Particle in a box: commuting energy and momentum operators

[mel11]

Hi,
I've been thinking about the following:
In an infinitely deep box a particle's energy operator can be written as E = p^2/2m, and the momentum operator as p = -i hbar dx. (particle moves in x direction)
I can see that the commutator of E and p is 0, so the operators commute, and should have a common set of eigenfunctions. But e.g. A sin(kx) with some A and k is an eigenfunction of E but not of p. I don't get where I'm going wrong.
Thanks for any answers!

 

[strangerep]

[...] A sin(kx) with some A and k is an eigenfunction of E but not of p.
I don't get where I'm going wrong.

Try thinking instead about functions like $e^{-ikx}$ ... :-)

[Edit: Hmm, I suppose I should give a better hint. Think about
both sin(kx) and cos(kx). Do they have the same energy eigenvalue?
If so, you have a degenerate case, so the eigenfunctions of one
operator are in general a linear combination of the (degenerate)
eigenfunctions of the other.]

 

[K^2]

Right. Commuting operators doesn't guarantee that all eigen states are the same. Only that there is a set of common eigen states.

And just for clarity, that isn't the energy operator. It's a Hamiltonian operator. Energy operator involves a time derivative.

$$H = \frac{p^2}{2m} + V(x)$$

$$E = i \hbar \frac{d}{dt}$$

With time-dependent Shroedinger Equation

$$H\Psi = E\Psi$$