Assignment 8
8.1 (Pathria 6.2) (a) Fill in the details for the derivations of Eqs. (6.3.10) and (6.3.11).
(b) Using the probabilities pǫ(n) directly, calculate the variances ∆n2
ǫ=hn2
ǫi − hnǫi2for both Bose and
Fermi systems. Verify Eq. (6.3.9).
(c) For both types of particle, show that
∆n2
ǫ=kT ∂hnǫi
∂µ T
Note the similarity to Eq. (4.5.3).
8.2 (Pathria 6.3) Refer to Sec. 6.2 and show that, if the occupation number nǫof an energy level ǫis
restricted to the values 0,1, . . . , ℓ, then the mean occupation number of that level is given by
hnǫi=1
z−1eβǫ −1−ℓ+ 1
(z−1eβǫ)ℓ+1 −1
Check that ℓ= 1 leads to the Fermi-Dirac distribution and ℓ→ ∞ leads to the Bose-Einstein distribution.
8.3 (Pathria 6.10) (a) Show that the momentum distribution of particles in a relativistic Boltzmann gas,
with ǫ=c(p2+m2c2)1/2, is given by
f(p)d3p=Cexp h−βc p2+m2c21/2ip2dp,
with the normalization constant
C=β
m2cK2(βmc2),
where Kν(z) is the modified Bessel function, with integral representation
Kν(z) = √π(z/2)ν
Γ(ν+ 1/2) Z∞
1
e−zt t2−1ν−1/2dt.
(b) Check that in the nonrelativistic limit (kT ≪mc2) we recover the Maxwell distribution
f(p)d3p=β
2πm 3/2
exp −βp2
2m4πp2dp.
(c) Verify explicity that in all cases, hpui= 3kT , as seen already in problem 6.3 (Pathria 3.24).
8.4 (Pathria 7.2) Deduce the virial expansion (7.1.13) from Eqs. (7.1.7) and (7.1.8), and verify the quoted
values of the virial coefficients in (7.1.14) up to third-order. (You don’t need to do the a4term.)
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