Coupled Quantum Harmonic Oscillator

[stumpedstuden]

Homework Statement

I need to transform the Hamiltonian of a coupled Harmonic Oscillator into the sum of two decoupled Hamiltonians (non-interacting oscillators).

Homework Equations

H = H1 + H2 + qxy, where H1=0.5*m*omega^2*x^2+0.5m^-1P_x^2 and H2=0.5*m*omega^2*y^2+0.5m^-1P_y^2, and q is the coupling constant

The Attempt at a Solution

I have tried a number of variable transformations, etc as well as attempted to complete the square to deive the proper variables that will allow me to rewrite the Hamiltonian properly. All without success. I thought completing the square and making the proper substitions would work but I still end up with a coupled term. Once this step is complete solviing the rest of the problem should be pretty straightforward.

Any help that gets me started would be greatly appreciated.

 

[vela]

To get rid of the cross term, you don't want to complete the square; you need to use a rotation of the coordinates.