Expectation value of a wave function

[bjogae]

Homework Statement

The wave function of a state is Psi(x)= N*a(x)exp(i*p0*x/h)where a(x) is a quadratically integrable real valued function Show that the expectation value of the function is p0.

Homework Equations

The Attempt at a Solution

The only thing I'm having a problem with is how to integrate the square of the wavefunction so that I could normalize N and put the operator p=-ih(d/dx) in the integral. So I think I know what to do, just not sure how to do it.

 

[Feldoh]

First off the expectation value of what? Position?

Also what is p0, just a real-valued constant?

 

[bjogae]

It's the expectation value of the momentum I'm after. Sorry, forgot to put that in. And yes, p0 is a real-valued constant.

 

[Feldoh]

All right so to normalize the wave function the wave function squared (probability) must be one. In this case you need to integrate $\Psi^{*} \Psi$ over all possible values (-inf to inf) and set that equal to 1, then solve for N.

 

[bjogae]

But that's the thing. I have no idea of how to do the integral.

 

[Feldoh]

Well what have you tried? Start by setting up the integral and working as far as you can.

 

[bjogae]

well I get stuck when I'm supposed to integrate a(x)^2*exp(i*2*p0*x/h). How do I do that?

 

[Feldoh]

$$1 = \int_{- \infty}^{\infty} \Psi^{*} \Psi dx$$

Psi* is the complex conjugate of Psi. Do you know how to take a complex conjugate?

exp(i*2*p0*x/h) makes it seem like you're trying to do:

$$1 = \int_{- \infty}^{\infty} \Psi \Psi dx$$

When you need to be doing:

$$1 = \int_{- \infty}^{\infty} \Psi^{*} \Psi dx$$

 

[Feldoh]

For instance the complex conjugate of $$f(x) = A e^{i x}$$ is $$f(x)^{*} = A e^{-i x}$$ where A is a real constant.

 

[bjogae]

Thanks, I think I got it right.