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1. Maximize the Gibbs entropy (with kB = 1) subject to the constraints
hxi = ∑
i
xip(xi)
hx
2
i = ∑
i
x
2
i
p(xi),
to find the probability distribution of xi
. Here, p(xi) is the probability of xi
.
Show that this becomes the normal distribution when x is allowed to be continuous and run from −∞ to ∞, i.e.
p(x) = 1
√
2πσ2
e
−(x−µ)
2/2σ
2
with µ = hxi and σ
2 = hx
2
i − hxi
2
. Note that for continuous x, p(x) is a probability density, so that the normalization condition is given by R ∞
−∞
dxp(x) = 1,
for instance.
2. Consider a system in the grand canonical ensemble with two single-particle
energy states 0 and e.
(a) Assuming the particles are Fermions, calculate hNi as a function of µ and
T. Show the limits as T → 0 and ∞.
(b) Assuming the particles are Bosons, calculate hNi as a function of µ and
T. Show the limits as T → 0 and ∞.
3. Consider a system with two single-particle states 1 and 2. This system could
also be in the mixed states
|ψ±i =
1
√
2
[|1i ± |2i]
(a) Write down the density matrix for the system in the basis defined by 1
and 2 when the system is in state |ψ+i and verify ρˆ
2 = ρˆ.
(b) Now consider the density matrix
ρˆ = ∑α=±
pα|ψαihψα|,
p+ + p− = 1. Find the value of p+ which minimizes the purity of the
ensemble, Tr[ρˆ
2
].
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4. In some cases, we might want to talk about systems that have a net macroscopic angular momentum in a particular direction. If the system could exchange angular momentum and energy with the world around it, then it would
be natural to describe its properties in terms of an ensemble subject to the constraints hLi = ∑i piLi
, ∑i pi = 1, and hEi = ∑i piEi
, but with every ensemble
member having fixed volume V and particle number N.
(a) Write down the normalized probability pi
for drawing an ensemble member in state i and define the normalization coefficient (or partition function) for this ensemble ZL.
(b) Consider a system of N distinguishable quantum rotors that rotate about
the same fixed axis, with single particle energies em = h¯
2
2I m2 and single
particle angular momenta ` = hm¯ , where m = −∞, …, −1, 0, 1, …∞. I
is the moment of inertia of a single rotor. Calculate ZL for this system
(eventually assuming that the energy levels are closely spaced enough to
take sums over m to integrals).
(c) Calculate the average angular momentum of the system of quantum rotors and show that the Lagrange multiplier associated with the angular
momentum constraint multiplied by T can be interpreted as the net angular velocity of the system.
(d) Calculate the average energy of the system of quantum rotors.
5. 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
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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