Similarly, from the condition [D, B] = 0,

we get that if Sp E 1l;t, then Dip E H. Therefore, if cp E f mn, then

Dcp E xrrara. By assumption, the spaces flmn are one dimensional,

and DScrnn is only a numerical multiple, which can be different from

Omn; that is, D;p,nn = XrrznjOmn. Choosing the function f (x, y) such

that Xmn = f (Am, Pn), we see that D = f (A, B).

Let us proceed to the more complicated case when the subspaces

xrrtn are not one dimensional

A'rrrnn = Am'prnn,

B"pmmn = inSO(k)

k = 1121.

We consider the collection of eigenvectors corresponding to the

pair Am, Ian . For brevity we omit the index mn. The operators CU)

200 L. D. Faddeev and 0. A. Yakubovskil

and C(ji) are introduced by specifying that

(k)

_ Sjk(P(k).

CUO

(k)

cu)

P =

Cw) O

= 0,

where f

is the orthogonal complement of the. subspace H,nn. It is

Irri

easy to verify that all the operators CO) and COO commute with A

and B. From the condition [D, CU)] = 0 we get that

DC(j) p(j) = C{j) AP(j )

from which it follows that D' p(j) is proportional to Sp(j) :

We show that all the numbers x(j) coincide:

X(j)(PU) = DW(j)

C(jt)Dcp(d)

= CUO X(0)

0(l)

= X(l) cP(3 }

Thus, the vectors

are

are eigenvectors of D, and the eigenvalues

do not depend on the index k: D'pmn =

Therefore, D -

f (A. B) as in the first case.

We point out that commutativity of D with just A and B would

not be sufficient here. What is more, if the commutativity of D with

A and B implies that D = f (A, B), then we can assert that to each

pair of eigenvalues Am and jn there corresponds a single eigenvector

'P,nn .

Indeed, if there are several such vectors, then we can always

construct an operator that commutes with A and B but is not a

function of them: for example, the operator C(pl).

We can now define the important concept of a complete system of

commuting operators. A system of self-adjoint operators A1, ... , A,,,

is called a complete system of commuting operators if:

1) the operators Ai commute pairwise, that is, [As. A3] = 0 for

i, = 192,...,n;

2) none of the operators Ai is a function of the others;

§ 45. Representation of the state space

201

3) any operator commuting with all the Ai is a function of these

operators.

From the assertions proved above and the conditions 1) and 3) of

the definition of a complete collection A1, ... ,

An, it follows that there

is a common complete system of eigenvectors of all these operators,

A1'Pai

.Q n = a1 'Pa i , ass

i=1,2,...,n,

and to each collection of eigenvalues a1,.

.. , an there corresponds a

single eigenvector Vai,

a,,.

The condition 2) implies that the last

property is not enjoyed by a common complete system of eigenvectors

for only a part of the operators A1,.. . ,

An. In fact, 2) means that

there are no "superfluous" operators among A1,

... , An-

If as a result of measurements it is known that the numerical

values of a complete set of observables A1,.. .

, Ay, in some state are

certainly equal to a,, ... , an, then we can assert that this state is

described by the vector 'pai,

an .

In conclusion we note that if a set of pairwise commuting inde-

pendent operators is not complete, then it can be supplemented

many ways -to form a complete system.

§ 45. Representation of the state space with

respect to a complete set of observables

Suppose that A1,. .. ,

A,7 is a complete set of operators with pure

point spectrum. These operators have a common complete set of

eigenvectors 'Pal,

an, and

corresponding to each collection of eigen-

values is a single eigenvector Vai ,

an .

An arbitrary vector /' E ?l

can be represented as a series

11 = E VI(al, ... , a,t} (pat,

are ,

2/(al,

... ,

an) = (Z/J, SPai,

a,,

This formula determines a one-to-one correspondence between the

vectors 0 and the functions '(al, ..

an) defined on the spectrum of

the operators A1,

... , An:

(ai,...,an).

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

Obviously,

(PI,'2) _

ii (a, .

.. , an) 12(a] , . .. I an)Y

al a

Aiy) H aii/J(al,

... ,

an);

that is, the representation constructed is an eigenrepresentation for

all the operators A1, ... , An (the action of these operators reduces to

multiplication by the corresponding variables).

The function ,&(a1,.. . , an) is called a wave function. To clarify

its physical meaning we construct, as in the preceding section, an

operator R such that Ai = F i (R) f o r i = 1, 2, ... , n,

i ,

an. = ra, ,

.a n SPu, ,

,a,:

where ra,,

are distinct real numbers and ai = Fi (ra, ,

know that J ,a,,) I2 = l'I'(ai..

. .

an) j2 is the probability of ob-

taining the numerical value ra,,

a,z

for the observable R by a mea

surentent. Therefore, liI'(ai,..

, an) I2 is the probability of obtaining

the values al

, .

. . , an as a result of simultaneous measurement of the

observables A 1,

. . . ,

Are.

All these results generalize to the case of a complete set of opera-

tors A1, . . .

