Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
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3.8. PROBLEMS FOR CHAPTER 3 65<br />
Problem 4.<br />
The st<strong>at</strong>e of a many body system is characterized by two quantum numbers,<br />
n and m. The possible values of the quantum number n are 0, 1, 2, · · · , ∞, while<br />
the values of m are in the range 0, 1, · · · , n. The energy of the system in the st<strong>at</strong>e<br />
(n, m) is nω and the number of particles is m. Evalu<strong>at</strong>e the grand partition<br />
function for this system.<br />
Problem 5.<br />
An ideal gas of <strong>at</strong>oms with mass m is contained in a cylinder th<strong>at</strong> spins<br />
around with angular frequency ω. The system is in equilibrium. The distance<br />
to the axis of the cylinder is r. The radius of the cylinder is R. Calcul<strong>at</strong>e the<br />
density of the gas as a function of r.<br />
Problem 6.<br />
Extremely rel<strong>at</strong>ivistic particles obey the rel<strong>at</strong>ion E( k) = c| k|. Assume we<br />
have a gas of these identical particles <strong>at</strong> low density, or n ≪ nQ(T ).<br />
(A) Calcul<strong>at</strong>e the partition function Z1(T, V ) for N=1.<br />
(B) Calcul<strong>at</strong>e Z(T, V, N).<br />
(C) Calcul<strong>at</strong>e p(T, V, N), S(T, V, N), and µ(T, V, N).<br />
Problem 7.<br />
The chemical potential of an ideal gas is given by 3.101. Suppose n ≪<br />
nQ(T ). In this case we have µ < 0. A bottle contains an ideal gas <strong>at</strong> such an<br />
extremely low density. We take one molecule out of the bottle. Since µ < 0 the<br />
Helmholtz free energy will go up. The equilibrium of a system is reached when<br />
the Helmholtz free energy is minimal. This seems to favor molecules entering the<br />
bottle. Nevertheless, we all know th<strong>at</strong> if we open the bottle in an environment<br />
where the density outside is lower than inside the bottle, molecules will flow<br />
out. Wh<strong>at</strong> is wrong with the reasoning in this problem?<br />
Problem 8.<br />
The only frequency of the radi<strong>at</strong>ion in a certain cavity is ω. The average<br />
<br />
number of photons in this cavity is equal to e ω −1 kB T − 1 when the temper<strong>at</strong>ure<br />
is T. Calcul<strong>at</strong>e the chemical potential for this system.<br />
Problem 9.