Description
1. Show that in the grand canonical ensemble, for a three-dimensional gas of
spin S particles with single-particle energies e = |~p|
2/2m, the pressure can be
written as
P = (2S + 1)
Z
d
3 p
(2πh¯)
3
|~v||~p|
3
f∓(e(~p)), (1)
where ~v = ~p/m.
2. Find the density of single-particle states for particles trapped in three-dimensional
parabolic potential. Assume the single particle energy levels are e = h¯ ω(mx +
my + mz) where mi = 0, 1, 2, …, ∞, i.e. neglect the zero point energy of the
quantum harmonic oscillator. How does this differ from the density states of
a gas with energy momentum relation e = |~p|c trapped in a box with side
lengths L, where c is a constant?
3. Consider a gas of N non-relativistic electrons (spin = 1/2) confined to a twodimensional area A with mass m in contact with a reservoir with temperature
T and chemical potential µ.
(a) Find the Fermi energy, eF of the system.
(b) Calculate the two-dimensional “pressure” (i.e. −
∂F
∂V
T,N
) of the system
when T = 0.
(c) What is the heat capacity of the electrons at fixed N (
∂E
∂T
N,V
) when T
eF, to first order in T?
(d) What is the heat capacity of the electrons at fixed µ (
∂E
∂T
µ,V
) when T
eF, to first order in T?
4. (From last week:) Assume there are N random variables labeled by i = {1, …, N}
that each obey the arbitrary normalized probability distribution g(x), so that
they have averages hx
n
i
i =
R
dxxng(xi). Assume that hxii = 0 and hx
2
i
i = σ
2
for all i. Show that the distribution of the average of these random variables,
x¯ = 1
N ∑i xi
, in the large N limit is given by
P(x¯) = 1
√
2πσ2/N
e
− Nx¯
2
2σ
2
,
which is essentially the central limit theorem, which says that the probability
distribution of the sum of a large number of random variables tends to a Gaussian (or normal) distribution. Therefore, it is maybe not so surprising that this
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distribution shows up quite often in statistical mechanics. This also shows the
standard deviation of x¯ is ∝ 1/√
N, since we have R ∞
−∞
dxx2
exp(−x
2/2σ
2
) =
√
2πσ3/2, and the distribution of x¯ goes to a delta function in the large-N limit
[Hints: Use δ(x) = 1
2π
R ∞
−∞
dyeixy and R ∞
−∞
exp(iay − by2
) = √
π/b exp(−a
2/4b).]
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