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

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208

L. D. Faddeev and 0. A. YakubovskiI

and in the space C2 as the operators U(g) = e-i(s1a1

}.S2a2+.S3a3). The

mapping g ---+ WS (g) with W,5 (g) = W(g)

U (g) is a representation

on the space 1-Is, the product of the representations W and U.

An operator acting in 1-CS is said to he spherically symmetric if

it commutes with all the operators Ws(g). If the Schrodinger oper-

ator H is spherically symmetric, then the operators WS (g) and the

infinitesimal operators

i)wv

as j

= -?VLj Z9 1 + I '11)j),

= 11Z151

are integrals of motion. Therefore. for a system with spherically sym-

metric Schrodinger operator we have the law of conservation of the

total angular moment urn, whose projections are Jj = L j + SS.

We remark that in the general case of a spherically symmetric op-

erator 11, there are no separate laws of conservation of orbital angular

momentum and spin angular momentum. However, if a spherically

symmetric Schrodinger operator acting in Ns commutes also with

all the operators W (g) 0 1, then it commutes with all the operators

I 0 U (g), and there are

laws of conservation for the observables L j

and Sj separately. An example of such a Schrodinger operator is the

operator 1101, where H is the Schrodinger operator for a particle in

a central field.

§ 47. Spin of a system of two electrons

The space C2 introduced in the previous section is often called the

spin space for the electron. For a system of two electrons the spin

space is the space C4 = C2 0 C2. In C2 we choose a basis consisting

of the eigenvectors

rl; =

(0I)

and u_

of the operator S3 with eigenvalues 1/2 and -1 /2, respectively. As

basis vectors in C4 we can take U(1)U(2), U(1)U(2),

U+1)U(2),

and

(1) U(2)

, where the

indices (1) and (2) below indicate the spin sub-

spaces of the electrons.

§ 47. Spin of a system of two electrons

209

However, it turns out to be more convenient to use another or-

thonormal basis consisting of the vectors

Wi = U+

iV2 = 0)

U(1) U (2)

W3 =

72=

W4 =

IU(I)U(2) _ U(I)U(2))

(' +

The convenience of the new basis consists in the fact that the vectors

Wi, i = 1, 2,3,4, are eigenvectors of the operators

S3 = S31) +

53(2)

S2=S +52+53.

Here the operator S3 is the third projection of the total spin of the

two electrons, 52 and S1 and S2 have the analogous meanings. The

operator S2 is the square of the total spin.

To verify our assertion about the vectors Wi, i = 1,2,3,4, we

find the results when the operators U1, o2, 03 are applied to the basis

vectors U+ and U-:

Ulu+ =

0 1) (1

) -

U-I

0) O

(1)

cr1 U_ = U+,

a2 U+ = W-,

U2 U_ _ -- i U4-,

U3U+ = U,,

U3U_ _ -U--

52 More precisely, S3

= z Q3 0 I + 119 03 The operators a j 9 I and 19 oj will

be denoted below by Q 1 and Q 2 ) , respectively

210

L. D. Faddeev and 0. A. Yakubovskii

Using these formulas, we obtain

S3W1 =

1 (T(1)

+ Q(2))U(')U(2) = U(1)U(2} = 1W

1 3 + + +

(2)

S3W2 = -1W2.

S3W3 = OW3,

S3W4

= OW4.

Along with (2) we have the formulas

S2 W3 = 2W31

= 1,2;3,

S2W4 = OW4

We verify (3) for the vector W;:

S2W3=(52+

-1 "32 + -

S3 W3

4Rai1)

(0,0)

+ 022)2 + (o1) +v32))2]Ws

+ 1

(011

al + U2

172

.(1)x(2)

U(1)U(2) + U(1)U(2)

2 2(1

a 2

3 3)

(+ +}

= 3W

+ 1W3 + 1W -1W3=

2W3.

2 3 2 2

Here we have used (1) and the equalities rrj2= 11 j = 1, 213. Thus, the

first three vectors W1, W2, W3 describe states in which the square of

the total spin of the system of two electrons is equal to 2. The number

2 can be written in the form 2 = S(S + 1), where S = 1, and therefore

the total spin in these states is equal to 1. In correspondence with

the general properties of the angular momentum, the projection of

the total spin takes the values

±1 and 0 in these states. The vector

W4 describes a state with total spin equal to zero. One sometimes

says that in the states W1, W2, W3 the spins of the electrons are

parallel, while in the state W4 they are antiparallel.

