Fitts D.D. - Principles of Quantum Mechanics[c] As Applied to Chemistry and Chemical Physics (1999)(en)

212 Systems of identical particles

has coordinates q1 and which q2. Thus, only the linear combinations ØS and ØA are suitable wave functions for the two-identical-particle system. We note in passing that the two probability densities are not equal, even though ØS and ØA correspond to the same energy value E. We conclude that in order to incorporate into quantum theory the indistinguishability of the two identical particles, we must restrict the allowable wave functions to those that are symmetric and antisymmetric, i.e., to those that are simultaneous eigenfunc-

tions of ^ (1, 2) and ^.

H P

Three-particle systems

The treatment of a three-particle system introduces a new feature not present in a two-particle system. Whereas there are only two possible permutations and therefore only one exchange or permutation operator for two particles, the three-particle system requires several permutation operators.

We ®rst label the particle with coordinates q1 as particle 1, the one with coordinates q2 as particle 2, and the one with coordinates q3 as particle 3. The

Hamiltonian operator	^
	H(1, 2, 3) is dependent on the positions, momentum

operators, and perhaps spin coordinates of each of the three particles. For identical particles, this operator must be symmetric with respect to particle interchange

^	^	^	^	^	^
H(1, 2, 3)	H(1, 3, 2)	H(2, 3, 1)	H(2, 1, 3)	H(3, 1, 2)	H(3, 2, 1)
If Ø(1, 2, 3) is a solution of the time-independent SchroÈdinger equation
	^				(8:16)
	H(1, 2, 3)Ø(1, 2, 3) EØ(1, 2, 3)

then Ø(1, 3, 2), Ø(2, 3, 1), etc., and any linear combinations of these wave functions are also solutions with the same eigenvalue E. The notation Ø(i, j, k) indicates that particle i has coordinates q1, particle j has coordinates q2, and particle k has coordinates q3. As in the two-particle case, we seek

eigenfunctions of

^

H(1, 2, 3) that do not specify which particle has coordinates

qi, i 1, 2, 3.

^

We de®ne the six permutation operators Páâã for á 6â 6ã 1, 2, 3 by the

relations

^

Ø(i, j,

k)

Ø(i, j, k)

P123

>

P^132Ø(i,

j,

k)

Ø(i, k, j)

9

^

>

=

P^231

Ø(i,

j,

k)

Ø(j, k, i)

>

6 6

213

>

1, 2, 3

(8:17)

P^ Ø(i, j, k)

Ø(j, i, k)

>

i j k

>

>

>

P312Ø(i,

j,

k)

Ø(k, i, j)

>

P^321Ø(i, j, k)

;

Ø(k, j, i) >

^

with>

coordinates

q1

(the ®rst position)

The operator Páâã replaces the particle

>

8.1 Permutations of identical particles

213

by the particle with coordinates qá, the particle with coordinates q2 (the second position) by that with qâ, and the particle with coordinates q3 (the third

position) by that with qã. For example, we have
^	Ø(1, 2, 3)	Ø(2, 1, 3)	(8:18a)
P213
^	Ø(2, 1, 3)	Ø(1, 2, 3)	(8:18b)
P213
^	Ø(3, 2, 1)	Ø(2, 3, 1)	(8:18c)
P213
^	Ø(1, 2, 3)	Ø(2, 3, 1)	(8:18d)
P231
^	Ø(2, 3, 1)	Ø(3, 1, 2)	(8:18e)
P231

^	because it leaves the
The permutation operator P123 is an identity operator
function Ø(i, j, k) unchanged. From (8.18a) and (8.18b), we obtain
^2
P213Ø(1, 2, 3) Ø(1, 2, 3)

so that	^2	equals unity. The same relationship can be demonstrated to apply
	P213
		^	^				^
to the operators P132 and P321, as well as to the identity operator P123, giving
		^2	^2	^2	^2	^	(8:19)
		P213	P132	P321	P123	P123 1
						^	^

Any permutation corresponding to one of the operators Páâã other than P123

is equivalent to one or two pairwise exchanges. Accordingly, we introduce the

linear hermitian exchange operators

^

^

^

P12,

P23, and P31 with the properties

^

Ø(i, j, k)

Ø( j, i, k)

