Description
Problem 1:
A system is composed of a large number N of one-dimensional quantum harmonic oscillators whose angular frequencies are distributed over the range ωa ≤ ω ≤ ωb with a frequency
distribution function D(ω) = Aω−1
, where A is a real constant.
Let us assume that the
quantum oscillators can be treated as distinguishable quantum particles.
(a) Calculate the specific heat per quantum oscillator at temperature T.
Hint: It is convenient to use the canonical ensemble.
(b) Evaluate your result from part (a) in the high temperature limit (clearly define what
“high T” and “low T” mean).
(c) Make a plot of your result from part (a) and compare with the single-frequency case.
Problem 2:
The “baloneyon” is an imaginary fermion with spin-1/2 and the relationship E = B|~p|
4
between the energy E and the momentum ~p (as an aside, such dispersion curves can be
engineered to a very good approximation using cold atoms),
E = B|~p|
4
. (1)
Consider a non-interacting gas of baloneyons in two spatial dimensions.
(a) What units does B have?
(b) Determine the Fermi energy of a non-interacting gas of baloneyons as a function of the
particle density.
(c) Explicitly check the units of your result obtained in part (b).
(d) Provide a physical interpretation of the Fermi energy.
Problem 3:
Consider a single electron with mass m, intrinsic spin 1
2
h¯
ˆ~σ, and spin magnetic moment
ˆM~
s, where
ˆ~σ =
σˆx
σˆy
σˆz
. (2)
Using the eigen states | ↑i and | ↓i of ˆσz as basis, we have the following matrix representations:
σx =
0 1
1 0 !
, (3)
σy =
0 −ı
ı 0
!
, (4)
and
σz =
1 0
0 −1
!
. (5)
The spin of the electron has two possible orientations, up and down, with respect to
an applied magnetic field B~ . Letting the B-field point along the negative z-direction
(B~ = −Bzeˆz with Bz = |B~ |), the quantum mechanical Hamiltonian Hˆ takes the form
Hˆ = −
ˆM~
s · B~ = µB
ˆ~σ · B~ = −µBBzσˆz, (6)
where
µB =
eh¯
2mc
. (7)
Use the canonical ensemble to treat this problem.
Express the density matrix ˆρ in terms of the eigen states | ↑i and | ↓i of ˆσz and calculate
the thermal expectation value hσˆzi.
Problem 4:
Consider a three-dimensional free particle in a box of length L. Assume periodic boundary
conditions. Use the canonical ensemble to treat this problem.
(a) Find a compact expression for the density matrix ˆρ in the coordinate representation,
i.e., find a compact expression for the quantity h ~r | ρˆ| ~r 0
i; “compact” means that the expression should not contain any (infinite) sums.
Hint: Consider converting the sum over ~k into an integral.
(b) Evaluate and interpret the quantity h ~r | ρˆ| ~r i.
(c) Calculate hHi ˆ .