We discuss the result obtained from the point of view of group

theory. We know that the irreducible representation of the rota

tion group by the operators U(g) = e 2 (o,,a,+12a2+a3as) acts in the

space C2. It is clear that the mapping g -+ U(g) = U (g) 0 U (g)

e -

i (S 1 a 1 + S2 a.2 S3 a3) is

a representation of the rotation group on the

space C4. The representation U is the tensor product of the same

two representations U, and it is reducible. According to results in

§ 47. Spin of a system of two electrons 211

§ 29, the space C4 is representable as a direct sum of two subspaces

that are invariant under the operators U (g), and irreducible repre-

sentations act in these subspaces. The first subspace is spanned by

W1, W2, W3 and the second by W4. Using the notation in § 29, we

can write this result in the form

D2 , x D2 , = Doc3Di.

We remark that we have proved a particular case of the theorem on

decomposition of a tensor product of irreducible representations of

the rotation group. We state this theorem without proof:

If the irreducible representations Dj, and D j, of the rotation

group act in the spaces E1 and £2. then the tensor product of the

representations can be represented as the direct sum

Dj, ®Dj2 = Dlji-j21 0 DIj1

-j2I+1 e ... ®Dj1 +j2

of irreducible representations.

The last formula is called the Clebsch-Gordan decomposition.

This decomposition is obtained by passing from the basis consisting

of the vectors e j

1 nt, ej2m2, mk = -

jk, -- jk + I.... ti,,

k= 1, 2, where

the e jk m k are eigenvectors

of the operators (J(k))2 and J3k) , to the

basis of vectors ej

1 j2 /A17

J = h- j21,

hi - j21 + 1, ... J1 +j2, M

-J, -J + 1, .... J. The vectors f'j1J2JM are eigenvectors for the four

commuting operators (J(1)) 2, (J(2) )2,

J2 =

(j(l)+ 1(2))2+( J21)+ J22)}Z+( J31}+ J22)}Z,

= Al)

3+J32)

The vectors e j1 j 2 , f M are represent

able in the form

(4)

ejlj2JM = E Cjij2 1M,j1j2rra1rn2ejlrn, ej2rn27

m1,m2

where the summation indices Mk run through the values -3k, -3k +

1, ... , j,, k = 1, 2, and the coefficients C in the decomposition (4)

are called the Clebsch-Gordan coefficients. We remark that in passing

to the basis W1, W2, W3, W4 we found these coefficients for the case

j1=j2=1/2.

From the stated theorem it follows also that if in some state the

angular morrienturn J M 1) has the value j, and the angular momentum

j(2) has the value j2, then the total angular momentum J

can take

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

the values 131 - j21. hi -- j21 + 1, .... jl + j2. Both the orbital or

spin angular rrlorrlenta for different particles and the orbital and spin

angular momentum for a single particle can combine in this way.

In concluding this section we mention a property of the basis

elements I'V1. i = 1, 2, 3, 4, that, is important for what follows. Intro-

ducing the spin variables s

31)

and s3Z), each of which can take the two

values ±1/2, we write the W as

W'V1(.531), S(2))

U+(3

}3 )).

U'2(eS31},.532})

U_(.5(31)) 1_(8(32)).

(5)

s(2)

U+ U- 91U .S(2(S(1) )

(41))

(.3(2))

)

+( 3

W, s(1)

s(2)

(2)

(S3

3 3

U+ (.s3

where U+.(2) = 1, U+(-2) = O. U ()=O,U_(-)=1.

It follows from (5) that the functions W1, ti'V2, W3 are symmetric

functions of the spin variables; that is,

Wi(83l),.531)}

tiili(s31), s3Z)},

i = 1, 2, 3.

while W4 is an antisymmetric function

4(eS3 . 831})

-W4(s31} 532}

§ 48. Systems of many particles. The identity

principle

So far we have studied mainly the behavior of a single quantum par-

ticle. In the coordinate representation, the state space of a spinless

particle is the space L2 (R3), and for a particle with spin 1/2 it is

L2 (R3) ® C2

A natural generalization of such spaces to the case of

a system of n particles is the space

L2 (R3n) = L2 (R3) ,

... ® L2

(R3

)

for spinless particles, and the space

L2(R3n) g C2n

L2(R3) o C2 ...

