Partition function of electrons in a magnetic field

[MMS]

Hello everyone,

How can I calculate the partition function of N classical electrons (forgetting about the spin) in a box of volume V with Hamiltonian

(The Hamiltonian is missing a factor of 1/(2m))
?
I tried calculating the partition function of one electron first in the canonical ensemble but the integral seems to be quite hard to solve as I have 3 coordinates of momentum as well as 3 coordinates of vector potential but it doesn't seem to work...
Any ideas?

Thanks in advance.

 

[eljose79]

Hello there,

Calculating the partition function for N classical electrons in a box can indeed be a challenging task. However, there are a few steps you can follow to simplify the process.

First, let's define the Hamiltonian for one electron in the box as H = p^2/2m + V(x), where p is the momentum, m is the mass, and V(x) is the potential energy. As you mentioned, the Hamiltonian is missing a factor of 1/(2m), so let's include that in our calculations.

Next, we can use the canonical ensemble to calculate the partition function. This involves taking the trace of the Boltzmann factor e^(-βH), where β = 1/kT and k is the Boltzmann constant. In this case, the trace will involve integrating over all possible momentum and position states.

To simplify the integral, we can use the fact that the Hamiltonian is separable in terms of momentum and position. This means we can write the trace as a product of two integrals: one over momentum and one over position.

For the momentum integral, we can use the fact that the momentum states are discrete in a box (due to the quantization of momentum), so the integral becomes a sum over all possible momentum states.

For the position integral, we can use the fact that the potential energy is constant in a box, so the integral becomes a simple volume factor.

Putting all of this together, we end up with the partition function Z = (1/V)^N * Σ e^(-βp^2/2m), where the sum is over all possible momentum states.

This expression can be further simplified using the fact that the momentum states are equally spaced, so the sum can be replaced by an integral. This gives us Z = (1/V)^N * (2πmkT)^N/2.

I hope this helps you in your calculations. Good luck!