Particle in an infinite potential box- expected values of energy

[Rorshach]

Could anybody please respond?

 

[TSny]

Hello, Rorshach. I don't understand "by leveraging development coefficients" in part (c), so I don't think I can help much.

You've already calculated $<\hat{p}^2>$, so once you determine $<p>$ you will easily be able to get $\Delta p$.

Note that the wavefunction $\psi$ is an even function about $x = a/2$. Since $\hat{p}$ is proportional to the derivative operator, you should be able to argue that $\hat{p} \psi$ is an odd function about $x = a/2$. So, if you set up the integral for $<\hat{p}>$ you should be able to conclude what the answer is without actually carrying out the integration. But that doesn't appear to be the way you are expected to do it.

I will bring this problem to the attention of some of the other helpers.

I guess you don't have access to the person who made the question. It would be nice if you could get clarification on what he/she meant for you do do here.

 

[Rorshach]

Unfortunately, I have no contact with the person who made this question. I also don't want to integrate anything, since it was clearly stated in the problem to not to do it, however user tannerbk gave a formula a few replies above for expectated value for hamiltonian <H>, so maybe there is an analogic one for <p>?

 

[TSny]

Maybe they want something like this. You know that you can expand your wave function in terms of the energy eigenstates:

$\psi(x) = \sum c_n \sin(k_n x)$ where $k_n = n\pi/a$.

Write each $\sin(k_n x)$ as $\frac{1}{2i}(e^{ik_nx} - e^{-ik_nx})$.

Suppose you could show that each exponential $e^{ik_nx}$ is a momentum eigenstate corresponding to momentum $\hbar k_n$ while each exponential $e^{-ik_nx}$ is a momentum eigenstate corresponding to momentum $-\hbar k_n$.

Can you then construct an argument showing $<\hat{p}> = 0$?

Note, however, that this type of argument might not be sound. There are difficulties with defining a momentum operator for the particle in a box. The functions $e^{ik_nx}$ don't satisfy the boundary conditions of the particle in a box, so I'm not sure they qualify as legitimate states for the particle in a box.

See for example http://academic.reed.edu/physics/fa...ous Essays/Generalized Momentum Operators.pdf

or this thread https://www.physicsforums.com/showthread.php?t=285694