# Polyatomic quantum harmonic oscillator

1. May 11, 2009

### 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!

Last edited: May 11, 2009
2. May 12, 2009

### alxm

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.

3. May 13, 2009

### 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

4. May 13, 2009

### alxm

Yes, IIRC, it's separable in some relatively specific (but useful) circumstances, when you have the same masses and force constants all around, and when the system is all symmetrical.

It requires some clever changes of variables and such. You should probably be able to find the full derivation in any good textbook that handles phonons and lattice vibrations, but I don't remember it offhand. Someone else here might, though.