Prof. Raghuveer Parthasarathy
University of Oregon; Spring 2008
Physics 353: Problem Set 7 – SOLUTIONS
1 General properties of fermion and boson distributions.
Changing the particles from fermions to bosons only changes the density of states,
()
D
ε
, by a
numerical factor corresponding to the spin degeneracy. The rest of
(
)
D
ε
is determined by the “particle-in-
a-box” energy levels, which are unchanged. Therefore statement B is true.
The distribution function,
f
, changes if we change the particle type. However both the Fermi and
Bose distributions reduce to the classical (Boltzmann) distribution at high temperature. Therefore statement
D is true.
For Bosons, as 0
τ
→, all the particles will occupy the lowest, zero-energy orbital. Therefore
statement E is true.
Statements A and C are false.
2 Another general question.
The distribution function,
f
, (for Fermions or Bosons) depends on temperature. The density of
states does not. Since U depends on
f
via () ()UfDd
ε
εε
∫
=
, this means that no matter what
(
)
D
ε
Mr. K. creates, U will still depend on temperature.
3 Liquid helium.
The BE condensation temperature
2/3 2
12
2.612
C
N
Vm
π
τ
⎛⎞
⎛⎞
=⎜⎟
⎜⎟
⎝⎠
⎝⎠
=. The number density N
nVm
ρ
==,
where
ρ
is the density, 145 kg/m3. The boson mass 4
P
mm
=
, where
P
m is the proton mass.
Therefore 23
4.3 10
CJ
τ
−
=× , or more clearly 3.1
C
T
=
Kelvin.
