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Two fermions perturbation

  1. Jan 27, 2016 #1
    1. The problem statement, all variables and given/known data
    Screen Shot 2016-01-27 at 4.34.21 PM.png

    2. Relevant equations


    3. The attempt at a solution
    I suppose to determine if a hamiltonian is rotational invariant, we check if [H(1),L^2], however, I am not sure how to do it if the hamiltonian is operate on a two particle wave function.
    Is it just to evaluate [S1z Z2 +S2z Z1, L1^2+L2^2]?

    What is the ground state wave function for this system?
    I suppose it is

    A (|0,1,1/2,-1/2> - |0,1,-1/2,1/2>), if you apply S1z, you get 0. It is also true for S2z. I think I am just lost here.
     
  2. jcsd
  3. Jan 28, 2016 #2

    blue_leaf77

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    The rotation operator in this system will be associated with the angular momentum operator ##J^z = L_1^z + S_1^z + L_2^z + S_2^z##. To check if the perturbation alters the rotation invariant of the system, calculate the commutation ##[H^{(1)},J^z]##.
     
  4. Jan 28, 2016 #3
    Thank you fore replying.
    How should I reprent ## L_z ##? is it ## XP_y - YP_x ##? Get r in spherical coordinate using raising and lowering operator and projection z axis? seems like a lot of work.
     
  5. Jan 29, 2016 #4

    blue_leaf77

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    Yes. ##L_z## is the orbital angular momentum of one of the fermions.
    That's too much than needed. Just plug in each expression for ##H^{(1)}## and ##J_z## into the commutator ##[H^{(1)},J_z]##. It should be simple if you use the property of commutator between angular momentum and position.
     
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