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The similarity between the master equation and the Boltzmann transport equation may be noted, although we should remember that the latter refers to }L space whereas the former refers to r space. The random-phase assumption here is similar to the assumption of molecular chaos in the Boltzmann transport equation. In both cases the solution for t ~ 00 is relatively easy to obtain, but the relaxation time is difficult to calculate. The approach involving the master equation seems to hold greater promise for a satisfactory derivation of statistical mechanics and the concomitant understanding of general nonequilibrium phenomena. Further discussion of the master equation, however, is beyond the scope of this book. *
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Find the density matrix for a partially polarized incident beam of electrons in a scattering experiment, in which a fraction f of the electrons are polarized along the direction of the beam and a fraction 1 - f is polarized opposite to the direction of the beam.
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8.3 Prove (7.14) in quantum statistical mechanics. 8.4 Verify (8.49) for Fermi and Bose statistics, i.e., the fluctuations of cell occupations are
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8.2 Derive the equations of state (8.67) and (8.71), using the microcanonical ensemble.
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Solution. By (8.65),
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(n k)
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Differentiating this again with respect to k leads to
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(nk) - (nk)
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1 8 -Ii 8 k (nk)
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from which we can deduce
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with the plus sign for Bose statistics, and the minus sign for Fermi statistics. (For Fermi statistics the results is obvious because n~ = n k .) The fluctuations are not necessarily small. Note, however, that (A) refers to the fluctuations of the occupation of individual states, and not the cell occupations. As a calculation useful for later purposes, we note 1 8 (p"* k) (nknp) - (nk)(n p) = 8 k (n p)'
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The right side is zero because (n p) depends only on
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Thus we have
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(nk)(n p ),
"* k)
*For a general discussion of the master equation, see N. G. Van Kampen, in Cohen, op. cit. An improvement on the random phase approximation is described by L. Van Hove, in Cohen, op. cit.
In the infinite-volume limit the spectrum of states becomes a continuum. The physically interesting question concerns the fluctuations in the occupation of a group of states, or a cell. Let
ni = Lnk
where the sum extends over a group of states in cell i. We are interested in
(nl> - (nY
(~nkr) - (~nk)2
(n i > L(nk
By using (B), it is easily shown that the right side is equal to
L((n1> - (nk )
Hence using (A) we obtain
(n > - (nY
where the plus sign holds for Bose statistics, and the minus sign for Fermi statistics. In the infinite-volume limit, the k sum is replaced by an integral over a region in k space. No matter how small this region is, the integral is proportional to the volume Vof the system. (This is equivalent to the statement that a finite fraction of the particles occupies a cell.) Thus the left side is of order v 2 , but the right side is only of order V.
8.5 Calculate the grand partition function for a system of N noninteracting quantum mechanical harmonic oscillators, all of which have the same natural frequency woo Do this
for the following two cases:
(a) Boltzmann statistics (b) Bose statistics.
Suggestions. Write down the energy levels of the N-oscillator system and determine the degeneracies of the energy levels for the two cases mentioned.
8.6 What is the equilibrium ratio of ortho- to parahydrogen at a temperature of 300 K
What is the ratio in the limit of high temperatures Assume that the distance between the protons in the molecule is 0.74 A. The following hints may be helpful. (a) Boltzmann statistics is valid for H 2 molecules at the temperatures considered. (b) The energy of a single H 2 molecule is a sum of terms corresponding to contributions from rotational motion, vibrational motion, translational motion, and excitation of the electronic cloud. Only the rotational energy need be taken into account. (c) The rotational energies are 1i 2 (l = 0,2,4, ... ) Epara = 2I1(I + 1)
= 211(1 + 1)
1i 2
(I = 1,3,5, ... )
where 1 is the moment of inertia of the H 2 molecule. Answer. Let T = absolute temperature and f3 = l/kT. Then