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!

 

[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.

 

[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

 

[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.