=

P12

6 6

P^31

Ø(i, j, k)

Ø(k, j,

i)

P^23

Ø(i,

j, k)

Ø(i, k,

j)

9

i

j

k

1, 2, 3

(8:20)

^

the particles with coordinates

qá

and

The exchange operator Páâ interchanges;

^

qâ. It is obvious that the order of the subscripts in Páâ

is immaterial, so that

^

^

^

^

^

are the same as those

Páâ Pâá. The permutations from P213, P132, and

P321

^

^

^

from P12

, P23, and P31, respectively, giving

^

^

^

^

^

^

P213 P12,

P132

P23,

P321 P31

The permutation from

^

may also be obtained by ®rst applying the exchange

P231

operator

^

^

P12 and then the operator P23. Alternatively, the same result may be

^

^

^

obtained by ®rst applying P23

followed by P31

or by ®rst applying P31 followed

^

by P12. This observation leads to the identities

^

^

^

^

^

^

^

(8:21)

P231

P23 P12

P31 P23

P12 P31

A similar argument yields

^

^

^

^

^

^

^

(8:22)

P312

P31 P12

P23 P31

P12 P23

These

permutations

of

the

three

particles

are

expressed in terms

of the

minimum number of pairwise exchange operators. Less ef®cient routes can

also be visualized. For example, the permutation operators	^	and	^	may
	P132		P231
also be expressed as

214 Systems of identical particles

^	^	^	^		^ ^	^
P132	P31 P23 P12			P12 P23 P31
^	^	^	^	^	^	^ ^ ^
P231	P12 P23 P31 P12				P31 P12 P31 P12

However, the number of pairwise exchanges for a given permutation is always

	^		^	,	^	are even permutations	and	^
either odd or even, so that P123,			P231		P312			P132,
^	^
P213	, P321 are odd permutations.						^
Applying the same arguments regarding the exchange operator
							P for the
two-particle system, we ®nd that
	^2		^2		^2	1
	P12	P23			P31

giving real eigenvalues 1 for each operator. We also ®nd that each exchange

								^
operator commutes with the Hamiltonian operator H
		^		^	^	^	^	^			(8:23)
		[P12,		H]	[P23,	H]	[P31,	H] 0
^	and	^						^		^	possess
so that P12		H possess simultaneous eigenfunctions, P23							and H
					^	^
simultaneous eigenfunctions, and P31						and H possess simultaneous eigenfunc-
				^	^	^	do not commute with each other.
tions. However, the operators P12, P23, P31
For example, if we operate on					the wave		function Ø(1, 2, 3)			®rst	with the
^	^					^	^
product P31 P12 and then with the product P12 P31, we obtain
	^	^	Ø(1, 2, 3)		^
	P31 P12				P31Ø(2, 1, 3) Ø(3, 1, 2)
	^	^	Ø(1, 2, 3)		^
	P12 P31				P12Ø(3, 2, 1) Ø(2, 3, 1)

The wave function Ø(3, 1, 2) is not the same as Ø(2,3,1), leading to the conclusion that

	^	^	^	^
	P31 P12		6P12 P31
				^			^
Thus, a set of simultaneous eigenfunctions of H(1, 2, 3) and							P12 and a set of
	^			^
simultaneous eigenfunctions of H		(1, 2, 3) and P31 are not, in general, the same
set. Likewise, neither set are simultaneous eigenfunctions of							^
							H(1, 2, 3) and
^
P23.				^
There are, however, two eigenfunctions of H(1, 2, 3) which are also simul-
taneous eigenfunctions of all three pair exchange operators						^		^	^
						P12,		P23	, and P31.
These eigenfunctions are ØS and ØA, which have the property
^			á 6â 1, 2						(8:24a)
PáâØS ØS ,
^				á 6â 1, 2					(8:24b)
PáâØA ÿØA,
To demonstrate this feature, we assume				that Ø(1, 2, 3)		is	a	simultaneous
^				^	^		^
eigenfunction not only of H(1, 2, 3), but also of P12,					P23, and P31. Therefore,
we have
^	Ø(1, 2, 3)		ë1Ø(1, 2, 3)
P12
^	Ø(1, 2, 3)		ë2Ø(1, 2, 3)						(8:25)
P23

