Description
Problem 1:
This is a quantum statistical mechanics problem.
Consider N identical bosons, each of which is treated as an isotropic three-dimensional
harmonic oscillator.
Using the recursion relation from P. Borrmann and G. Franke, J. Chem. Phys. 98, 2484
(1993) [you can access this paper through the OU library] for the partition function in the
canonical ensemble, write a little code that plots (or allows you to plot) the internal energy
per particle as a function of the temperature for N = 10.
WARNING 1: I suggest that you test your code with N = 2, 3, · · · to get results quickly.
For large N, the calculations may take hours (please do not run this for N = 1, 000).
WARNING 2: If you set your loops up incorrectly, you might end up writing a code that
runs infinitely long or eats up tremendous amounts of memory. Please carefully test your
code to ensure that this is not happening.
Hint: How to approach this problem?
• Step 1: Find a compact analytical expression for S(k) defined in Eq. (2) of the paper
[you can get rid of the infinite sum(s)—note that the subscript j is hiding three
infinite sums and note that k is an integer and not Boltzmann’s constant…].
• Step 2: Implement the finite sum for Z(N) given in Eq. (1) of the paper. Note that Z
is used to denote the canonical partition function; in class, we used Q or QN . It can
be recognized that Z(N) is defined recursively—this is where you might encounter
problems if your code is not set up correctly… To give you a sense, my Mathematica
notebook is seven lines long/short.
Problem 2:
In class we discussed configuration integrals. For three particles, e.g., there exist four different configuration integrals, three with two “connections” (two of the “circled” 1, 2, and
3 are connected) and one with three “connections” (the three “circled” 1, 2, and 3 are
connected). We can refer to this loosely as two “topologically distinct” classes, containing
respectively three graphs and one graph.
For four particles, work out the number of topologically distinct classes as well as the
number of graphs per class. Do not just write down your answers—please also explain how
you arrived at your answers.
Problem 3:
This problem considers the cluster expansion using classical statistical mechanics.
(a) Calculate a2 for the hard sphere potential, which is given by v(r) = ∞ for r < σ and
v(r) = 0 otherwise.
(b) Calculate a2 for the square well potential, which is given v(r) = ∞ for r < σ, v(r) = −
for σ ≤ r ≤ ασ ( > 0 and α > 1), and v(r) = 0 otherwise.
(c) What are realistic parameters for σ, , and α? And how do you know? What is the
temperature regime in which you might expect the description to work reasonably well.
Problem 4:
In looking at the virial equation of state, we faced the following mathematical problem.
Given the expansions
x = t + a2t
2 + a3t
3 + · · · (1)
and
y = t + b2t
2 + b3t
3 + · · · , (2)
where a2, a3, · · · and b2, b3, · · · are assumed to be known, one needs the expansion coefficients
An that appear in the expansion
y = x + A2x
2 + A3x
3 + · · · (3)
Task of this problem: Obtain explicit expressions for A2, A3, and A4.
Note: Obtaining expressions for all An is, in general, non-trivial.