# Canonical Ensemble and potential energy

1. Apr 2, 2013

### Abigale

Hi guys ,

I regard an ideal diatomic gas which is in a Volume x,z and got a angular $\phi$:
$0 \le x \le L$, $~~~~~~~~~~~~ 0 \le z \le \infty$, $~~~~~~~~~~~~0 \le \phi \le 2 \pi$

The hamiltonian for the single particle is:
$~~~~~~H= \frac{p^{2}_{z} + p^{2}_{x} }{2M} + \frac{P^{2}_{\phi}}{2I}+mgz$
($m$ is the mass of a diatomic particle, $g$ the gravitational constant, $p_{x,y,\phi}$ are momentums and angular momentum.)

I am interessted in the average value of th potential energy!

At first i have callculated the canonical partition function for a single particle: $\text{z}=\frac{-L^{2} N^{2} k^{3} T^{3} I }{\pi \hbar^{5}}$
The partition function for the whole system can be calculated by: $Z=\frac{\text{z}^{N}}{N!}$
The average internal energy of the canonical System can be callculated with the formula: $\bar{E}=(\frac{\partial \ln{Z}}{\partial \beta})$.

But the problem is, I want to callculate the average of the potential energy.
There should be two capabilities of callculating it.

The one solution is with the average internal energy,
the other with the definition of the expected value.

With the definition of expected value, i can callculate the average potential energy of a single particle.

$\bar{E}_{potential}= \frac{\int \int d^{3N} q ~ d^{3N} p ~~~ mgz~~ \exp{- \beta H(q,p) } } { \int \int d^{3N} q ~ d^{3N} p ~~~~~ \exp{- \beta H(q,p) }}$

But i don't know how to callculate the average potential energy for the whole System of $N$ particles.