Problem Set 5

Problem Set 5

DQ 5 -- Due Tuesday, March 9, by 5pm

Consider N, indistinguishable, 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?

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Problems -- Due Thursday, March 11, by 5pm.

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Problem 1 [15 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 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.
Problem 2 [20 points total]
Consider a classical gas of N indistinguishable non-interacting particles with ultrarelativistic energies, i.e. their kinetic energy - momentum relation is given by ε = pc, with c the speed of light and p the magnitude of the particle's momentum. The gas is confined to a box of volume V.
(a) [5 pts] Compute the canonical partition function for this system.
(b) [5 pts] Show that this system obeys the usual ideal gas law, pV = Nk_BT.
(c) [5 pts] Show that the total average energy is, E = 3Nk_BT (and hence using (b) gives, E/V = 3p).
(d) [5 pts] Show that the ratio of specific heats is, C_p/C_V = 4/3.