What is the expected value of x^2 for a wave packet in momentum representation?

[rgalvao]

Homework Statement

My teacher made up this question, but I think there's something wrong.

Consider the wave packet in momentum representation defined by Φ(p)=N if -P/2<p<P/2 and Φ(p)=0 at any other point. Determine Ψ(x) and uncertainties Δp and Δx.

Homework Equations

Fourier trick and stuff...

The Attempt at a Solution

I found Ψ(x)=(2ħN/x√2πħ)sin(Px/2ħ), where N=±√1/P
<x>=0, <p>=0, <p^2>=(P^2)/12

But when I try to calculate <x^2>, I get a strange integral, which goes to infinity. Am I doing anything wrong?

 

[BvU]

Hello rgalvao, welcome to PF !

I see you didn't get a reply yet, so perhaps I can put in my five cents:

Your wave function $sin x\over x$ is the Fourier transform of a rectangular function and you can indeed see that $\int \Psi^* \,x \, \Psi dx$ yields zero, but $\int \Psi^* \, x^2 \, \Psi dx$ diverges.

I don't see anything wrong with what you do. The wave function simply doesn't fall off fast enough with |x| to give a finite expectation value for x².
Let us know if you or teacher finds otherwise (i.e. correct me if I am wrong ... ) !

 

[mfb]

I'm not even sure about <x>. Sure, you can argue with symmetry, but the integral is not well-defined.