Partition Function for N Quantum Oscillators

[Daniel Sellers]

Homework Statement

For 300 level Statistical Mechanics, we are asked to find the partition function for a Quantum Harmonic Oscillator with energy levels E(n) = hw(n+1/2). No big deal.

We are then asked to find the partition function N such oscillators. Here I am confused.

Homework Equations

Z_N = (Z^N)/N! Where Z is the partition function for a single oscillator or particle.

This equation shows up a lot when I look for information on partition functions for N particles, but it seems to only apply when the particles are indistinguishable, non-interacting, and unlikely to occupy the same energy levels, basically an ideal gas. A system of oscillators seems to meet only one of these conditions (indistinguishable).

Z_N = Z^N

This also shows up a lot but only in the context of distinguishable particles.

The Attempt at a Solution

I have tried to search for as many sources as possible and reason my way through this problem, but I can't come up with an answer in which I am confident.

Can anyone provide an answer and convince me that it is correct? Thanks

 

[Daniel Sellers]

Perhaps I'm overthinking this? If the oscillators are allowed to occupy the same energy levels then the partition function (which I understand to be the sum of probabilities of all possible states of the system) would simply be a string of statistical 'and' statements. So I could say that Z_N = Z^N?

Thoughts? Anyone want to tell me I'm wrong? Is there some subtlety I'm missing involving over-counting states with the same energy level distributions?