An with arbitrary spectrum. We formulate the theorem

wit hout proof.

Theorem. Let A1,. .. ,

An be a complete set of commuting operators.

Then there exists a representation of the state space such that a vector

b E fl is represented by a function *(a,, ..,an) defined on some set

'A (a = (al, ..,an) E 2). A measure duc(a) is given on the set %.,

and a scalar product is defined by the formula

(11, )2) =

jIPI(a)1k2(a)d/2(a).

In this representation the operators A, , ... , An are the operators of

multiplication by the corresponding variables

AiVY(al,...,an) = ai (ai,...,an), i = 1,2,...,n.

The function I?J(al, ...,an) 12

.. , an

)12 is the density of the common distribu-

tion function for the observables Al, .... An with respect to the mea-

sure dp(a).

§ 46. Spin

203

Above we already had examples of complete sets of commuting

operators and corresponding representations of the state space R.

For a structureless particle the coordinate operators Q1, Q2. Q3

form a complete set. The coordinate representation corresponds to

this set The momentum representation is constructed similarly from

the complete set P1, P2, P3. A complete set is also formed by the oper-

ators H, L2, I,3, where H is the Schrodinger operator for a particle in

a central field. The representation corresponding to this complete set

was described in § 31. For a one-dimensional particle the Schrodinger

operator II for the harmonic oscillator by itself represents a complete

set. The corresponding representation was constructed in § 18.

§ 46. Spin

tip to this time we have assumed that the electron is a material point

with mass ni and charge -e; that is, it is a structureless particle whose

state space H can be realized, for example, as the space L2 (R3) of

square-integrable functions O(x). On the basis of this representation

of the electron, we calculated the energy levels of the hydrogen atom

and obtained results that coincide with experimental results to a high

degree of accuracy. Nevertheless, there are experiments showing that

such a description of the electron is not complete.

We have already mentioned the experiments of Stern and Gerlach.

These experiments showed that the projection on a certain direction

of the magnetic moment of the hydrogen atone in the ground state

can take two values. In § 34 we constructed the quantum observable

"project ion of the magnetic moment" of a structureless charged par-

ticle, and we saw that the third projection of the magnetic moment

is proportional to the projection L3 of the angular momentum. From

the calculation of the hydrogen atone, we know that the numerical

value of L3 is zero for the ground state.

Therefore, the magnetic

moment of the atom in the ground state must also be zero.

This

contradiction can he explained if we assume that the electron itself

has a magnetic moment and a mechanical moment whose projections

on a certain direction can take two values

204 L. D. Faddeev and 0. A. Yakubovskiii

The electron's own angular momentum is called its spin, in con-

tradistinction to the angular momentum associated with its motion

in space, which is usually called its orbital angular momentum.

The existence of an observable, which can take two numerical

values, leads to the necessity of considering that the electron can

be in two different internal states independently of the state of its

notion in space. This in turn leads to a doubling of the total number

of states of the electron. For example, to each state of the electron in

a hydrogen atom (without counting spin) there correspond two states

that, differ by the projection of the spin on some direction.

If we assume that there is no additional interaction connected

with the spin, then the multiplicity of all the energy eigenvalues turns

out to be twice that of a spinless particle. But if there are such inter-

actions, then the degeneracies connected with the spin can disappear,

and there can be a splitting of the energy levels. Experiments show

that such a splitting does indeed take; place.

In § 32 we described a model for an alkali metal atomn, based

on the assumption that the valence electron of the. atom moves in a

central field. This model is not had for describing the configuration

of the energy levels of alkali metal atoms, but neither this model itself

nor any of its refinements can explain the observed splitting of the

levels into two nearby levels when 1 0. The hypothesis of spin

enables us to easily explain this splitting. In atomic physics there are

many other phenomena which can be explained on the basis of this

hypothesis. We shall see later that only the doubling of the number

of electron states that is connected with spin makes it possible to

explain the length of the. periods in the periodic table

We would like. to stress that the hypothesis of spin for the electron

is a hypothesis about the nature of a concrete elementary particle

and does not concern the general principles of quantum mechanics.

The apparatus of quantum mechanics turns out to be suitable for

describing particles with spin.

Let us begin with the construction of the state space for an elec-

tron. Without spin taken into account, L2 (R3) is the state space f

in the coordinate representation. The introduction of spin requires

§ 46. Spin 205

us to extend the state space, since the number of states of a particle

with spin is larger than for a spinless particle.

A doubling of the number of states without changing the phys-

ical content of the theory is easily attained by replacing the state

space 'H = L2 (R3) by HS = L2 (R3) 0 C2 and associating with each

observable A acting in R the observable A 0 I acting in 7-Is.

We explain this in more detail. The elements of the state space

of a particle with spin are pairs of functions

01 (X)

()

The scalar product in the space 7_Is is given by

(2)

(q, f) =

V)1 (x) W, (x) dx + ?F12 (X) 2 (X) dX.

