Degeneracy of multiple electron states

[Ahmes]

Homework Statement

A seemingly trivial part of a Homework I've gotten is to find the degeneracy of the ${}^1\! S_0, {}^1\! D_2, {}^3\! P_2, {}^3\! P_1, {}^3\! P_0$ in an $O^{++}$ ion.

Homework Equations

The Attempt at a Solution

an $O^{++}$ ion has 4 valance electrons and when $\ell=1$ for example, there are 2x2x2x2x3=48 states all together. I know how to add angular momenta, and technically can add all electrons and the orbital one by one, finding explicitly the states with their Clebsch-Gordan-like coefficients. The thing is that it would take about a week, and I have the feeling that if I just want to find the degeneracy (and not necessarily the states) then it would be much easier.

Any help will be appreciated!

 

[Jhero]

Hello! Finding the degeneracy of these states can definitely be done more efficiently than manually calculating each state. First, let's clarify the notation used in the homework statement. The {}^1\! S_0, {}^1\! D_2, {}^3\! P_2, {}^3\! P_1, {}^3\! P_0 states refer to the electronic configurations of the O^{++} ion, where the superscripts represent the total spin and the subscripts represent the orbital angular momentum. The degeneracy of a state refers to the number of states with the same energy and quantum numbers.

To find the degeneracy of these states, we can use the formula:

g = (2S + 1)(2L + 1)

where S is the total spin and L is the orbital angular momentum. For the {}^1\! S_0 state, S=0 and L=0, so the degeneracy is (2x0 + 1)(2x0 + 1) = 1. For the {}^1\! D_2 state, S=0 and L=2, so the degeneracy is (2x0 + 1)(2x2 + 1) = 5. For the {}^3\! P_2 state, S=1 and L=2, so the degeneracy is (2x1 + 1)(2x2 + 1) = 15. Similarly, for the {}^3\! P_1 state, the degeneracy is (2x1 + 1)(2x1 + 1) = 9, and for the {}^3\! P_0 state, the degeneracy is (2x1 + 1)(2x0 + 1) = 3.

Therefore, the total degeneracy for an O^{++} ion is 1+5+15+9+3 = 33. This method is much quicker and more efficient than manually calculating each state. I hope this helps with your homework! Let me know if you have any further questions.