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[u Pal (1) U Qal (1)] ... [u Pa) N) U Qa) N)]
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For fermions we must have P
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Q, because the a i are all distinct. Therefore
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(<I>a' <I>a)
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For bosons the a i are not necessarily all distinct. Suppose that among a 1, ., aN there are n a having the value a. Then (bosons)
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which is not necessarily unity.
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The integer n a is called the occupation number of the single-particle level a. Obviously we have the conditions
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n a = 0,1, ... , N n a = 0,1
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Instead of {a 1 , .. , an} we can equally well label the wave function by the occupation numbers {no, n 1 , ... }. Thus we also introduce the notation <I>n' defined by (A.20) <I>n = <I>a where n stands for the set {no, n b . .. }. If we wish, we may use in place of <I>a the wave function <I>'=
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which is normalized to unity for both bosons and fermions. It is, however, neither necessary nor convenient to do this. The reason is as follows. Suppose we are to calculate the trace of an operator. We may write Tr (2 =
(<I>~, (2<I>~)
where the sum extends over all distinct sets {a 1, ... , aN}' A convenient way to calculate this sum is to sum over each a i independently and to take into account the fact that a permutation of the a i must not be counted as a new term in the sum. That is, we write Tr(2 By (A.21) we have (A.23) Thus it is actually more convenient to use the wave functions <I>a as defined in (A.14). Free-Particle Wave Functions A useful choice of ua(r) is the single-particle wave function for a free particle of momentum p. The quantum number a is now explicitly p, and we have
ai' .. ,a N
TI (n))
(<I>~, (2<I>~)
The allowed values of P are determined by the periodic boundary conditions
up(r + nL)
where nand L are defined in (A.ll). This implies that the allowed values of pare
(A.26) L These values form a cubic lattice in momentum space with the lattice constant 27T iii L, which approaches 0 as V ~ 00. In this limit a volume element of size d 3p in momentum space contains (Vlh 3 ) d 3p lattice points. Thus as V ~ 00 a sum over P may be replaced by an integration over P in the following manner: V L ~ 3 d 3p (A.27)
= -p
The functions defined by (A.24) obviously form an orthogonal set. The completeness of this set follows from the fact that any function can be Fourier analyzed. The N-body wave functions built up from up(r) according to (A.14) are the N-body free-particle wave functions. They are denoted by <Pp(I, ... , N) and are eigenfunctions of the kinetic-energy operator K: 1 K<Pp(I, ... , N) = 2m (pi + ... +p~)<Pp(I, ... , N) (A.28) where PI" .. , PN are the momenta of the N-single-particle wave functions contained in <Pp.
Example of Calculation: System of Bosons
We calculate (<P a, Q<P a) for a system of bosons:
(<p a, Q<P a)
* r <Pa I>ij<P a = ~N(N - 1)
r <Pa 12<Pa *v
where the second equality is obtained by renaming the integration variables r I , , r N in an appropriate fashion in each term of the sum Lv ij . Using (A.14) we have
(<p a, Q<Pa)
N(N - 1) NI2
LLf d Q
r [uPaJI) ... uPa)N)]v12[uQa ,(l) uQa)N)]
N(N - 1) N I2 L L(Pal> Pa21v1QaI, Qa 2>( ~Pa3,Qa3 ... ~paN,QaJ
. P Q
where (A.30)
In (A.29) only two terms in the sum the conditions (a) or (b):
are nonzero, namely the terms satisfying
Qa 1 = Pal' Qa 1 = Pa2'
Qa 2 = Pa2' Qa 2 = Pal'
Qa= PaJ J Qa j
, N) , N) (A.31)
(j = 3,
(<I>a' [2<1>a) =
N(N - 1)
TI (n,.!) a
+ (Pal' Pa2lulPa2, Pal ) (A.32)
X L(Pa1' Pa 2lulPa 1, P(2)
As P ranges through all the N! permutations of the set {a 1, ... , aN}' the pair {Pal' Pa 2 } takes on all possible pairs of values {a, 13} chosen from the set { a 1, ... , aN}' Suppose the occupation numbers for the single-particle states a,13 are respectively n a' n p. Then the number of ways in which the pair { a, 13} can be chosen from the set {a1> ... , aN} is
(a*13) (a=13)
(1 - ~aP)nanp + t~apna(na - 1)
Furthermore, there are (N - 2)! permutations that affect only the quantum numbers { a3' ... , aN} and thus leave { a1, a2} unchanged. Changing the label of <I> a to occupation numbers we obtain
(<I>~,[2<1>~) = t N (N-1)
(N - 2)!