Problem 1.5 – Griffith’s Intro to QM

Problem 1.5

Consider the wave function

$\Psi (x,t) = Ae^{-\lambda |x|}e^{-i \omega t}$

where $A$ , $\lambda$ , and $\omega$ are positive real constants. (We’ll see in Chapter 2 for what potential (V) this wave function satisfies the Schrödinger equation.)

(a) Normalize $\Psi$ .
(b) Determine the expectation values of $x$ and $x^2$ .
(c) Find the standard deviation of $x$ . Sketch the graph of $|\Psi|^2$ , as a function of $x$ , and mark the points $(\langle x \rangle + \sigma)$ and $(\langle x \rangle - \sigma)$ , to illustrate the sense in which $\sigma$ represents the “spread” in $x$ . What is the probability that the particle would be found outside this range?