Free eneryg of a harmonc oscillator (1 Viewer)

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tau = boltzmann constant times absolute temperature
A one dimensional harmonic oscillator has an infinite series of equally spaced energy states, with [itex] \epsilon_{s} = s \hbar \omega [/itex] where s is a positive integer or zero and omega is the calssical frequency of the oscillator.

a) Find the free energy of the system

b) Find the entropy an Heat capacity

well ok the parititon function here is

[tex] Z = \sum_{s=0}^{\infty} \exp(\epsilon_{s}/ \tau) = 1 + e^{\frac{\hbar \omega}{\tau}} + e^{\frac{2\hbar \omega}{\tau}} + e^{3\frac{\hbar \omega}{\tau}} + ... = \frac{e^{\frac{\hbar \omega}{\tau}}}{e^{\frac{\hbar \omega}{\tau}} -1}[/tex]

so to find the free energy i simply have to find the natural log of Z and multiply that by -tau, which becomes
[tex] -\tau \left(\frac{\hbar \omega}{\tau} - \log (e^{\frac{\hbar \omega}{\tau}} -1 )\right) [/tex]

can this simplify furhter though?

b) to find the netropy we simply find the partial derivative of the free energy wrt tau while holding V contsnat and we get the negative entrpy

for the specific heat capacaity i am not usre....
isnt [itex] C_{V} = \frac{3}{2} [/itex] since N = 1 as there is only one one dimension??

Please help

thank you for the help!

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