Expectation Value of Momentum for Wavepacket

[torq123]

Homework Statement

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.

Homework Equations

$\hat{p}\rightarrow -i\hbar\frac{d}{dx}$

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...

 

[strangerep]

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

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...

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

 

[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.

 

[strangerep]

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

That's the idea.

 

[torq123]

Awesome. Thanks for the help.