Simultaneous observables for hydrogen

[bobred]

Homework Statement

Is there a state that has definite non-zero values of $E, L^2$ and $L_x$

Homework Equations

$L^2$ and $L_z$ commute with the Hamiltonian so we can find eigenfunctions for these

The Attempt at a Solution

I would say that there is a state with simultaneous eigenfunctions of $L_x,L_y,L_z$ and $L^2$, but with eigenvalues equal to zero. This being the state with $l=0$ and $m=0$, so there are no definite non-zero values of $E, L^2$ and $L_x$. For other states $L_x,L_y,L_z$ and $L^2$ do not commute.

 

[TSny]

Hello.

Something to think about. Should the z axis be special in the hydrogen atom? That is, if there exist states with definite non-zero eigenvalues of $E, L^2,$ and $L_z$, why shouldn't there exist states with definite non-zero eigenvalues of $E, L^2,$ and $L_x$?

Suppose you had a wavefunction $\psi(r, \theta, \phi)$ that represents an eigenstate of $E, L^2,$ and $L_z$. Can you think of how you could transform $\psi(r, \theta, \phi)$ into another function $\psi'(r, \theta, \phi)$that would be an eigenstate of $E, L^2,$ and $L_x$ with the same eigenvalues for $E$ and $L^2$ and with an eigenvalue of $L_x$ equal to the eigenvalue that $\psi$ had for $L_z$?

[Edit: It might be easier to think in terms of Cartesian coordinates $\psi(x, y, z)]$

 

[bobred]

Thanks, z is an arbitrary choice.