8.1 Permutations of identical particles			215
^	Ø(1, 2, 3)	ë3Ø(1, 2, 3)
P31

where ë1 1, ë2 1, ë3 1 are the respective eigenvalues. From equations (8.21) and (8.25), we obtain

^	^ ^	^ ^		^ ^	Ø(1, 2, 3)
P231	Ø(1, 2, 3) P23 P12	Ø(1, 2, 3) P31 P23	Ø(1, 2, 3) P12 P31
or	Ø(2, 3, 1)
	ë2ë1Ø(1, 2, 3) ë3ë2Ø(1, 2, 3) ë1ë3Ø(1, 2, 3)
from which it follows that
		ë1 ë2 ë3
Thus,	the simultaneous	eigenfunctions Ø(1, 2, 3)		are either	symmetric

(ë1 ë2 ë3 1) or antisymmetric (ë1 ë2 ë3 ÿ1).

The symmetric ØS or antisymmetric ØA eigenfunctions may be constructed from Ø(1, 2, 3) by the relations

ØS 6ÿ1=2[Ø(1, 2, 3) Ø(1, 3, 2) Ø(2, 3, 1) Ø(2, 1, 3) Ø(3, 1, 2)

Ø(3, 2,	1)]	(8:26a)
ØA 6ÿ1=2[Ø(1, 2, 3)	ÿ Ø(1, 3, 2) Ø(2, 3, 1) ÿ Ø(2, 1, 3) Ø(3, 1, 2)
ÿ Ø(3, 2, 1)]		(8:26b)

where the factor 6ÿ1=2 normalizes ØS and ØA if Ø(1, 2, 3) is normalized. As in the two-particle case, the functions ØS and ØA are orthogonal. Moreover, a wave function which is initially symmetric (antisymmetric) remains symmetric (antisymmetric) over time. The probability densities ØS ØS and ØAØA are independent of how the three particles are labeled. The two functions ØS and

		^
	ØA are, therefore, the eigenfunctions of H(1, 2, 3) that we are seeking.			^
	Equations (8.26) may be expressed in another, equivalent way. If we let
				P be
	any one of the permutation operators	^
		Páâã in equation (8.17), then we may
	write
	X		^
	ØS,A 6ÿ1=2	äP P^Ø(1, 2, 3)	(8:27)

P

where the summation is taken over the six different operators Páâã, and äP is either 1 or ÿ1. For the symmetric wave function ØS , äP is always 1, but for the antisymmetric wave function ØA, äP is 1 (ÿ1) if the permutation

^
operator P involves the exchange of an even (odd) number of pairs of particles.
^	^	^
Thus, äP is ÿ1 for P132, P213		and P321.

N-particle systems

The treatment of a three-particle system may be generalized to an N-particle

216 Systems of identical particles

system. We begin by labeling the N particles, with each particle i having coordinates qi. For identical particles, the Hamiltonian operator must be symmetric with respect to particle permutations

^	^	^	. . . , 1)
H(1, 2,	. . . , N) H(2, 1,	. . . , N) H(N, 2,

There are N! possible permutations of the N particles. If Ø(1, 2, . . . , N) is a solution of the time-independent SchroÈdinger equation

^	. . . , N)Ø(1, 2,	. . . , N) EØ(1, 2,	. . . , N)	(8:28)
H(1, 2,

then Ø(2, 1, . . . , N), Ø(N, 2, . . . , 1), etc., and any linear combination of

these wave functions are also solutions with eigenvalue E.

We

next introduce the

set

of linear

hermitian

exchange

^

operators Páâ

(á 6â 1, 2, . . . , N). The exchange operator

^

interchanges the pair of

Páâ

particles in positions á (with coordinates qá) and â (with coordinates qâ)

^

â , . . . ,

) Ø( , . . . ,

á, . . . ,

, . . . ,

)

:

PáâØ(i, . . . , j, . . . ,

l

j

l

k

i

k

(8 29)

á

â

^

is immaterial.

As in the three-particle case, the order of the subscripts on Páâ

Since there are

N choices for the ®rst particle and (N ÿ 1) choices for the

second

particle

(á 6â) and

since each

pair

is to be counted

only once

^

^

^

^

(Páâ Pâá), there are N(N ÿ 1)=2 members of the set Páâ.

