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Spin 1/2 system

  1. Feb 2, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the energy levels of a 2-spin 1/2 system with spinoperators S1 and S2 in an external magnetic field. The hamiltonian is of the form,

    H= A ( 1-[tex]\frac{2S_{1}}{h}[/tex] . [tex]\frac{2 S_{2}}{h}[/tex] )+ [tex]\frac{\mu B}{h}[/tex](S[tex]_{1,z}[/tex]+S[tex]_{2,z}[/tex])

    The h is a h-bar, constants A, B, and S1 and S2 the spin operators

    2. Relevant equations

    I have to solve the equation H l[tex]\psi[/tex]> = El[tex]\psi[/tex]>

    3. The attempt at a solution

    The spin system is has a basis, l[tex]\uparrow\uparrow>,\left| \uparrow\downarrow>,\left|\downarrow\uparrow>,\left|\downarrow\downarrow>[/tex]

    so any [tex]\left| \psi[/tex]> is a linear combination of the basis above, but i don't know how i can calculate the eigenvalues of the above equation. I have a feeling i have to use the Pauli matrices but iam not sure. Anyone has an idea? It should be a 3 level system...
  2. jcsd
  3. Feb 3, 2009 #2
    I know the matrices S1 and S2 commute, is also know that S1,z + S2,z = S,z

    couldn't that help?
  4. Feb 3, 2009 #3


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    Homework Helper
    Gold Member

    Why not try expanding your wavefunction into the spin basis and then using that to calculate [tex]H|\psi\rangle[/tex]? Under what circumstances is your result a constant multiple of [tex]|\psi\rangle[/tex]?
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