Exploring the Origins of Quantum Energy Level Degeneracy

[soxymoron]

I'm not going to follow the form because this is more of a hand-wavy question, I don't need it for a specific homework question but it's confusing me, I hope it's still okay to post here.

I understand the splitting of energy levels due to j (dependant on l and s) and I understand that each of the split states contains multiple degenerate states. What I don't understand is where they come from.

For an example of l=1 the energy is split into a j=1/2 and a j=3/2 levels. I know 6 states in total are contain within this split 2:4. This is the bit I don't understand, I'm assuming that the degenerate states are caused by the quantum number m but if that's the case why aren't they split evenly 3 to each energy level?

Thanks for any help you can give me

 

[borgwal]

It's because the interaction that splits the energies of the state is the spin-orbit coupling, proportional to $$\vec{L}\cdot\vec{S}$$, which can be rewritten as being proportional to the difference $$\vec{J}^2-\vec{L}^2-\vec{S}^2$$, which is dependent only on the quantum numbers j and l (s=1/2 in either case): so that's why the l=1 levels split according to their j quantum number, and not m.

In a magnetic field, on the other hand, the additional splitting would depend on m.

 

[Marco_84]

right... but i f you want to go deeper... see the rapresentations of angular momentu in QM, see what are the wieight of a reprs ;)

 

[soxymoron]

Okay so m only comes into play when a magnetic field is added, that makes sense. Though I'm still not sure why j=1/2 contains 2 degenerate states and j=3/2 contains 4. Also, I have a question which asks me to 'estimate the weak Zeeman splitting of a system', my main problem with that is I'm not sure what the weak Zeeman splitting of a system is. I could calculate it for each state but I don't know what that would mean for the whole thing.

 

[borgwal]

For j=1/2 there are two states: m=+/- 1/2, and for j=3/2 there are 4 different m-states.

The "weak" Zeeman effect just refers to a situation where the energy shift due to the magnetic field is small and can be treated with perturbation theory: the unperturbed Hamiltonian has split the l=1 level into 2: j=1/2 and j=3/2, and then the Zeeman effect splits each of those into 2 and 4, respectively, different energy levels.