L2(R3) g C2

for particles with spin 1/2.

§ 48. Systems of many particles. The identity principle 213

A comparison of theory with experiment shows, however, that

this assumption about the stat e spaces of systems of rt particles turns

out to be valid only in the case when no two particles in the system

are alike. In the presence of identical particles there are peculiarities

in the behavior of quantum systems that can be explained on the

basis of the so-called identity principle. We note at once that the

identity principle is a new principle of quantum mechanics

It cannot

be derived frorri the other fundamental tenets of quantum mechanics

formulated earlier and must be postulated.

We use the single symbol f to denote the spaces L' (R3,) and

L2 (RU) 0 C2*,, , and we write the elements of these spaces in the form

( , .. , r

}, where z = x(i) for a spinless particle and j = ((), 8(`) )

for a particle with spin, s being the spin variable of the ith particle.

For a particle with spin 1/2, the variables s3(i) take the two values

±1/2.

The scalar product in the spaces R can also be written in the

uniform way

(TI D) = f

T( I..

. I jj 0 * .. -rj) 41, * a

where it is assumed that, for a particle with spin the integration with

respect to the variable & is integration with respect to the space

variable x(a) and summation with respect to the spin variable

Before passing to the statement of the identity principle, we con-

sider the permutation group S, also called the symmetric group. The

elements of this group are the permutations

1, 2,

...

il,

i2.

...

int

and the identity element is the identity permutation

I-

2% ..., n

2, ..., n

The product it = 7r2 7r1 of two permutations is the permutation

obtained as a result of the successive application of the permutations

7r1 and 7r2.

214

L. D. Faddeev and 0. A. Yakubovskii

It is easy to construct a representation of the group S" on the

space W. We introduce the operators P, by

P,r'P(fit, .....n) =

Obviously, 7 is a unitary operator, and the mapping 7r --+ P,

is a

representation of the symmetric group on the space W.

In N we can immediately distinguish two invariant subspaces with

respect to the operators P,r, namely, the subspace 7-(s of symmetric

finict ions, for which

Src) ='P(f 1, i

n)i

and the subspace NA of antisynmmet ric functions, for which

P,r"P

i - - - 9 W = (-1)[7r] qP

(f1,...7Wi

where [7r] denotes the parity of the permutation 7r.

is clear that

R=NA _-?is.

In the case of two particles. H = 7-HA 0 N. Indeed, any function

1, 2) can be written in the form

2) +T( 2, 1) l

1 2} -

`1- 2}

+2 2

where %PS E Ns and

%PA E NA.

in the case of a large number of particles there are also invariant

subspaces more complicated than NS and HA, but these subspaces

are not of interest. The identity principle asserts that the state space

of a system of n identical particles is either the space WA or the space

7-Is. The choice of one of these spaces as the state space depends only

on the type of particles.

one says that particles whose states are described by symmet-

ric functions are subject to Bose- Einstein statistics, while particles

described by antisymmetric functions are subject to Fermi-Dirac sta-

tistics. The first particles are called bosons, and the second particles

are called fermions

It turns out that the spin of a particle determines the statistics to

which it is subject. Particles with integer spin (including those with-

out spin) are bosons, and those with half-integer spin are fermions.

in nonrelativistic quantum mechanics there is no explanation for the

§ 49. Symmetry of the coordinate wave functions

215

connection between spin and statistics. This connection is explained

in part in relativistic quantum mechanics. Electrons, protons, and

neutrons are fermions with spin 1/2, photons are bosons with spin

1, and mesons are bosons with spin zero. The statistics of identical

compound particles (for example, atomic nuclei) is determined by the

parity of the fermions in their structure, since a permutation of the

same complex particles is equivalent to a permutation of several pairs

of elementary particles. For example, deuterons, which are composed

of a neutron and a proton, are bosons. We note that a deuteron has

integer spin,53 since the spins of a proton and a neutron are equal to

1 2.

We write the Schrodinger operator for a system of pairwise inter-

acting particles in an external field:

1I=

Ai+E V(x{i} ) + 1: U(x(i) - x{k)).

2rn

i^x

i=1

i<k

The first term is the kinetic energy operator of the system of particles,

the second term describes the interaction of the particles with the

external field, and the third term is the interaction of the particles

with each other. We point out that the masses of all the particles are

the same, and the interaction potentials V (x) and U(x) do not depend

on the numbers of the particles. As a consequence, the Schrodinger

operator 11 commutes with all the operators P,.- [H, P,.] = 0.

