Suppose I have a harmonic oscillator with frequency $\Omega_1$ and another one with frequency $\Omega_2$. Is there a simple relationship between the eigenstates of the two? Especially, how would the ground-state of one of them be expressed in terms of eigenstates of the other one?
An application of my question would be a harmonic oscillator whose frequency can be controlled. Suppose then I start out in the ground state and then suddenly change the frequency. I'd expect that I'm then not in a ground state of the (new) oscillator any more, and I'd be interested in the time evolution of my state. For that, I need to do a basis transform of my groundstate.
The problem seems basic enough to me that there should be previous work done on it. A brute force solution would probably be to perform integrals over the eigenstates in real space, but I have hope that an algebraic solution in terms of creation and destruction operators exists.