Controlled Condition Exam: 3 Hours exam
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module in the following academic year(s):
Year
2020/21
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count
TURN OVERAnswer ALL THREE questions
The numbers in square brackets show the provisional allocation of
maximum marks per question or part of question.
[Part marks]
1. Consider a one-dimensional harmonic oscillator system that
consists of a point mass m moving in a potential of the form 1 2 k x 2,
with k a constant and where x denotes the deviation from its
equilibrium position. We defifine ! = pk/m as the natural
frequency of the oscillator.
(a) Assume that the oscillator is classical and ignore thermal
flfluctuations.
i. Assume that the oscillator is isolated from its
environment and has a constant energy. How does the
entropy of the oscillator change with variations in x?
Explain your answer.
[2]
ii. Now assume instead that as the point mass moves, heat
is generated due to friction between the oscillator and the
environment. The point mass is released at a position
x
6
= 0. Assuming that the environment is at a constant
temperature Tenv , use the fifirst and second laws of
thermodynamics to explain the subsequent behaviour of
the point mass.
[4]
(This question continues on the next page)
PHAS0024/2021-22 CONTINUED
1[Part marks]
(Question continued from previous page)
(b) Now assume that the oscillator is quantum-mechanical.
i. The oscillator is isolated from its environment and has a
constant energy E. What is the entropy of the oscillator?
Explain your answer.
[2]
ii. Now assume that the oscillator is weakly coupled to
another quantum-mechanical oscillator having the same
natural frequency !. The system of the 2 oscillators is
isolated from the environment and has a total energy of
4~!. Calculate the entropy of this system.
[3]
(c) Next consider the quantum mechanical oscillator to be
weakly coupled to a large number of other quantum
mechanical oscillators with identical !, and assume there are
a total of N ” 1 energy quanta. Let P(n) be the probability for
the fifirst oscillator to have n quanta, with n ⌧ N.
i. Defifine the relevant inverse temperature β = (kBT) −1 in
terms of: ~!, n, N, and the total microstate multiplicity of
the other oscillators ⌦0 (N, n). In doing so, demonstrate
that P(n) is a canonical distribution.
[6]
ii. Argue how the effective temperature T would change if
we start to further increase n, such that the
approximation n ⌧ N becomes less accurate.
[3]
PHAS0024/2021-22 CONTINUED
2[Part marks]
2. Consider a system of two independent single-particle quantum
levels characterised by energies 0 and ✏, in equilibrium at an
inverse temperature β = (kBT) −1 and chemical potential µ.
(a)
i. We know that the total probability of two uncorrelated
events is given by the product of the probabilities of the
individual events. Starting with that observation, explain
why the grand canonical partition function ZG of this
system can be written as the product of the grand
canonical partition functions ZG (0) and Z
(✏)
G
for the states
with energies 0 and ✏, respectively.
[3]
ii. Noting that we are considering quantum particles, explain
if ZG = ZG (0) ⇥ Z
(✏)
G
still applies (or not) when the particles
are bosons or fermions.
[2]
iii. Referring to the same system and making use of the
observations above, give expressions for ZG for bosons
and for fermions that are in the same spin state.
[2]
(This question continues on the next page)
PHAS0024/2021-22 CONTINUED
3[Part marks]
(Question continued from previous page)
(b) Now assume that the quantum particles in this system are
spin 1 2 particles, i.e., can be in spin state ±1 2 .
i. Derive expressions for ZG and for the mean energy of this
specifific system.
[4]
ii. Determine ZG for the case in which the particles interact
with each other via a pair-wise interaction energy U.
[6]
iii. If the particles in this system are electrons, which
physical interaction would give rise to this U? Brieflfly
explain how this interaction is accounted for in the free
electron model for metals and why this may be done.
[3]
PHAS0024/2021-22 CONTINUED
4[Part marks]
3. Consider non-interacting quantum-mechanical particles in a large
two-dimensional, square box of area L ⇥ L = L2.
(a) Show that the single-particle density of states as a function of
energy can be written as
[4]
g(✏) = (2s + 1) L2 2m
⇡
h
2
.
(b) Using this density of states, show that the one-particle
canonical partition function is given by
[2]
Z1 = (2s + 1) L2 2m
⇡
β
h
2 .
(c) Based on your result for the single-particle canonical partition
function above, determine the length scale of L at/below
which quantum effects become noticeable, and compare it
with the three-dimensional case.
[3]
(d) Assume that the particles have spin s = 1 2 . Use the
degenerate Fermi gas approximation to express the mean
energy of the gas as a function of the number of particles.
Show that this energy is inversely proportional to the area L2. [5]
(e) Assume a tension on the boundaries, acting as a constant
force, with direction locally aligned with the boundaries, that
reduces the perimeter of the square area. This will cause the
square area to shrink. Considering the energetics of this
process, explain if/how this shrinkage will come to a halt.
[3]
(f) Explain if/how the shrinkage of the area would come to a halt
if the particles had integer spins rather than half-integer spins. [3]
PHAS0024/2021-22 END OF EXAMINATION PAPER
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