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 + iГ (with E0 andГ real), and show that if Equation 1.20 is to hold for all t, Г 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 Schrodinger 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 ψ’s 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)
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