Prove the following three theorems:
(a) For normalizable solutions, the separation constant E must be real. Hint: 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 ψs that are real. Hint: If ψ(x) satisfies Equation 2.5, for a given E, so too does its complex conjugate, and hence also the real linear combinations (ψ+ψ∗) and i(ψ−ψ∗).
(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. Hint: If ψ(x) satisfies Equation 2.5, for a given E, so too does ψ(−x), and hence also the even and odd linear combinations ψ(x)±ψ(−x).