Question

Consider two particles of masses m₁ ≠ m₂ interacting via the Hamiltonian

(a) Find the exact solutions.

(b) Sketch the spectrum in the weak coupling limit K << \(\mu \omega^2\), where p is the reduced mass.

Answer 1

(a) Let

It can be reduced to two independent oscillator equations with energy states and wave functions

(b) For K << \(\mu \omega^2\), if we can take

This means that the degeneracy of the Nth energy level is N + 1:

If K is not so small and we have to take

then the energy levels with the same m will move upward by

and so the degeneracy is destroyed.