Problem 2.1 – Griffith’s Intro to QM

Problem 2.1

Prove the following three theorems:

(a) For normalizable solutions, the separation constant $E$ must be real. Hint: Write $E$ (in Equation 2.7) as $E_0 + i \Gamma$ (with $E_0$ and $\Gamma$ real), and show that if Equation 1.20 is to hold for all $t$ , $\Gamma$ must be zero.

(b) The time-independent wave function $\psi (x)$ can always be taken to be real (unlike $\Psi (x,t)$ , 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 $\psi$ s that are real. Hint: If $\psi (x)$ satisfies Equation 2.5, for a given $E$ , so too does its complex conjugate, and hence also the real linear combinations $(\psi + \psi^*)$ and $(\psi - \psi^*)$ .

(c) If $V(x)$ is an even function (that is, $V(-x) = V(x)$ ) then $\psi (x)$ can always be taken to be either even or odd. Hint: If $\psi (x)$ satisfies Equation 2.5, for a given $E$ , so too does $\psi (-x)$ , and hence also the even and odd linear combinations $\psi (x) \pm \psi (-x)$ .