- #1

fog37

- 1,568

- 108

For the particle in the 1D box of width ##L##, the time invariant Schrodinger equation is cast in the form of the Hamiltonian operator and automatically leads to the energy eigenfunctions

$$\Psi(x) = \sqrt{\frac 2L} sin(n\pi x/L)$$

I know that these energy eigenfunctions ##\Psi(x)## have each a well defined energy and are NOT simultaneous eigenfunctions of the linear momentum operator ##\hat p##. The energy eigenfunctions ##\Psi(x)## are simultaneous eigenfunctions of the magnitude squared of the momentum operator ##\hat p##.

To find the eigenfunctions ##\Phi## of linear momentum operator, should we simply apply the operator ##-i \hbar \frac \partial {\partial x}## to the energy eigenfunctions? The result would be a new function that represents the linear momentum eigenfunction ##\Phi(x)##...

What other eigenfunctions can we find for the particle in the box? What other operators can we apply?

I know that the particle in the box does not have any "rotational attributes" so the angular momentum operator is not applicable...But what if we applied the angular momentum operator (in position space) to the functions ##\Psi(x)##? What would we obtain?

Thanks!

Fog37