Spectroscopic Notation Problem

[njdevils45]

Question

(a) Write down the quantum numbers for the states described in spectroscopic notation as ²S_3/2, ³D₂ and ⁵P₃.
(b) Determine if any of these states are impossible, and if so, explain why. (Please note that these could describe states with more than one electron.)

My Attempt

A)
I came up with the quantum numbers for everything that I'm pretty sure is correct. Was hoping I could get a double check on this.
²S_3/2 l = 0, s = 1/2, j = 3/2
³D₂ l = 2, s = 1, j = 2
⁵P₃ l = 1, s = 2, j = 3

B)
I'm almost 100% confident that ²S_3/2 is impossible because when l = 0, j must = 1/2.

I'm not sure about the other two because I don't know all the rules for what s can be compared to l and j.

Any help would be greatly appreciated. Thank you!

 

[DrClaude]

I'm almost 100% confident that ²S_3/2 is impossible because when l = 0, j must = 1/2.

Not exactly. When $L=0$, $J=S$.

I'm not sure about the other two because I don't know all the rules for what s can be compared to l and j.

What are the rules of addition of angular momenta? In other words, given $L$ and $S$, what are the possible values of $J=L+S$?

 

[kuruman]

(A) is correct
You are correct that ²S_3/2 is impossible.

The rule is that the maximum value of J is L+S, the minimum value of J is |L-S| and J takes all values in between in steps of 1.

 

[njdevils45]

Not exactly. When $L=0$, $J=S$.

So to say that when l = 0, j must = 1/2 for the first state is correct right?

What are the rules of addition of angular momenta? In other words, given $L$ and $S$, what are the possible values of $J=L+S$?

Does this mean that j could only equal 1,2, or 3 for the second and third one? Therefore the value for j is possible for those two?
Is the value for s possible for the other two though? Is s allowed to be integer? I know for a fact that the values for l is possible for all three of them, and assuming that s is correct, the value for j is possible for the 2nd and third state, but are the values of s possible for the 2nd and third state?

 

[njdevils45]

(A) is correct
You are correct that ²S_3/2 is impossible.

The rule is that the maximum value of J is L+S, the minimum value of J is |L-S| and J takes all values in between in steps of 1.

So for the second and third states, j is correct assuming the value for s is correct? In that case is the value for s possible for those two states?

 

[DrClaude]

So to say that when l = 0, j must = 1/2 for the first state is correct right?

Correct.

Does this mean that j could only equal 1,2, or 3 for the second and third one? Therefore the value for j is possible for those two?

Yes on both.

Is the value for s possible for the other two though? Is s allowed to be integer? I know for a fact that the values for l is possible for all three of them, and assuming that s is correct, the value for j is possible for the 2nd and third state, but are the values of s possible for the 2nd and third state?

The spectroscopic notation is based on $S$, the total spin of the electrons. It can have a half-integer or an integer value.

 

[njdevils45]

The spectroscopic notation is based on $S$, the total spin of the electrons. It can have a half-integer or an integer value.

So in that case the 2nd and third states for the problem are both definitely possible?

 

[DrClaude]

So in that case the 2nd and third states for the problem are both definitely possible?

Yes.

 

[njdevils45]

Yes.

Thank you!