Prove the following three theorems:
(a) For normalizable solutions, the separation constant E must be real. Write E (in Equation 2.7) as E₀ + iΓ (with E₀ and Γ real), and show that if Equation 1.20 is to hold for all t, Γ must be zero.

(b) The time-independent wave function Ψ (x) can always be taken to be real (unlike Ψ (x, t), which is necessarily complex). This doesn’t mean that every solution to the time-independent Schrödinger equation is real; what it says is that if you’ve got one that is not, it can always be expressed as a linear combination of solutions (with the same energy) that are. So you might as well stick to s that are real.

(c) If V(x) is an even function (that is V (-x) = V (x) ), then Ψ (x) can always be taken to be either even or odd.

(x, t) = (x)e-iEt/h (2.7)