Applying the same arguments regarding the exchange operator

P for the

two-particle system, we ®nd that Páâ2 1, giving real eigenvalues 1. We also
^	^
®nd that Páâ and H commute
	^	^	á 6â 1, 2, . . . , N	(8:30)
	[Páâ, H] 0,

so that they possess simultaneous eigenfunctions. However, the members of the

^
set Páâ do not commute with each other. There are only two functions, ØS and
	^	pairwise
ØA, which are simultaneous eigenfunctions of H and all of the
^
exchange operators Páâ. These two functions have the property
^	á 6â 1, 2, . . . , N	(8:31a)
PáâØS ØS ,
^	á 6â 1, 2, . . . , N	(8:31b)
PáâØA ÿØA,
and may be constructed from Ø(1, 2, . . . , N) by the relation
^	X
ØS,A (N!)ÿ1=2	äP P^Ø(1, 2, . . . , N)	(8:32)

P

In equation (8.32) the operator P is any one of the N! operators, including the identity operator, that permute a given order of particles to another order. The summation is taken over all N! permutation operators. The quantity äP is always 1 for the symmetric wave function ØS , but for the antisymmetric

wave function Ø , ä is 1 (ÿ1) if the permutation operator ^ involves the

A P P

8.2 Bosons and fermions

217

exchange of an even (odd) number of particle pairs. The factor (N!)ÿ1=2 normalizes ØS and ØA if Ø(1, 2, . . . , N) is normalized.

Using the same arguments as before, we can show that ØS and ØA in equation (8.32) are orthogonal and that, over time, ØS remains symmetric and ØA remains antisymmetric. Since the probability densities ØS ØS and ØAØA are independent of how the N particles are labeled, the two functions ØS and

ØA are the only suitable eigenfunctions of	^	. . . , N) to represent a
	H(1, 2,
system of N indistinguishable particles.

8.2 Bosons and fermions

In quantum theory, identical particles must be indistinguishable in order for the theory to predict results that agree with experimental observations. Consequently, as shown in Section 8.1, the wave functions for a multi-particle system must be symmetric or antisymmetric with respect to the interchange of any pair of particles. If the wave functions are not either symmetric or antisymmetric, then the probability densities for the distribution of the particles over space are dependent on how the particles are labeled, a property that is inconsistent with indistinguishability. It turns out that these wave functions must be further restricted to be either symmetric or antisymmetric, but not both, depending on the identity of the particles.

In order to accommodate this feature into quantum mechanics, we must add a seventh postulate to the six postulates stated in Sections 3.7 and 7.2.

7.The wave function for a system of N identical particles is either symmetric or antisymmetric with respect to the interchange of any pair of the N particles. Elementary or composite particles with integral spins (s 0, 1, 2, . . .) possess

symmetric wave functions, while those with half-integral spins (s 12, 32, . . .) possess antisymmetric wave functions.

The relationship between spin and the symmetry character of the wave function can be established in relativistic quantum theory. In non-relativistic quantum mechanics, however, this relationship must be regarded as a postulate.

As pointed out in Section 7.2, electrons, protons, and neutrons have spin 12. Therefore, a system of N electrons, or N protons, or N neutrons possesses an antisymmetric wave function. A symmetric wave function is not allowed. Nuclei of 4He and atoms of 4He have spin 0, while photons and 2H nuclei have spin 1. Accordingly, these particles possess symmetric wave functions, never antisymmetric wave functions. If a system is composed of several kinds of particles, then its wave function must be separately symmetric or antisymmetric with respect to each type of particle. For example, the wave function for

218	Systems of identical particles

the hydrogen molecule must be antisymmetric with respect to the interchange of the two nuclei (protons) and also antisymmetric with respect to the interchange of the two electrons. As another example, the wave function for the oxygen molecule with 16O nuclei (each with spin 0) must be symmetric with respect to the interchange of the two nuclei and antisymmetric with respect to the interchange of any pair of the eight electrons.

