Expectation Value of Momentum for Wavepacket

1. Jan 16, 2014

torq123

1. The problem statement, all variables and given/known data

What is the average momentum for a packet corresponding to this normalizable wavefunction?

$\Psi(x) = C \phi(x) exp(ikx)$

C is a normalization constant and $\phi(x)$ is a real function.

2. Relevant equations
$\hat{p}\rightarrow -i\hbar\frac{d}{dx}$

3. The attempt at a solution

$\int\Psi(x)^{*}\Psi(x)dx = \int C^2 \phi(x)^{2}dx= 1$

Plugging in the momentum operator and using the chain rule:

$<\hat{p}> = \hbar k \int C^2 \phi(x)^2 dx - i \hbar \int C^2 \phi^{'}\phi dx$

The second term is always imaginary since $\phi(x)$ is real, so I said the momentum is $\hbar k$ which I think might be right, but for the wrong reasons? I didn't think Hermetian operators could give imaginary expectation values...

2. Jan 16, 2014

strangerep

Try working on the 2nd term a bit more. Hint: use integration by parts.

3. Jan 16, 2014

torq123

Are you saying that the second term must be zero since $\phi$ vanishes at ±∞ and the integral evaluates to $\phi^2(x)/2$? That makes sense to me.

4. Jan 16, 2014

strangerep

That's the idea.

5. Jan 16, 2014

torq123

Awesome. Thanks for the help.