Introduction to Quantum Mechanics · Problem 1.9

The Wave Function — David J. Griffiths

Physics — Quantum Mechanics · uncertainty_principle

Problem

A particle of mass

m

has the wave function

\Psi(x, t) = Ae^{-a\left[(mx^2/\hbar) + it\right]},

where

A

and

a

are positive real constants. (a) Find

A

. (b) For what potential energy function,

V(x)

, is this a solution to the Schrödinger equation? (c) Calculate the expectation values of

x

x^2

p

, and

p^2

. (d) Find

\sigma_x

and

\sigma_p

. Is their product consistent with the uncertainty principle?