The Fine Structure of Hydrogen

[samjohnny]

Homework Statement

Homework Equations

N/a

The Attempt at a Solution

Hi all,

For part a) I simply discussed the fact that the contributions to the fine structure arise from the Darwin term, relativistic corrections to the kinetic energy, and the spin-orbital coupling with a bit of detail.

As for part b) I'm not entirely sure on how to proceed with it. Is it essentially asking that a diagram be sketched similar to the following:

The only difference really being that the states are to be labelled as $^{2s+1}L_j$.

In which case for part c) we find that there are 6 possible transitions from n=3 to n=2.

 

[mfb]

In which case for part c) we find that there are 6 possible transitions from n=3 to n=2.

Do you expect to see all those 6 transitions, if you consider the selection rules?

 

[blue_leaf77]

As for part b) I'm not entirely sure on how to proceed with it. Is it essentially asking that a diagram be sketched similar to the following:

Yes, that's right.
In part c) the question first asks about the possible transitions between n=3 and n=2, therefore every possible transition must be identified. Then the question also asks about how many transitions will be observed experimentally, for this you should consider that some transitions between different pair of levels coincide in frequency.

 

[samjohnny]

Thanks for the replies.

Just to clarify one point; in considering how many possible, and how many allowed, transitions there are, are we to consider transitions as being between each degenerate state, even if they are in the same energy level?

For example, consider the $3D_{5/2}$ state, are there two possible transitions (to the $[2P_{3/2}]$ and the $[2S_{1/2}, 2P_{1/2}]$ levels), or three possible transitions (to the $[2P_{3/2}]$, the $[2S_{1/2}]$ and the $[2P_{1/2}]$ states where we consider the latter two states separately despite their degeneracy)?

 

[blue_leaf77]

Is your considering the transition from $3D_{5/2}$ to $2S_{1/2}$ only an example or indeed one of your answers? This transition obviously violates selection rule for $L$. Realizing this, is it possible that you list all possible transitions in your answer? I think there should be 7 possible transitions in the problem; between n=3 and n=2, you can list 15 pairs of level but only 7 of them are allowed.

 

[samjohnny]

Is your considering the transition from $3D_{5/2}$ to $2S_{1/2}$ only an example or indeed one of your answers? This transition obviously violates selection rule for $L$. Realizing this, is it possible that you list all possible transitions in your answer? I think there should be 7 possible transitions in the problem; between n=3 and n=2, you can list 15 pairs of level but only 7 of them are allowed.

Ah I think I've got it.

Essentially I'm trying to figure out what counts as a transition without applying selection rules. Is a transition between energy levels, or between states in the n=3 and n=2 bands.

If the latter, then there would be 15 ways in which one could go from n=3 to n=2 (disregarding selection rules). And then to get the number of transitions that would be observed, we'd have to apply the selection rules, in which case I also count 7 ways, which is reassuring.

 

[blue_leaf77]

Seems that it's cleared up now. Anyway, there are 7 allowed transitions but some of them have the same energy difference, such transitions should be counted once when answering how many transition observed, because they will coincide in your spectrometer.

 

[samjohnny]

Ah yes indeed. Thanks a lot.