49. Symmetry of the coordinate wave

functions of a system of two electrons.

The helium atom

Electrons are fermions, and therefore the wave function for a system

of two electrons must be antisymmetric:

T (6 1 W = -'P (6 1 W -

531t is known from experiments that the spin of a deuteron is I

216

L. D. Faddeev and 0. A. Yakubovskii

We decompose the function

with respect to the basis

functions W1, W2, W,-3, 1V4 introduced in §47:

3 3

), x(2)S(2))

qji (X(l). X(2)) IV,. (.531). 832}

(1)

T(X(1 ).s(1

i=1

The first three terms in this sump correspond to the states with total

spin 1. and the fourth tei in describes a state with total spin zero. The

functions T i (x(1) , x(2) ), i = 1,2,3,4, introduced by the relation (1)

are called the coordinate wave functions in coat rast to qP (fl , 2), which

is called the total wave function. In § 47 we saw that the W; (s31),

42))

with i = 1.2.3 are symmetric with respect to a permutation of the

spin variables, and W4 (s31), s32)) is an ant isymurietric function. Then

it follows from the antisymmetry of the total wave function that

i(X21X(l )=-Tj(xl ,X2 ), i=1,2,3,

4(x(2)

, x(1))

= W4 (X(1),

X(2) )j

t hat is, the coordinate wave functions for the states with spin I are

ant isyrmrietric. and those for states with spin zero are symmetric.

Let us apply this result to the helium atom. The Schr{ dinger

operator for a helium atom without considering spin interactions has

the form 54

H _

-1Q - 1Q _

2 1

r2 r12

If is a solution of the equation

then the coordinate functions 'i (x(1) , x(2)) also satisfy the Schrodinger

equation with the same eigenvalue E. Therefore, the problem reduces

to finding solutions of the equation

(2)

Hip (X(1), X(2)) =

Eq'(x(1).

X(2))

in subspaces of symmetric or antisymmetric functions. It is clear that

there are fewer solutions of (2) in each such subspace than in the space

L2 (R6).

The values of E for which (2) has a solution W (x(') , X(2))

54 In a precise formulation of the problem, the Schrodinger operator contains terms

depending on the spin, but it is possible to derive an expression for the spin interaction

only in relativistic quantum mechanics Moreover, for a helium atom these terms play

the role of small corrections and are always taken into account according to perturba-

tion theory

§ 50. Multi-electron atoms. One-electron approximation 217

in the subspace of antisymrnetric functions correspond to states with

spin 1, and the values of E for which there are symmetric solutions

of (2) correspond to states with spin zero.

We see that the energy levels of a helium atom depend on the

total spin even when we ignore the spin interactions in the Schrodinger

operator. This dependence is a consequence of the identity principle

and arises through the symmetry of the coordinate wave functions.

It can be shown that a symmetric coordinate wave function cor-

responds to the ground state of a helium atom: that is, the spin of

the helium atom in the ground state is equal to zero.

It is interesting to note that transitions with emission or absorp-

tion of quanta between states with S = 0 and S = I turn out to have

small probability. Therefore, the optical spectrum of helium is as if

there are two sorts of helium, with S = 0 and S = 1. The first sort

of helium is called parahelium, and the second sort is called orthohe-

lium. Corresponding to each energy level of parahelium is the single

spin state W4, and corresponding to an energy level of orthohelium

are the three spin states W1, W2, W3. Therefore, the states of para-

helium are called ringlet states, and those of orthohelium are called

triplet states. Taking into account the spin interactions leads to a

splitting of the triplet energy levels into three nearby levels. 5'5

§ 50. Multi-electron atoms.

One-electron approximation

The Schrodinger operator of a multi-electron atom or ion with charge

Z of the nucleus has the form

1 rt n Z

H = -

+ > - .

2 r

i-1

i=1

i<,7

L.-

This operator is written neglecting the motion of the nucleus and the

spin interactions. Neglecting the motion of the nucleus of a complex

atom is completely valid, since the corrections arising when it is taken

into account are less by several orders of magnitude than the errors

55Tt can be shown that such a splitting takes lace if the total orbital angular

rrioinentiirn L of the two clue troiis is tionzero, and therefore it is not for all triplet

energy levels that splitting is observed