# Stat mech: partition functions for N distinguishable harmonic oscill-

1. Mar 4, 2013

### kd215

1. The problem statement, all variables and given/known data
Consider a system of N distinguishable, non-interacting harmonic oscillators. The Hamiltonian is given (shown below).
Assuming that the oscillators obey Schrodinger's equation, determine the canonical partition function for the system. Then assume the oscillators obey Newton's equations of motion and determine the partition function for the system.

2. Relevant equations
$H$ = Ʃ$\frac{p_i^2}{2m}$ +Ʃ$\frac{1}{2}k|r_i-r_i^o|^2$

$r_i^o$ is the equilibrium position of the ith oscillator particle

3. The attempt at a solution

So for the first question, Schrodinger's equation means that we can only exactly know the position or the momentum for a particle, right? I've been trying to factorize the partition function as two functions that are dependent on either position or momenta. So if I do this correctly, would I obtain two functions that are both partition functions? And the one that's a function of momenta would be what I am supposed to find in the second question?

I'm just now even sure if I'm interpreting this correctly. I feel like my partition function that is a function of momenta is just the Boltzmann weighted sum.

Please just let me know if I'm even thinking in the right direction.
Thanks!

2. Mar 5, 2013

### clamtrox

I don't understand what Schrodinger's equation has to do with that.

Do you remember that the energy eigenvalues for a quantum harmonic oscillators are $E_n = \hbar \omega (\frac{1}{2}+n)$ ?