Hydrogen Atom Spin Orbit Coupling

[foranlogan5]

hi guys,could anyone please help with this question?


How many levels will n=4 hydrogen atom split into due to the spin orbit coupling?

i know J = L + S
i don't know if i have understood the concepts properly - i know if n = 1,2,3...
l=0,1,2 ... n-1
m= -l,-l+1 ..0.. l-1,l
and aslo thas s = +-1/2

so for instance n=1 level,will have l=0 so hence m=0 and hence j = l + s (which is 0 +or-1/2) so is it right in me saying j=-1/2 or j=1/2 but j cannot be -ve so i state from eigenfunctions at level n=1 which is j=1/2

now for n=2 , we have 1 state in l=0 level as before one state ,but now for l=1 we have 3 m states(m=-1,0,+1) and for each of these in j = 1 (+or - 1/2) we have j=1/2 or j=3/2 so we have 7 states all together

COULD ANYONE HELP ME AND TELL ME IF IM ON THE RIGHT LINES AND IF I HAVE GONE WRONG ANYWHERE?
THANKS LOGAN

 

[Jhero]

Hi Logan,

Based on your understanding of the concepts, you are on the right track. However, there are a few things to clarify. Firstly, for n=1 level, since l=0, the only possible value for m is 0. Therefore, j can only be 1/2 or -1/2. However, as you correctly stated, j cannot be negative, so the correct value for j is 1/2.

For n=2 level, you have correctly identified that there are 3 possible m states for l=1. However, for each of these states, there are two possible values for j, which are 1/2 and 3/2. This means that there are a total of 6 possible states for the n=2 level. Adding this to the 1 state from n=1 level, we get a total of 7 states, as you stated.

In general, for any given n level, there will be (2l + 1) possible m states, and for each of these states, there will be two possible values for j. Therefore, the total number of states for a given n level will be (2l + 1) x 2 = 2(2l + 1) = 2n^2.

For the n=4 level, this means there will be 2(2(3) + 1) = 14 possible states due to spin orbit coupling.

I hope this helps clarify your understanding. Let me know if you have any further questions.