JCP322H5S-Statistical Mechanics
Problem Set 4
Due: Monday April 8th, 2024, 11:59pm
There are 5 problems. Please clearly show all steps leading to your final solution.
Marks will not be granted for showing just the final result. This should include well labelled
diagrams and step-by-step explainations. You may choose to write out your solutions
neatly by hand, or choose to type out the solutions via software such as Latex. While
general discussions are permissable amongst classmates, final solutions for each of the
problems must be written independently.
1. Fluorescence Two state Model:
Consider the two state model of an excited atom discussed in class, which served
as a model for fluorescence. Here an electron can be excited from the ground state
S0 to an excited state S1 at rate kon , and may decay back to the ground state at
rate kof f . Now consider the inclusion of a third state S2 that can only be reached by
excited electrons. These electrons will irreversibly transition into S2 at rate kp . This
additional state serves as a model of a process known as ‘photobleaching’.
(a) Write the dynamic equations for the probabilities p0 , p1 , and p2 , which are the
probabilities of being in states S0 , S1 , and S2 , respectively.
(b) Solve for the equilibrium probabilities.
2. Phonons and photons are bosons:
Problem 7.2 from Entropy, Order Parameters and Complexity (Sethna, James)
3. Diatomic Molecules Part I:
Consider a diatomic molecule made up of two different atoms separated by a distance d. Such as molecule can exist in quantum states of different orbital angular
momentum given by the following relation:
ϵl =
ℏ2 l(l + 1)
,
2I
where I is the moment of inertia through the axis of the molecule and l = 0, 1, 2, 3 . . ..
Note, each energy level is multiply degenerate with a degeneracy of gl = 2l + 1.
(a) Find the partition function for this system.
Hint: you need to simply consider the product of the degenercy of the system
and the canonical partition function
(b) Determine the Helmholtz Free Energy at low temperatures. From this relation,
find the entropy and heat capacity at low temperatures.
(c) Find the entropy and heat capacity in the limit of high temperatures. Note, in
this limit
∞ in the partition function can be approximated by an integral
P∞the sum
(i.e., l=0 ≈ 0 dl).
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4. Diatomic Molecules Part II:
To a good approximation, the vibrational states of a diatomic molecule can be approximated by that of a harmonic oscillator with energy:
ϵn = ℏω(n + 1/2),
where ω is the angular frequency of vibration and n = 0, 1, 2, . . ..
(a) Find the partition function.
Hint: Look up a geometric series to help simplfy your partition function
(b) For high temperatures, find the vibrational contribution to the free energy. Then
solve for the entropy and heat capacity.
(c) Solve for the contribution to the entropy and heat capacity at low temperatures.
5. Fermi Gas Systems:
Based on our discuss in class about Fermi Gas, we derived a series of expressions
for the fermi energy, and the total kinetic energy for a Fermi Gas in the ground state.
Using these results as a basis, answer the following questions.
(a) Find an expression for the pressure exerted by a Fermi electron gas in the
ground state.
(b) Find an expression for the entropy of an electron gas in the limit that kB T ≪ ϵf .
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