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Three spin 1/2

  1. Apr 27, 2010 #1
    1. The problem statement, all variables and given/known data

    three distiguishable spin 1/2 particles interact via

    [tex] H = \lamda ( S_1 \cdot S_2 + S_2 \cdot S_3 + S_3 \cdot S_1 ) [/tex]

    a) What is the demension of the hilbert space?

    b) Express H in terms of [tex] J^2 [/tex] where [tex] J = S_1 + S_2 + S_3 [/tex]

    c) I then need to find the energy and eigenstates, but i think i can due this once i know the hamilitonian.

    2. Relevant equations

    3. The attempt at a solution

    a) would this be 6D, 2 from each particle? or
    (2*3/2 + 1) = 4

    or am i all wrong?

    b) [tex] J^2 = S_1^2 + S_2^2 + S_3^2 + 2S_1S_2 + 2S_3S_2 + 2S_1S_3 [/tex]

    [tex] H = \lamda (1/2 J^2 - _1^2 - S_2^2 - S_3^2) [/tex]

    but i still have S's in my H. Is this ok? I feel like its not.

    should i use.. [tex] J^2 + \hbar J_z + J_+J_- [/tex]?

    why cant i just write everything as a 2x2 matrix, for the energies, then solve it that way with out using J^2?
    Last edited: Apr 27, 2010
  2. jcsd
  3. Apr 27, 2010 #2
    so i know now that a) is 8D

    and i think that H = J^2 - S^2 - S^2 - S^2
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