Polyatomic quantum harmonic oscillator

GOLDandBRONZE
Hi!
Would anyone be able to point me toward a detailed explanation of determining the Hamiltonian of a polyatomic quantum oscillator? My current text does not explain the change of coordinates ("using normal coordinates or normal modes") in detail.
All I can find is material on a diatomic quantum oscillator...

Thanks!

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Well a one-dimensional Harmonic oscillator has the Hamiltonian:
$$H = \frac{1}{2m}\nabla^2 + \frac{1}{2}m\omega^2x^2$$

Generalizing this depends on how many 'springs' you want. If every particle is connected to every other particle with a harmonic potential:
$$H = \sum_{i=0}^N\frac{1}{2m}\nabla_i^2 +\sum_{i=0}^N\sum_{j>i}^N \frac{1}{2}m\omega^2(x_i - x_j)^2$$

Where N is the number of particles and x denotes their coordinates.

GOLDandBRONZE
Thanks for your reply. That certainly makes sense, but from what I understand it is possible to make that equation separable, through a change of coordinates or something. The form you wrote will have cross-multiplied terms. How would you go about showing that it can be transformed into a separable Hamiltonian?

Thanks again