The behavior of a multi-particle system with a symmetric wave function differs markedly from the behavior of a system with an antisymmetric wave function. Particles with integral spin and therefore symmetric wave functions satisfy Bose±Einstein statistics and are called bosons, while particles with antisymmetric wave functions satisfy Fermi±Dirac statistics and are called fermions. Systems of 4He atoms (helium-4) and of 3He atoms (helium-3) provide an excellent illustration. The 4He atom is a boson with spin 0 because the spins of the two protons and the two neutrons in the nucleus and of the two electrons are paired. The 3He atom is a fermion with spin 12 because the single neutron in the nucleus is unpaired. Because these two atoms obey different statistics, the thermodynamic and other macroscopic properties of liquid helium-4 and liquid helium-3 are dramatically different.

8.3 Completeness relation

The completeness relation for a multi-dimensional wave function is given by equation (3.32). However, this expression does not apply to the wave functions ØíS,A for a system of identical particles because ØíS,A are either symmetric or antisymmetric, whereas the right-hand side of equation (3.32) is neither. Accordingly, we derive here1 the appropriate expression for the completeness relation or, as it is often called, the closure property for ØíS,A.

For compactness of notation, we introduce the 4N-dimensional vector Q

with components qi for i 1, 2, . . . ,	N. The permutation operators	^	are
		P

allowed to operate on Q directly rather than on the wave functions. Thus, the

^	. . . , N) is identical to	^
expression PØ(1, 2,		Ø(PQ). In this notation, equation
(8.32) takes the form	X
		äPØí(P^Q)
	ØíS,A (N!)ÿ1=2		(8:33)

P

We begin by considering an arbitrary function f (Q) of the 4N-dimensional vector Q. Following equation (8.33), we can construct from f (Q) a function F(Q) which is either symmetric or antisymmetric by the relation

1 We follow the derivation of D. D. Fitts (1968) Nuovo Cimento 55B, 557.

8.3 Completeness relation

219

X
F(Q) (N!)ÿ1=2 äP f (P^Q)	(8:34)

P

Since F(Q) is symmetric (antisymmetric), it may be expanded in terms of a complete set of symmetric (antisymmetric) wave functions Øí(Q) (we omit the subscript S, A)

	F(Q) Xí			cíØí(Q)	(8:35)
The coef®cients cí are given by		…	Øí (Q9)F(Q9) dQ9
cí					(8:36)

because the wave functions Øí(Q) are orthonormal. We use the integral notation to include summation over the spin coordinates as well as integration over the spatial coordinates. Substitution of equation (8.36) into (8.35) yields

F(Q)		F(Q9)	í	Øí (Q9)Øí(Q)	dQ9	(8:37)
		"			#
	…		X

where the order of summation and the integration over Q9 have been interchanged. We next substitute equation (8.34) for F(Q9) into (8.37) to obtain

F(Q)		(N!)ÿ1=2	äP	f (P^Q9)	"X	Øí (Q9)Øí(Q)	dQ9	(8:38)
			X	…			#

Pí

We now introduce the reciprocal or inverse operator ^ÿ1 to the permutation

P

operator ^ (see Section 3.1) such that

P

^ÿ1 ^ ^ ^ÿ1 1

P P PP

We observe that
Øí(P^ÿ1Q) äPÿ1 Øí(Q) äPØí(Q)	(8:39)

The quantity ä equals ä because both ^ÿ1 and ^ involve the interchange

Pÿ1 P P P

of the same number of particle pairs. We also note that
X
ä2P N!	(8:40)

P

because there are N! terms in the summation and each term equals unity.

We next operate on each term on the right-hand side of equation (8.38) by

^ÿ1. Since ^ in equation (8.38) operates only on the variable 9 and since the

P P Q

order of integration over Q9 is immaterial, we obtain						#
		(N!)ÿ1=2	X …	"X	(P^ÿ1Q9)Øí(Q)
F(Q)			äP f (Q9)	Øí		dQ9 (8:41)
			P	í

220

Systems of identical particles

Application of equations (8.39) and (8.40) to (8.41) gives

F(Q)

(N!)1=2

f (Q9)

Ø

í

(Q9)Øí(Q)

dQ9

(8:42)

…

" í

#

X

Since f (Q9) is a completely

arbitrary

function of Q9,

we may

compare

equations (8.34) and (8.42) and obtain

äPä(P^Q

Q9)

(8:43)

Øí

(Q9)Øí(Q)

(N!)ÿ1

ÿ

í

P

X

X

where ä(Q ÿ Q9) is the Dirac delta function

ä(Q ÿ Q9)

N

ä(ri ÿ r9i)äó iói9

(8:44)

i 1

Y

Equation (8.43) is the completeness relation for a complete set of symmetric (antisymmetric) multi-particle wave functions.

