Partition function for harmonic oscillators

[Silviu]

Homework Statement

Calculate the partition function, the entropy and the heat capacity of a system of N independent harmonic oscillators, with hamiltonian $H = \sum_1^n(p_i^2+\omega^2q_i^2)$

Homework Equations

$Z = \sum_E e^{-E/kT}$

The Attempt at a Solution

I am not really sure what to do. From what I see this is not a quantum oscillator (and we haven't covered that in class anyway) so the E in the formula for partition function would have any value from 0 to infinity so It would turn into an integral. But this can be argued for a single classical harmonic oscillator, too, so I don't know where to use the fact that there are N of them.

 

[BvU]

It looks like the Hamiltonian for a quantum mechanical HO, I would say. $m$ has been set to 1 for convenience. The link tells you energy levels, which you will need.

What are you summing over to get your partition function ? I don't see the oscillators (for example if there are 100 oscillators, how does that end up in your summation?)

Be sure to keep the numbering of the energy levels of a single oscillator and the numbering of the oscillators themselves well distinguished.

 

[vela]

Homework Statement

Calculate the partition function, the entropy and the heat capacity of a system of N independent harmonic oscillators, with hamiltonian $H = \sum_1^n(p_i^2+\omega^2q_i^2)$

Homework Equations

$Z = \sum_E e^{-E/kT}$

The Attempt at a Solution

I am not really sure what to do. From what I see this is not a quantum oscillator (and we haven't covered that in class anyway) so the E in the formula for partition function would have any value from 0 to infinity so It would turn into an integral. But this can be argued for a single classical harmonic oscillator, too, so I don't know where to use the fact that there are N of them.

Check out https://en.wikipedia.org/wiki/Partition_function_(statistical_mechanics) and read about the canonical partition function. The fact you have $N$ oscillators is reflected in the Hamiltonian.