Introduction to Quantum Mechanics · Problem 1.5

The Wave Function — David J. Griffiths

Physics — Quantum Mechanics · expectation_values

Problem

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?