8.4 Non-interacting particles

In this section we consider a many-particle system in which the particles act independently of each other. For such a system of N identical particles, the

Hamiltonian operator	^	. . . , N) may be		written as the	sum of one-
	H(1, 2,
particle Hamiltonian operators		^		. . . , N
		H(i) for i 1, 2,
^		^	^	^	(8:45)
H(1, 2,	. . . , N) H(1)		H(2)	H(N)

In this case, the operator	^	. . . , N) is obviously symmetric with respect
	H(1, 2,
to particle interchanges. For the N particles to be identical, the operators			^
			H(i)

must all have the same form, the same set of orthonormal eigenfunctions øn(i),

and the same set of eigenvalues En, where
^	i 1, 2, . . . , N	(8:46)
H(i)øn(i) Enøn(i);

As a consequence of equation (8.45), the eigenfunctions Øí(1, 2, . . . , N) of

^ (1, 2, . . . , ) are products of the one-particle eigenfunctions

H N

Øí(1, 2, . . . , N) øa(1)øb(2) . . . ø p(N)		(8:47)
and the eigenvalues	^
	Eí of H(1, 2, . . . , N) are sums of one-particle energies
	Eí Ea Eb Ep	(8:48)
In equations (8.47)	and (8.48), the index í represents the set of one-particle
states a, b, . . ., p and indicates the state of the N-particle system.

The N-particle eigenfunctions Øí(1, 2, . . . , N) in equation (8.47) are not properly symmetrized. For bosons, the wave function Øí(1, 2, . . . , N) must be symmetric with respect to particle interchange and for fermions it must be antisymmetric. Properly symmetrized wave functions may be readily con-

8.4 Non-interacting particles

221

structed by applying equation (8.32). For example, for a system of two identical particles, one particle in state øa, the other in state øb, the symmetrized twoparticle wave functions are

Øab,S (1, 2) 2ÿ1=2[øa(1)øb(2) øa(2)øb(1)]	(8:49a)
Øab,A(1, 2) 2ÿ1=2[øa(1)øb(2) ÿ øa(2)øb(1)]	(8:49b)

The expression (8.49a) for two bosons is not quite right, however, if states øa and øb are the same state (a b), for then the normalization constant is 12 rather than 2ÿ1=2, so that

Øaa,S (1, 2) øa(1)øa(2)

From equation (8.49b), we see that the wavefunction vanishes for two identical fermions in the same single-particle state

Øaa,A(1, 2) 0

In other words, two identical fermions cannot simultaneously be in the same quantum state. This statement is known as the Pauli exclusion principle because it was ®rst postulated by W. Pauli (1925) in order to explain the periodic table of the elements.

For N identical non-interacting bosons, equation (8.32) needs to be modi®ed in order for ØS to be normalized when some particles are in identical singleparticle states. The modi®ed expression is

	Na!	N		1=2	X
ØS		b!				P^øa(1)øb(2) . . . ø p(N)	(8:50)
		N!			p

where Nn indicates the number of times the state n occurs in the product of the single-particle wave functions. Permutations which give the same product are included only once in the summation on the right-hand side of equation (8.50). For example, for three particles, with two in state a and one in state b, the products øa(1)øa(2)øb(3) and øa(2)øa(1)øb(3) are identical and only one is included in the summation.

For N identical non-interacting fermions, equation (8.32) may also be

expressed as a Slater determinant
					øa(1)	øa(2)		øa(N)
					ø p(1)	ø p(2)		ø p(N)
ØA		(N!)ÿ1=2								(8:51)
					øb(1)	øb(2)		øb(N)
The expansion of	this		determinant			is identical		to equation		(8.32) with

Ø(1, 2, . . . , N) given by (8.47). The properties of determinants are discussed in Appendix I. The wave function ØA in equation (8.51) is clearly antisymmetric because interchanging any pair of particles is equivalent to interchan-