Statistical Mechanics - Physics at Oregon State University

244 APPENDIX A. SOLUTIONS TO SELECTED PROBLEMS.
Problem 4.
The state of a many body system is characterized by two quantum numbers, n
and m. The possible values of the quantum number n are 0, 1, 2, · · · , ∞, while
the values of m are in the range 0, 1, · · · , n. The energy of the system in the state
(n, m) is nω and the number of particles is m. Evaluate the grand partition
function for this system.
or
Problem 5.
Z =
Z =
1
1 − e βµ
Z =
Z =
Z =
Z =
1
1 − e βµ
∞
n=0
∞
n=0 m=0
n
e −β(nω−µm)
∞
e −nβω
n=0
∞
n=0
1
1 − eβµ
n
e mβµ
m=0
−nβω 1 − e(n+1)βµ
e
1 − eβµ e −nβω − eβµ
1 − e βµ
1 eβµ
−
1 − e−βω 1 − eβµ 1
−
1 − e−βω ∞
e −nβ(ω−µ)
n=0
1
1 − e β(µ−ω)
1
e−βµ − e−βω
An ideal gas of atoms with mass m is contained in a cylinder that spins around
with angular frequency ω. The system is in equilibrium. The distance to the
axis of the cylinder is r. The radius of the cylinder is R. Calculate the density
of the gas as a function of r.
Reasoning similar to problem 1 .
Problem 6.
µ = kBT log( n(r) 1
) +
nQ(T ) 2 mω2r 2
n(r) = nQ(T )e βµ mω2 − 2k e B T r2
Extremely relativistic particles obey the relation E( k) = c| k|. Assume we have
a gas of these identical particles at low density, or n ≪ nQ(T ).