Hello everyone. I have the following problem.
I have The Hamiltonian of the 1D Harmonic Oscillator (hbar=m=w=1)
H= x^2+p^2
with the known solutions
psi(x)=exp(-x^2/2)*H_n(x)
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.
Hi Here is my opinion ,I
Hi
Here is my opinion ,I don‘t think the new hamiltonian is H=s^2+t^2, because both x and p are q-numbers. After a transformation (and your matrix is not a unitary matrix), the formula of the new hamiltonian will not identical with the old one.