Spin orbit coupling/orbital degeneracy

[granpa]

disregarding the hyperfine structure and the lamb shift for the moment, spin orbit coupling says that 2S_1/2 and 2P_1/2 have the same energy. do 2S₀ and 2P₀ have the same energy?

if 2P splits into 3/2 and 1/2 does the 3D orbital split into 5/2, 3/2, and 1/2 and the 4F orbital into 7/2, 5/2, 3/2, and 1/2? google gets plenty of hits but they only talk about 2P.


this is just my own personal interest but you can move it to homework if you are so inclined.

 

[Meir Achuz]

The subscript1/2 is the total J=L+S (QM addition). That limits J to L+1/2 and L-1/2.
There is no 0 subscrlpt for one electron.
A D orbital (L=2) can have only 3/2 or 5/2.
An F orbital (L=3) can have only 5/2 or 7/2.
The G...

 

[granpa]

thank you for responding.

the web says that those numbers only hold if the number of electrons is odd. I was wondering what happens if the number of electrons is even. that is why I asked about 2S₀.

 

[malawi_glenn]

"the web says" is a pretty non specific statement, why not state where you read it? Maybe you have misunderstood something?

Which atoms are you considering?

nLj referes to single electron-orbitals, a single electron orbital can not have j = half integer. 2So does not exists as single orbital, are you suggesting 2So to be notation for what??

 

[granpa]

because I read it on many sites. they all say the same thing. that the formula only works of odd number of electrons. I'll try to find the exact source.

 

[granpa]

I know that the zeeman effect is different for even and odd and all the sites talk about 2P1/2 and 2S1/2. maybe I assumed it was different. if the number of electrons was even then why would the S orbital have spin 1/2?

 

[granpa]

from the web:
The nlj quantum numbers are, then, again appropriate for a single electron outside closed subshells.

so what do I use for closed shells or nonclosed shells with more than one electron? especially for calculating fine structure.

 

[granpa]

https://www.physicsforums.com/showthread.php?t=114363

They are still valid. But that's not the point. The point is whether the *perturbation* hamiltonian commutes with these operators. One finds that the spin orbit hamiltonian commutes with L^2, S^2, J^2 and J_z, but not with L_z and S_z. Therefore, m_s and m_l are not good quantum numbers but must be replaced by m_j and j. So the states of definite energy when the spin-orbit interaction are taken into account are the states labelled by the quantum numbers l,s,j, m_j (instead of the usual l,m_l,s,m_s that one uses to label the unperturbed hydrogenic wavefunctions).

unperturbed hydrogenic wavefunctions?

 

[granpa]

http://www.pha.jhu.edu/~rt19/hydro/node9.html [Broken]

http://www.pha.jhu.edu/~rt19/hydro/img194.gif [Broken]

 

[malawi_glenn]

yes what about them, and what about the picture? what do you want?

 

[granpa]

I thought others might find it helpful.

 

[malawi_glenn]

but you explicit ask "unperturbed hydrogenic wavefunctions?"

 

[granpa]

that was 3 days ago. I figured you just weren't going to answer.

I guess that 'unperturbed hydrogenic wavefunction' just means the simplified hydrogen model without taking spin-orbit coupling into account? or does it mean something else completely?

 

[malawi_glenn]

yes, that is correct.

 

[nrqed]

nLj referes to single electron-orbitals, a single electron orbital can not have j = half integer.

?? L is an integer and S= 1/2 so j must be a half integer for a single electron orbital

 

[malawi_glenn]

?? L is an integer and S= 1/2 so j must be a half integer for a single electron orbital

LOL correct, i did too many "not" in one sentence ;-) Sorry