Quadrature operators and Hermite polynomials
Hello everyone. I have the following problem.
I have The Hamiltonian of the 1D Harmonic Oscillator (hbar=m=w=1)
with the known solutions
where H_n are the Hermite polynomials of order n.
If I change the variables for the following ones (quadrature operators)
s=cos(y) x + sin(y) p
t=sin(y) x + cos(y) p
the new hamiltonian is H=s^2+t^2, so it is invariant. Now my question is: How can I change my old wavefunction psi(x) to the new space (s,t). I suposse that the new wavefunction must be something similar to the old ona, because the Hamiltonian is invariant, but I'm really don't sure.
Do you have any idea?
Thanks a lot.