For the observables A 0 I acting in 7-Is we use the same notation

and terminology as for the observables A acting in H. Thus, the

coordinate operators Q1, Q2, Q3, the momentum projection operators

P1, P2. P3, and so on, all act in 7-Is. For example,

P1w=P1

(?P1 (X)

0z(X)

h oiy i (x)

i ax

Ox,

To each pure state V)(x) in 7-I there now correspond the two orthogonal

stat es

7P(X)

T =

V)(x)

in ?-Is, or any linear combination of them. It is clear that the mean

value of any observable A in a state b will be equal to the mean

value of the observable A 0 I in the states IF1 and ' 2 . Therefore, by

introducing the space 7-Is and confining ourselves to observables of

the form A R I, we have indeed attained a doubling of the number of

states while preserving all the physical consequences of the theory.

However, along with the observables A 0 1 there are also other

observables acting in the space 7Is, for example, of the form 1 0

S, where S is a self-adjoins operator on C2. Of course, there are

observables acting in 7Is that cannot be represented in the form A ® I

or 10 S, for example, sums or products of such observables.

206

L. D. Faddeev and 0. A. Yakubovskii

Let us consider observables of the type I ® S. First of all, it is

clear that any such observable commutes with any observable A 0 1

and is not a function of observables of the latter type. Therefore, on

the space RS the observables Q1, Q2. Q3 do not, form a corriplete bet

of commuting observables, arid we will have to supplement this set to

obtain a complete set.

Any self-adjoint operator S on C2 is representable, as a self-adjoint

2 x 2 matrix and can be expressed as a linear cornbinat ion of four

linearly independent matrices. It, is convenient to take these matrices

to be the identity matrix I and the Pauli matrices ar , (72, ar3.

-i

a1 = 1 0 ,

a2 = i 0 , a2 = 0 -1

The properties of the Pauli matrices were discussed in § 27. The

matrices S3 = o-3/27 j = 1, 2, 3, have the same commutation relations

as for the orbital angular momentum:

[S1, S2] = iS3,

[S2. S3] = iSl,

[,S3- S1 ] = iS2.

We have seen that the commutation relations are one of the most

important properties of the angular momentum operators Therefore,

it is reasonable to identify the operators 1 ® Sj, j = 1, 2, 3, with the

operators of the spin pro jections.49 These operators will usually be

denoted by S3 in what follows. The operators S j have eigenvalues

±1/2, which are the admissible values of the projections of the spin

on some direction.`} The vectors %P ] _

(lp(x)) and W2

= (V (X) }

are eigenvectors of the operator S3; with eigenvalues + 1 /2 and - 1/2.

Therefore, these vectors describe states with a definite value of the

third projection of the spin.

49 De eper considerations are based on the fact that for a system with spherically

symmetric Schrodinger operator the operators I., 4 S. are quantum integrals of motion

We shall discuss this question later

`"We write all fornnmlas in a system of units in which h = 1 In the usual systemu

of units. the operators of the spin projections have the form SJ - (h/2}a;, and the

admissible numerical values of these projc ctions are equal to ±h/2 We remark that in

passing to classic al muecha,iics h -- 0, and correspondingly the projections of the spin

tend to

zero

Therefore, the spin is a specifically quantum observable

§ 46. Spin

207

For what follows, it is convenient for us to change the notation

and write a vector 'P E 7s as a function W (x, s3 ), where x E R3 and

83 takes the two values -+-1 /2 and --1 /2. Such notation is equivalent

to (1) if we let

T X1 -

7P, (X). X. - - = '02 (X) -

(

From the equality

it follows that

(Pi(x))

0l(X)-

1101W

\-2VYX))

S3' (X, s3) = S3 %P (X, s3);

that is, the operator S3, like the operators Q,, Q2, Q3, is an oper-

ator of multiplication by a variable. We see that the representation

constructed for the state space of a particle with spin is an eigenrep-

resentation for the operators Q1, Q2, Q3 and S3, and these operators

form a complete set of commuting operators acting in 7is.

It is now easy to understand the physical meaning of the func-

tions T (x, S3). According to the general interpretation, Is3) 12

is the density of the coordinate distribution function under the con-

dition that the third projection of the spin has the value s3, and

R3 W (x, 83) 1 2 dx is the probability of obtaining the value 83 as a

result of measurement.

Along with the observables S1, S2, S3 we can introduce the op-

erator S2 = Sj +S22 +S32 of the square of the spin.

Substituting

Si = o j in this expression and considering that oj = I, we get that

S2 = 4I. We see that any vector

E 7s is an "eigenvector" for the

operator S2, with eigenvalue 3/4. This eigenvalue can be written in

the form51 s (s + 1). where s = 1 /2 Therefore, one says that the spin

of the electron is equal to 1/2.

Let us construct a representation of the rotation group on the

space fs. We recall that the representation of the rotations g act in

the space H = L2(R3) as the operators W (g) = e- i(f,ial+L2a2+L3a3

511n §29 we showed that from the commutation relations for the angular mo-

mentum it follows that the eigenvalues of the operator of the square of the angular

momentum have the form j (j + 1), where j is an integer or half-integer For the spin

operator this number is commonly denoted by s