Exploring the Degeneracy and Effects of Stark and Zeeman in Atomic Systems

[learning_phys]

in the zeeman effect, there are degeneracy levels due to the the magnetic quantum number which when placed in a magnetic field, the degeneracy is broken.

is there a way to see the degeneracy of the angular momentum? since Energy is only effected by the principal quantum number, there should be a degeneracy for electrons in the n=2 shell, which can have angular momentum l=0 and l=1 right? is there an effect (like the zeeman effect) that breaks this energy degeneracy?

also, what quantum number does the start effect break in degeneracy? is it also the magnetic quantum number?

 

[Barny]

It's been a while since I have done the stark effect, so am a tiny bit sketchy...

However I was under the impression that the stark effect did break the degeneracy of the angular momentum states.

Having said this I know for a fact that when using degenerate perturbation theory to calculate the energy shift of the n=2 states of hydrogen, caused by a small dc electric field, the only two matrix elements which are non zero are between the n=2,l=0,m=0 and the n=2,l=1,m=0 states. This is the first order correction to the energy and is linear in the electric field strength.

As I said though, this shouldn't be taken as gospel as its been more then a while, more just a nudge in a certain direction.

Kind Regards

Barny

 

[learning_phys]

i'm confused as how an electric field influences the angular momentum? I'm guessing it's effects the electron's orbit. however, shouldn't the zeeman effect also break angular momentum degeneracies as well?

 

[Barny]

The electric field creates a perturbation, this takes the form of $$\hat{H_{p}}=-e\textbf{E.r}$$

If you plug this into the first order degenerate perturbation theory on the n=2 states of Hydrogen you end up with a linear shift in the energy levels equal to 3eaE where a is the bohr radius, e is the electron charge and E the electric field magnitude. This can be "poked at" classically as the dipole energy of an atom in the presence of an Electric field.

Now, I really am quite rusty on this next bit, and it should be verified/shot down by someone else. When calculating the energy shift due to the Zeeman effect you have to consider weather you are in the weak field limit or the strong field limit (Paschen-Back effect), relative to the energy of spin-orbit coupling for that particular atom.

In the weak field limit the Zeeman perturbation only slightly shifts the energy of the total angular momentum operator and has an energy correction of $$\DeltaE=\mu_{b}M_{j}B$$

However in the Strong field limit, when the perturbation must be applied before the spin-orbit correction, as it is greater in energy, we end up with a correction to the energy of $$\DeltaE=\mu_{b}(M_{L}+g_{s}M_{s})B$$.

Which, I think means the Zeeman effect does break the degeneracy of the angular momentum states, but only in the strong field limit.

Does that help at all?

Kind Regards

Barny

 

[learning_phys]

Thanks for the response. I guess I'm still confused about the stark effect

note: http://upload.wikimedia.org/wikipedia/en/4/44/Stark_splitting.png

It says: "First and second order Stark shifts in hydrogen, magnetic quantum number: m = 1. Each n-level consists of n-1 degenerate sublevels; application of an electric field breaks the degeneracy."

Lets take n=8 and m=1, the graph says there would be a break in degeneracy by 7 fold. Is this coming form the angular momentum quantum number? Because of m=1 and n=8, then the only thing that changes would be the angular momentum right? the l=0,1,2,...,7 states.

Would the same effect happen for zeeman effect?