# Simultaneous observables for hydrogen

## 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
Homework Helper
Gold Member
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)]##

Last edited:
• 1 person
Thanks, z is an arbitrary choice.