1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    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.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook