timetthe integral

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The number of terms in the :um

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is given by

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r W(I, ... , N)

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[(I!)m 1 (2!)m 1

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](m 1 !m 2 ! ... )

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(10.50) Therefore the partition function is given by

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QN(V, T)

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{mtl'~l

fI ~(~b,)m,

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(10.51)

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This is of precisely the same form as (10.25) for the classical partition function. The discussion following (10.25) therefore applies equally well to the present case and will not be repeated. We point out only the main differences between the quantum cluster integrals and the classical ones. F or an ideal gas we have seen in earlier chapters that

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0(0) -

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r5/2 (

(ideal Bose gas) (ideal Fermi gas)

(_1)1+1 /- 5/ 2

(10.52)

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Thus for a Bose and a Fermi gas 0, does not vanish for I > 1, even in the absence of interparticle interactions, in contradistinction to the classical ideal gas. The calculation of 0, in the classical case only involves the calculation of a number of integrals-a finite task. In the quantum case, however, the calculation of 0, necessitates a knowledge of lft, which in tum necessitates a knowledge of WN , for N' :::;; I. Thus to find b, for I > 1 we would have to solve an I-body problem. There is no finite prescription for doing this except for the case I = 2, which is the subject of the next section.

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STATISTICAL MECHANICS

10.3 THE SECOND VIRIAL COEFFICIENT

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To calculate the second virial coefficient a 2 for any system it is sufficient to calculate 02' since a 2 = -02' A general formula for 02 (in fact, for all bl) has already been given for the classical case. Only the quantum case is considered here.* To find 02 we need to know W2(l, 2), which is a property of the two-body system. Let the Hamiltonian for the two-body system in question be

h VI = - 2m (2 + V 22) + v(lr 1

r 2 1)

(10.53)

and let its normalized eigenfunctions be 'Ya(l, 2), with eigenvalues E a :

'Ya (l,

2) = Ea'Ya (l, 2)

(10.54)

Hr1 + r2 )

Then

'Ya (1,

(10.55)

IV eiP R l/Jn(r)

(10.56)

p2 E a = 4m

where the quantum number a refers to the set of quantum numbers (P, n). The relative wave function l/Jn(r) satisfies the eigenvalue equation (10.57) with the normalization condition (10.58) Using (10.56) to be the wave functions for the calculation of W2 (1,2), we find from (10.35) that

W2 (1,2) = 2A6 I: l'Ya (1,2) 1 e- pEa =

2 2 V I: I: Il/Jn(r) 1 e-pp2/4m e-P<n

2"A6

(10.59) In the limit as V

~ 00

the sum over P can be effected immediately:

- I: e-pp2/4m =

4'17

dP p 2 e- pp2 /4m = - -

2 3/ 2

(10.60)

'The following development is due to E. Beth and G. E. Uhlenbeck, Physica 4, 915 (1937).

APPROXIMATE METHODS

where A = /2w1z 2/mkT, the thermal wavelength. Therefore

W 2 (1,2)

= 25/ 2A3 L Il/;n(r) 12 e-fJ<n

(10.61)

If we repeat all the calculations so far for a two-body system of noninteracting particles, we obtain

W}0)(1,2)

2 3 = 25/ 2A L Il/;~O)(r) 1 e-fJ<~O)

(10.62)

where the superscript (0) refers to quantities of the noninteracting system. From (10.49) and (10.47) we have

O2 = -3-jd3rld3r2U2(1,2) = -3-jd 3Rd 3r [W2(1,2) -1] 2A V 2A V

Hence

= 2fi d 3r

L [1l/;n(r)1 2 e-fJ<n n

1l/;~0)(r)12 e-fJ<~O)]

(10.63)

where

0(0)

2- 5/ 2 (

2 -5/2

(ideal Bose gas) (ideal Fermi gas)

(10.64)

To analyze (10.63) further we must study the energy spectra (~O) and the noninteracting system, (~O) forms a continuum. We write

= __

;,2k 2

(10.65)

which defines the relative wave number k. For the interacting system the spectrum of (n in general contains a discrete set of values (B, corresponding to two-body bound states, and a continuum. In the continuum, we define the wave number k for the interacting system by putting

;,2k 2

(10.66)

Let g( k) dk be the number of states with wave number lying between k and + dk, and let g(O)(k) dk denote the corresponding quantity for the noninteracting system. Then (10.63) can be written in the fo~m

O2 -

b~O) = 23/ 2 { ~e-fJ<B + {Xl dk

[g(k) - g(O)(k)] e-fJIi2k2/m} (10.67)

where

denotes the energy of a bound state of the interacting two-body system.

STATISTICAL MECHANICS

We remark in passing that the factor 2 3/ 2 in front of (10.67) is the ratio (A/A cm )3/2, where A is the thermal wavelength, and Acm is the thermal wavelength of the center-of-mass motion of the two-body system. Let T//(k) be the scattering phase shift of the potential u(r) for the lth partial wave of wave number k. It will be shown that

g(k) - g(O)(k)