Consider the time-independent one-dimensional Schrdinger equation when the potential function is symmetric about the origin, i.e., when

Consider the time-independent one-dimensional Schrödinger equation when the potential function is symmetric about the origin, i.e., when U(x) = U(–x).

(a) Show that if ψ(x) is a solution of the Schrödinger equation with energy E, then ψ(–x) is also a solution with the same energy E, and that, therefore, ψ(x) and ψ(–x) can differ by only a multiplicative constant.

(b) Write ψ(x) = Cψ(–x), and show that C = ±1. Note that C = +1 means that ψ(x) is an even function of x, and C = –1 means that ψ(x) is an odd function of x.