By a straightforward calculation we obtain in Java

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By a straightforward calculation we obtain
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(13.117)
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Thus the one-phonon state is a superposition of unperturbed states in which there are any number of particles p, - p for all p, plus an additional particle of momentum k. The average number of particles that have momentum k in the perturbed ground state is
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--2 1 - ak
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(13.118)
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Therefore the total number of excited particles in the perturbed ground state is
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(13.119)
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The number of particles of zero momentum in the perturbed ground state is
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(13.120)
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which shows that the approximation (13.92) is justified. It is instructive to calculate the wave functions in configuration space. In each term of the sum in (13.114), the number n == L Ik is half the total number k>O of particles with nonzero momentum. Thus we must have N ~ 2n. We rewrite (13.114) as follows:
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L 'IE'k=n . '2""
[(-a 1 )"(-a 2 )'2 ]1/1 , /2 , )
(13.121)
We imagine N
~ 00
at the end of the calculation. The normalized configuration
SPECIAL TOPICS IN STATISTICAL MECHANICS
space wave function for an unperturbed state specified by the occupation numbers {no, n 1 , } is 1 1 "Pei(PI"rl+ ... +PN'rN) (13.12_~ ( r l' ... , r NIn) no, l' . . . V N/2 N! nk!) ';:
where among the N momenta PI' ... ,PN, no are 0, n i are k I , etc. The symbol P denotes a permutation of rl> ... , rN' Hence
(r 1 , , r N l/ 1 , 12 " , , ) 1 1
= V N/ 2 VN!(N - 2n)!
where among the n vectors PI' ... , Pn' II are kl> 12 are k 2 , etc. Now consider the sum appearing in (13.121):
IXn) =
This may be rewritten as
'I' '2" ..
[(-al)"(-a 2)" '"
]1 / 1,/ 2",,)
E'k =
where, in the first line, k 1 ~ k 2 ~ . . . denotes any ordering of the momenta. In the second line each momentum independently ranges through half of momentum space excluding k = 0 (as denoted by k > 0). The configuration space representation of (13.124) is obtained by substituting (13.123) for 1/1,/2",,): 1 1 Xn(r 1 , .. ,r N ) = V N / 2 n!VN!(N - 2n)! ~P
kl>O
(-al)
e ik "(r,-r2)
k,,>O
(-aJ
e ik ".(r 3 -r4 )]
Let us define
f(ij) = f(r i
f(r) = - -2
ak e
ik r '
fd k
ak e
ik r '
(13.125)
Since ak depends only on Ikl, it follows that f(r) depends only on Irl. We can then write
Xn(r I , .. , rN ) =
V~/2 n!VN!( ~ _ 2n )! (~ ~P [f(12) ... f(34)]
in which there are n factors f(ij).
SUPERFLUIDS
The number of distinct ways of choosing the arguments of j(ij) from the N coordinates f1> . , r N can be found by filling the n boxes in the following with N balls. The boxes are identical, each holding two balls, and the order of the two balls is irrelevant.
100100C8 00 1 00
1 2 n-l
000 ... 000
N - 2n balls left over
It is evident that there are
(N - 2n )!n!2 n
distinct ways to fill the boxes, whereas there are N! permutations of r 1,. . ,rN' Hence
Xn(r1, ,r N ) =
/(N-2n)! , NnI:[j(12) j(34)] (13.126) N.
where the sum is extended over all distinct ways of filling the "boxes." Therefore
'l'o(r1, ,r N) = Z
Xn(r1, ,r N )
= V NI2 {I + [/(12) + j(34) + ... ]
+ [j(12)j(34) + j(12)j(56) + .. , ]
+ [j(12)j(34)j(56) + ... J + ... }
It differs from the function
(13.127)
(13.128) only in that (13.128) contains extra terms of the type
j(12)j(13), j(12)j(13)j(34), ...
in which the same particle appears in more than one "box." These terms may be shown to belong to a higher order in the calculation than we have considered. Therefore we may take (13.128) to be the wave function to the order /a 3 ju. The function j(r) defined in (13.125) has the following asymptotic behavior:
j(r) "" - -
(r ro)
j(r) "" - 32 V
where
~ (~
(13.129)
(r ro)
(13.130)
SPECIAL TOPICS IN STATISTICAL MECHANICS
Thus although (13.128) vanishes at rij = a, (13.1;7) does so only approximatel~. but its value at rij = a is of a higher order than a 3/V . The one-phonon wave function (13.116) is easily shown to have the following form in configuration space:
(13.1311
which verifies the result of the last section. The extension of the present results to a dilute Bose gas with attractive interaction, which has a more realistic phase diagram, has also been worked out.'