Solving Puzzling Questions: Total Spin S, L and J

[oliver.smith8]

I'm just going through a past paper for an exam I am revising for and I can't work out how to solve this question!

it's
" Total spin S, total orbital angular momentum L and total angular moment J are specified according to spectroscopic notation by the term symbol $$^{S+1}L_{J}$$

i) For Helium show the the levels 1s2p and 1s3d include the terms $$^{1}P_{1}$$ and $$^{1}D_{2}$$
ii) Determine allowed optical transitions between the $$^{1}P_{1}$$ and $$^{1}D_{2}$$ levels in the presences of an external magnetic field. How many transitions occur? How many spectral lines will be observed"

Ive got no idea where to start

 

[nickjer]

This site can help out for part (i): http://en.wikipedia.org/wiki/Term_symbol

I will help you with the first part of (i). You know one electron will have S=1/2, L=0 and the other has S=1/2, L=1. So S_total can be 0 or 1 (singlet and triplet). And L_total = 1 (3 states). That gives you a total of 12 states. Pauli exclusion principle doesn't apply here since the electrons can never exist in the same state.

So you can have $^{1}P_{1}$ or $^{3}P_{J}$ (where J=0,1,2). If you add up all the states for each J, you get 12 states as before.

For part (ii), the magnetic field will interact with the J_z operator and the energies will split for different m_j. In the case of $^{1}P_{1}$, J=1 (a triplet), so you will see three split levels. Do the same for the other state. Then find all the transitions between them.

Similar to what can be seen here: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/sodzee.html#c2 (but this is a slightly more complicated example, but the picture is helpful)