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Homework Help: Quantum Energy Levels

  1. Nov 27, 2008 #1
    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
     
  2. jcsd
  3. Nov 27, 2008 #2
    It's because the interaction that splits the energies of the state is the spin-orbit coupling, proportional to [tex]\vec{L}\cdot\vec{S}[/tex], which can be rewritten as being proportional to the difference [tex]\vec{J}^2-\vec{L}^2-\vec{S}^2[/tex], 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.
     
  4. Nov 27, 2008 #3
    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 ;)
     
  5. Nov 27, 2008 #4
    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.
     
  6. Nov 28, 2008 #5
    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.
     
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