Stat phys problem

197
4
1. The problem statement, all variables and given/known data
I need to find ntropy S(E) for N independent HO's for both the classical and quantum cases,
using the [tex]\mu[/tex]-canoncal esemble

2. Relevant equations
For one classical HO
[tex]H={p^2 \over{2m}}+{m\omega^2 q^2 \over 2}[/tex]
For one quantum HO
[tex]H=\hbar \omega (n+{1\over 2})[/tex]


3. The attempt at a solution
I tried to find the partitn function for one HO.
[tex]W_1={1\over h^3}\int d^3q d^3p \delta(H-E)[/tex]
[tex]W_1={1\over h^3} [/tex] times volume of 6-sphere with radius E
[tex]W_1={1\over h^3} {{\pi^3 E^6}\over 6}[/tex]
So for N HOs
[tex]W={W_1^N \over {N!}}={1\over h^{3N}} {{\pi^{3N} E^{6N}}\over 6^N}[/tex]
Then I can use
[tex]S(E)=k ln(W)[/tex]
Can anyone verify if I'm on the right track??
 

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