Sorry for being late, I didn't notice that I had to post my result in the blogger, another student alerted me, so I thank him for telling me, here is the problem and my result.
The hamiltonian of the system is similar to the Hamiltonian of a harmonic oscillator but in this case we consider multiple frecuencies, for each frecuency we consider a vector p_j and a position vector q_j, so, the partition function is calculated using the formula we have in our lecture notes, here is the results.
Student: Carlos Daniel Cervantes García.
Your Hamiltonian function is not correct
ResponderBorrarIt misses the summation over the particles. It could be the single particle
Hamiltonian, but then you should have said so, and it would not make sense
to use vector notation like "\vec p" and "\vec q".
The partition function seems to be correct.