Can we find momentum in finite square well potential?

Question

Can we find momentum in finite square well potential? If so how can we find it? Does the eigenfunction of momentum operator is same as eigenfunction of Hamiltonian operator?

Answer 1

If the potential well is finite, then the Hilbert space is the same as for the free particle: $L^2(R)$. The momentum operator is the usual one. Its improper eigenvectors are not eigenvectors of the Hamiltonian.