PHY 418: Statistical Mechanics I
Prof. S. Teitel: stte@pas.rochester.edu ---- Spring 2022
DQ 9 -- Due Tuesday, April 5, 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.Post your response on the Discussion Board at this link: DQ9Assuming 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 kBT ≫ ℏω0 you recover the answer for a classical particle.
Upload your solutions to Blackboard at this link: PS9
In the grand canonical ensemble, the probability to have a given state "a" with total energy Ea and total number of particles Na is ("a" is labeling the full N-particle state),
Pa = [e-(Ea-µNa)/kBT]/Lwhere
L = ∑a [e-(Ea-µNa)/kBT] is the grand canonical partition function.
(a) For a quantum ideal gas, with single particle states i of energy εi, many particle states are specified by the occupation numbers {ni} and have energy E = ∑i [εini]. Show that the probability for the state with occupations {ni} is given by
P({ni}) = ∏i [pi(ni)]where pi(ni) is the probability that single particle state i has occupation ni, and pi(ni) is given by
pi(ni) = [e-(εi-µ)ni/kBT]/wiwhere
wi = ∑ni [e-(εi-µ)ni/kBT]can be thought of as the partition function for the single particle state i. The above factorization says that the number of particles ni in state i, is independent of the number of particles nj in state j.
(b) Using the above result, show that the Shannon definition of entropy can be written as
S = -kB ∑{ni} [P({ni}) ln P({ni})] = -kB ∑i ∑ni [pi(ni) ln pi(ni)]
(c) Using the above result, show that the following expressions apply for the entropies of an ideal gas of bosons and fermions, respectively
where <ni> = ∑ni [ni pi(ni)] is the average occupation number of state i.
bosons: S = kB ∑i [(1+<ni>) ln (1+<ni>) - <ni> ln <ni>] fermions: S = kB ∑i [-(1-<ni>) ln (1-<ni>) - <ni> ln <ni>]
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 ni that occupy the single-particle energy eigenstate i is statistically independent of the number of particles nj in that occupy state j, and you found the probability distribution pi(ni) that there are exactly ni particles in state i.
a) Using this pi(ni) rederive the average occupation number of particles <ni> in state i, for a gas of bosons and a gas of fermions.
b) Using pi(ni), derive the occupation number fluctuations <ni2> - <ni>2 for a gas of bosons and a gas of fermions.
c) If the total number of particles is N = ∑ini, show that the fluctuation in N is given by,
<N2> - <N>2 = ∑i [ <ni2> - <ni>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.]