444 Quantum ideal gases
11.6. Consider a system ofNidentical bosons. Each particle can occupy one of
two single-particle energy levels with energiesε 1 = 0 andε 2 =ε. Determine
a condition in terms ofN,β, andεthat must be obeyed if the thermally
averaged occupation of the lower energy level is twice that of the upper level.11.7. Derive expressions for the isothermal compressibility at zerotemperature for
ideal boson and fermion gases. Recall that the isothermal compressibility is
given by
κT=−1
V
(
∂V
∂P
)
T.
11.8. A cylinder is separated into two compartments by a piston thatcan slide freely
along the length of the cylinder. In one of the compartments is an ideal gas
of spin-1/2 particles, and in the other is an ideal gas of spin-3/2 particles. All
particles have the same mass. At equilibrium, calculate the relative density
of the two gases atT= 0 and at high temperature (Huang, 1963).11.9. a. Two identical, noninteracting fermions of massmare in a harmonic os-
cillator potentialU(x) =mωx^2 /2, whereωis the oscillator frequency.
Calculate the canonical partition function of the system at temperature
T.b. Repeat for two identical, noninteracting bosons.11.10. Consider a system with a HamiltonianHˆ 0 that has two eigenstates|ψ 1 〉and
|ψ 2 〉with the same energy eigenvalueE:Hˆ 0 |ψ 1 〉=E|ψ 1 〉Hˆ 0 |ψ 2 〉=E|ψ 2 〉with the orthonormality condition〈ψi|ψj〉=δij.Let a perturbationHˆ′be applied that breaks the degeneracy such that the
new eigenstates ofHˆ=Hˆ 0 +Hˆ′are|ψ+〉=1
√
2
[|ψ 1 〉+|ψ 2 〉]|ψ−〉=1
√
2
[|ψ 1 〉−|ψ 2 〉]with corresponding energiesE+andE−:Hˆ|ψ+〉=E+|ψ+〉