homework and exercises - Simple coupled quantum harmonic oscillator

I am having trouble finding the eigenvalues for the Hamiltonian

$$H = \frac{P_1^2}{2M} + \frac{P_2^2}{2m} + \frac{K}{2}x_1^2 + \frac{k}{2}(x_1 - x_2)^2$$

Even though I can find a basis where the $x_1$ and $x_2$ coordinates are decoupled I then get products of momenta in my new Hamiltonian. I think this problem should be transformable into two decoupled oscillators. Am I wrong about that?

Answer

Structurally, it looks like this:

• Start by rescaling one or both of the position/momenta pairs such that the kinetic-energy term has both masses equal (while also retaining $[x_i,p_i]=i\hbar$), and


• then find a rotation in the rescaled $x_1,x_2$ plane that will eliminate the coupling terms.


• That rigid rotation will be mirrored in the momentum plane, but because the masses are symmetric, it will no longer introduce momentum couplings.

Then you're done - you've got two decoupled oscillators whose position and momentum variables are canonically conjugate to each other and given as explicit linear combinations of the old ones.