Question

Consider a one-dimensional real-space wave-function ψ(x) and let Pˆ denote the parity operator such that P ψˆ (x) = ψ(−x).

a)Starting from the Rodrigues formula for Hermitian polynomials, Hn(y) = (−1)^n*e^y^2*(d^n/dy^n)e^-y^2 with n ∈ N, show that the eigenfunctions ψn(x) of the one-dimensional harmonic oscillator, with mass m and frequency ω, are also eigenfunctions of the parity operator. What are the eigenvalues?

b)Define the operator Π = exp [  iπ (( 1 /2α) *pˆ 2 + α xˆ 2/ (h/2π)^2-1/2)] , α ∈ R +, where ˆx and ˆp denote the position and momentum operators. Show that Π is a parity ˆ operator.

Answer 1