Answer to Sakurai 2.22 makes no sense

[QFT25]

Homework Statement

Consider a particle of mass m subject to a one-dimensional potential of the following form:

(1/2)k*x^2 x>0

Infinity for 0<x

Find <x^2> for the ground state

Homework Equations

<x^2>= <Psi|x^2|Psi>

The Attempt at a Solution

I know the answer is 3*h/(4*m*w) but to me that makes no sense. That answer was derived using the first excited state of the ground state for the potential in which there is no infinite wall at x=0. That state is normalized for -Infinity to Infinity. But in our circumstance the motion can only be from 0 to Infinity. That means I need to re-normalize the wave function so that it it's norm is 1 from 0 to Infinity. When I do that and calculate the expectation value I get 3*h/(2*m*w). I don't understand how you can use a wave function which is not normalized from 0 to Infinity to calculate < x^2> from 0 to infinity

 

[TSny]

Welcome to PF!

I think you are correct that the wavefunction should be renormalized and the answer for <x²> is (3ħ)/(2mω).

 

[DrClaude]

I know the answer is 3*h/(4*m*w) but to me that makes no sense.

Where did you get that answer?

 

[QFT25]

Where did you get that answer?

The solutions manual. I think it has an error.

 

[DrClaude]

The solutions manual. I think it has an error.

I agree. The wave function has to be renormalized.