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Homework Help: Hamiltonian being a function of either orbital and spin operators

  1. Jan 2, 2010 #1
    1. The problem statement, all variables and given/known data
    The title presents my problem. I know in principle how to find eigenvalues and eigenfunctions of the Hamiltonian if it depends only on orbital operators or in spin operators. On the other hand I have no clue how to solve it if there are both types of operators.


    3. The attempt at a solution
    The complete state space will be the cartesian (or direct?!) product of the orbital state space with the spin state space. Nevertheless, I have no idea how the hamiltonian will act on that complete space.
     
  2. jcsd
  3. Jan 2, 2010 #2

    diazona

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    Homework Helper

    Are you talking about something like this?
    [tex]H = \vec{L}\cdot\vec{S}[/tex]
    In that case, as you said, you represent the states as products of a spatial wavefunction with a spinor,
    [tex]\vert\psi\rangle\vert\chi\rangle[/tex]
    The orbital angular momentum acts on only the spatial wavefunction [itex]\vert\psi\rangle[/itex] and the spin operator acts on only the spinor [itex]\vert\chi\rangle[/itex].
    [tex]H\vert\psi\rangle\vert\chi\rangle = \bigl(\vec{L}\vert\psi\rangle\bigr)\cdot\bigl(\vec{S}\vert\chi\rangle\bigr)[/tex]

    If you have a particular example in mind, why don't you post it.
     
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