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Help me understand hamiltonian

  1. Jan 11, 2014 #1
    For an exercise I am given the attached Hamiltonian, but I don't understand it completely. We sum over spin -½ and ½ and the paulimatrices seem to be dependent on this since they are labeled by σσ'. What does this mean? I mean the pauli matrices are just operators for the spin in the x,y,z-direction, so what does it exactly have to do with our summation of spin eigenvalues.
     

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  3. Jan 11, 2014 #2

    tiny-tim

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    hi aaaa202! :smile:
    i'm not sure what you're asking

    each pauli matrix, here written as τ1 τ2 and τ3, is a 2x2 matrix, so it has four entries, labelled ½½ ½-½ -½½ and -½-½

    each entry of τi is written τiσσ'
     
  4. Jan 12, 2014 #3
    So in each component the spin operator S=(Sx,Sy,Sz) I add the entries of the corresponding pauli matrix?
     
  5. Jan 12, 2014 #4

    tiny-tim

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    i don't understand :confused:

    you add over every index (k k' σ and σ') in every ∑

    (x y and z are not relevant indices for the spinors: those are ½ and -½)
     
  6. Jan 12, 2014 #5
    Okay I misunderstood. I understand now - thanks :)
     
  7. Jan 14, 2014 #6
    Hi again tiny Tim, maybe you can help me further with a problem concerning this Hamiltonian. I am supposed to perform a thermal average with respect to H0 of HS1*HS2
    Now to perform a thermal average one simply just sums over:
    <Eil HS1HS2 lEi> , where Ei are eigenstates of H0.
    But what is the action of the product HS1HS2 on an eigenstate of H0?
    I can understand the electron operators, but how do I interpret ∑Si1∑Si2 on an eigenstate of H0?
    S1 and S2 simply just represent 2 different spin vectors. How do I use this on a single electron gas eigenstate (H0 describes an electron gas).
     
  8. Jan 14, 2014 #7

    tiny-tim

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    hi again aaaa202! :smile:

    sorry, can you ask this in a separate thread?
     
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