Problem Set 9

Problem Set 9

DQ 9 -- Due Tuesday, April 13, by 5pm

Consider a point particle of mass m attached to a harmonic spring with spring constant κ=mω₀². The particle moves in only one dimension given by the coordinate x. The rest position of the particle (when the spring in unstretched) is at x=0. The particle is in equilibrium at temparature T.
Assuming that the particle behaves quantum mechanically, compute the root-mean-square fluctuation in the position of the particle, Δx = (⟨x²⟩ - ⟨x⟩²)^1/2.
Hint: recall that for the quantum harmonic oscillator, the expected value of the kinetic energy equals the expected value of the potential energy (this is the quantum virial theorem).
Show that in the limit k_BT ≫ ℏω₀ you recover the answer for a classical particle.

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Problems -- Due Thursday, April 15, by 5pm.

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Problem 1 [20 points]
In the grand canonical ensemble, the probability to have a given state "a" with total energy E_a and total number of particles N_a is ("a" is labeling the full N-particle state),
P_a = [e^{-(E_a-µN_a)/k_BT}]/L
where
L = ∑_a [e^{-(E_a-µN_a)/k_BT}] is the grand canonical partition function.

(a) For a quantum ideal gas, with single particle states i of energy ε_i, many particle states are specified by the occupation numbers {n_i} and have energy E = ∑_i [ε_in_i]. Show that the probability for the state with occupations {n_i} is given by
P({n_i}) = ∏_i [p_i(n_i)]
where p_i(n_i) is the probability that single particle state i has occupation n_i, and p_i(n_i) is given by
p_i(n_i) = [e^{-(ε_i-µ)n_i/k_BT}]/w_i
where
w_i = ∑_{n_i} [e^{-(ε_i-µ)n_i/k_BT}]
can be thought of as the partition function for the single particle state i. The above factorization says that the number of particles n_i in state i, is independent of the number of particles n_j in state j.
(b) Using the above result, show that the Shannon definition of entropy can be written as
S = -k_B ∑_{{n_i}} [P({n_i}) ln P({n_i})] = -k_B ∑_i ∑_{n_i} [p_i(n_i) ln p_i(n_i)]

(c) Using the above result, show that the following expressions apply for the entropies of an ideal gas of bosons and fermions, respectively

bosons: S = k_B ∑_i [(1+<n_i>) ln (1+<n_i>) - <n_i> ln <n_i>]

fermions: S = k_B ∑_i [-(1-<n_i>) ln (1-<n_i>) - <n_i> ln <n_i>]

where <n_i> = ∑_{n_i} [n_i p_i(n_i)] is the average occupation number of state i.
Problem 2 [25 points]
Consider an ideal quantum gas of non-interacting identical particles in the grand canonical ensemble. The gas is in equilibrium at temperature T and chemical potential μ. In the previous problem part (a) you found the number of particles n_i that occupy the single-particle energy eigenstate i is statistically independent of the number of particles n_j in that occupy state j, and you found the probability distribution p_i(n_i) that there are exactly n_i particles in state i.
a) Using this p_i(n_i) rederive the average occupation number of particles <n_i> in state i, for a gas of bosons and a gas of fermions.
b) Using p_i(n_i), derive the occupation number fluctuations <n_i²> - <n_i>² for a gas of bosons and a gas of fermions.
c) If the total number of particles is N = ∑_in_i, show that the fluctuation in N is given by,
<N²> - <N>² = ∑_i [ <n_i²> - <n_i>² ].
Are the flutucations in N for the quantum gas bigger or smaller than they are for the corresponding clasical gas, for a gas of bosons? for a gas of fermions? [You can find the fluctuation in N for the classical ideal gas from Discussion Question 7.]