QM Questions: Operators, Eigenfunctions, and Hydrogen-like Atoms Explained"

[deep582]

I just had a couple quick questions while I'm reviewing for a test...

If two operators commute, what can we say about their eigenfunctions?

x commutes with pz, correct?

I understand that the energy for a hydrogen-like atom depends on n according to the equation ((-z^2 μe^4)/(2(4πϵ_o )^2 ℏ^2 ) 1/n^2 ), but why does it not depend on l also? Isn't a 3p electron supposed to have more energy than a 3s electron? Or is that only in many electron atoms?

Thanks, any help appreciated

 

[Hootenanny]

If two operators commute, what can we say about their eigenfunctions?

If two [hermitian] operators commute, then there exists a common eigenbasis for the two operators.

x commutes with pz, correct?

Indeed.

I understand that the energy for a hydrogen-like atom depends on n according to the equation ((-z^2 μe^4)/(2(4πϵ_o )^2 ℏ^2 ) 1/n^2 ), but why does it not depend on l also? Isn't a 3p electron supposed to have more energy than a 3s electron? Or is that only in many electron atoms?

You are correct. Since the Hydrogen atom has a purely Coulomb potential, the energy levels are indeed degenerate with respect to l. However, as you correctly note, this is not the case with many-electron atoms.

 

[borgwal]

Actually, even in hydrogen the 3p and 3s (or 2p and 2s) levels are not quite degenerate because of QED effects: the Lamb shift of 2p vs 2s being the most famous example.

But leaving out QED effects the levels with the same n and different l are indeed degenerate in H: this has to do with a symmetry that the H atom has, but no other: the Laplace-Runge-Lenz vector is conserved for an exact 1/r potential.

 

[jtbell]

Attention mna skt!

mna skt, I've moved your homework question to a new thread in one of the homework forums... click the following link to go to it:

https://www.physicsforums.com/showthread.php?t=271329

Everybody else, please carry on. I'll try to remember to delete this post in a few days.