Problem Set 4

Problem Set 4

Due Thursday, March 6, uploaded to Blackboard by 11:59pm

Problem 1 [10 points total]
In Notes Eq. (2.4.14) we derived the number of states Ω(E,V,N) for the idea gas of N point particles, which can be written as given below,

Ω(E,V,N) = [ V
h³ (2πmE)^3/2 ]^N 1
[(3N/2) - 1]! N! (ΔE/E)

Note, compared to Eq. (2.4.14) the above expression has the factor N! in the denominator, as needed for indistinguishable particles, as described in Notes 2-7.
By taking the Laplace transform of Ω(E,V,N), directly compute the canonical partition function, as in Notes Eq. (2.8.6),

Q_N(T,V) = ∞
∫
0 dE
ΔE Ω(E,V,N)e^-βE

Using the resulting canonical partition function, compute the Helmholtz free energy A(T,V,N) and compare it to what was found in Problem Set 1.
Problem 2 [10 points total]
Consider N, indistinguishable, non-interacting, non-relativisitic, particles moving in one dimension -- so there is only one coordinate variable x, and one momentum variable p, per particle. The particles move in a box of length L. There is a potential energy in the box U(x) such that U=0 for 0 ≤ x < L/2, and U=U₀ for L/2 ≤ x ≤ L. What is the probability that a particular particle will be found in the right half of the box? What is the probability to find M of the particles in the right half of the box?
Problem 3 [20 points total]
Consider the same system as in the previous Problem Set, a system of N distinguishable non-interacting objects, each of which can be in one of two possible states, "up" and "down", with energies +ε and −ε. Assume that N is large.
(a) Working in the canonical ensemble, find the Helmholtz free energy A(T, N) as a function of temperature T and number N. Note: since these degrees of freedom are distinguishable, there is no Gibbs factor 1/N! when you compute the canonical partition function Q_N(T).
Note: Having found Ω(E,N) in the previous Problem Set, you could compute the canoncial Q_N(T) by taking the Laplace transform of Ω(E,N) with respect to E. Don't do it this way! Instead, compute Q_N(T) by directly summing the Boltzmann factor over all states in the phase space.
(b) Starting from A(T, N) of part (a), find the canonical entropy, and then express it as a function of the average energy E and number N. Show that, in the large N limit, your result agrees with your answer for the entropy in the microcanonical ensemble, as computed in the previoius Problem Set.