Statistical Mechanics - Physics at Oregon State University

2.9. PROBLEMS FOR CHAPTER 2 41
(From Reif) The following describes a simple two-dimensional model of a
situation of actual physical interest. A solid at absolute temperature T contains
N negatively charged impurity atoms per cm 3 ; these ions replacing some of
the ordinary atoms of a solid. The solid as a whole, of course, is electrically
neutral. This is so because each negative ion with charge −e has in its vicinity
one positive ion with charge +e. The positive ion is small and thus free to move
between lattice sites. In the absence of an electrical field it will therefore be
found with equal probability in any one of the four equidistant sites surrounding
the stationary negative ion (see diagram in hand-out).
If a small electrical field E is applied along the x direction, calculate the
electric polarization, i.e., the mean electric dipole moment per unit volume
along the x direction.
Problem 7.
The probability of finding
a system in a state s is Ps. In this case the entropy
of the system is S = −kB Ps log(Ps). Assume that the system consists of two
s
independent subsystems and that the state s is the combination of subsystem 1
being in state s1 and system 2 being in state s2. Use the formula for the entropy
given above to show that S = S1 + S2.
Problem 8.
The energy eigenvalues for a single particle in a given potential are ɛn. These
energy values are independent of how many other particles are present. Show
that the partition function Z(T, N) obeys the relation Z(T, N) = (Z(T, 1)) N .
Problem 9.
The energy eigenvalues of a system are given by 0 and ɛ + n∆ for n =
0, 1, 2, · · ·. We have both ɛ > 0 and ∆ > 0. Calculate the partition function for
this system. Calculate the internal energy and the heat capacity. Plot the heat
capacity as a function of temperature for 0 < kBT < ɛ for (a) ∆ ≫ ɛ, (b) ∆ = ɛ
, and (c) ∆ ≪ ɛ.