Download Phys 210A — Spring 2016 Problem Set #3: Quantum Statistical

Phys 210A — Spring 2016
Problem Set #3: Quantum Statistical Mechanics
Due date: Wednesday May 11th
1. Relativistic Electron Gas — For an electron whose energy is much larger
than the rest energy me c2 , the relation between energy and momentum can be
approximated by the extreme relativistic form ε = p · c. This can happen at
very high electron density or very high temperature regardless of density.
(a) Compute (up to a constant) the classical partition function, i.e., using Boltmann statistics, for a gas of N electrons confined in a volume V maintained
at a temperature T .
[You may assume that a background of fixed positive charges exists
and that the Coulomb interaction between the particles are screened
out.]
(b) Obtain the mean thermal energy E(T ) and the specific heat CV (T ) of this
system. Compare to the result of the equipartition theorem. For what
range of temperatures is the use of the extreme relativistic form justified.
(c) Obtain the equation of state using the grand canonical ensemble and compare to that of the ideal nonrelativistic gas.
(d) At low temperatures, the relativistic form is still valid at sufficiently high
density, if the effect of quantum statistics is included. Compute the density
of states D(ε) and obtain the relation between the Fermi energy εF and
the electron density. For what temperature range is the use of quantum
statistics necessary?
[You may assume the electrons to be confined in a ‘‘periodic’’
cubic box.]
2. Ideal Fermi Gas — In class, we showed that in the limit T → 0, the chemical
potential µ of an ideal Fermi gas is given by the Fermi energy
h̄2
εF =
2m
!2/3
6π 2
ρ
g
,
where m is the mass of the fermion, g is the degree of degeneracy, and ρ is the
density of the gas. In this problem, you are asked to computer the leading order
correction to the following quantities at low temperatures: (a) the chemical
potential, (b) the pressure exerted by the gas, (c) average energy. From your
answer to (c), find the specific heat to leading order in T .
3. Ideal Gas in Two Dimensions — Consider an ideal gas consisted of N spin-0
particles of mass m in a two-dimensional periodic box of size L × L, kept at
a constant temperature T . In studying this problem. it will be convenient to
introduce the grand canonical ensemble as a mathematical device.
(a) Find the density of states, D(ε).
(b) Write down the pressure P in terms of the fugacity f ; write down also the
auxiliary condition which determines f (N ).
(c) In the limit of low density and high temperature, find the equation of state
P (N ) to leading nonlinear order.
(d) Show that in the thermodynamic limit, no Bose-Einstein condensation
occurs at any finite temperature. More specifically, show that TBE ∝
1/ log(N ).
4. Vibrational Specific Heat of Solids — In this problem, we will analyze the
contribution to the specific heat of a crystalline solid due to its quantized lattice
vibrations, phonons.
In the Einstein model of lattice vibration, one approximates a lattice of N
atoms by noninteracting (quantum) harmonic oscillators, each with frequency ω.
Three oscillators per atom are used to describe motion in the x, y, z directions.
(a) Compute the partition function of this system, and hence obtain the mean
energy ε(T ; ω) per atom.
(b) Find the specific heat CV (T ). Show that in the limit T → ∞, the DulongPetit law of classical physics is recovered. Show also that CV (T ) → 0 as
T → 0.
The explanation of vanishing specific heat at low T , as a result of quantum
mechanics, was a great triumph of the Einstein theory. A more quantitatively
accurate theory was due to Debye, who assumed that each oscillator can take
on a range of frequencies ωi , corresponding to the different normal mode k of
lattice vibration. Alternatively, one can view the quantized lattice vibrations
(phonons) as an ideal bose gas, with energies ε(k) = h̄ω(k). It is useful to
introduce the frequency distribution function D(ω), where D(ω)dω describes
the number of modes having frequency between ω and ω + dω.
(c) What is the mean energy per atom ε(T ) in term of D(ω) ? Find ε(T ) in
the limit of high temperature, thereby showing that the high temperature
behavior is indepedent ofR the choice D(ω). [Note the normalization
condition on D(ω) is 0∞ dωD(ω) = 3N .]
(d) Low temperature behavior does depend on the form of D(ω), which can
be obtained from the knowledge of the phonon dispersion relation ω(k).
Assuming that ω(k) = v·|k|, where v is the sound speed, show that D(ω) =
αV ω 2 , with V being the volume of the solid and α being a proportionality
constant.
To satisfy the normalization condition on D(ω), Debye introduced an approximation in which he took the result of (d) up to some cutoff frequency ωD , chosen
to give a total of 3N modes. For ω > ωD , Debye assumed that D(ω) = 0.
(e) Using Debye’s approximation, write down an expression for the mean energy per atom ε in terms of h̄ωD and a dimensionless parameter TD /T ,
where the Debye temperature TD is defined by kB TD = h̄ωD .
(f ) Find the vibrational specific heat CV (T ) for T TD . Show that the T dependence follows generally from the k → 0 limit of the dispersion relation
and is insensitive to Debye’s approximation. What is the T -dependence
for a d-dimensional solid ?