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How to select the good basis for the special Hamiltonian?

  1. Jan 14, 2014 #1
    How to select the good basis for the special Hamiltonian??

    For the Hamiltonian [itex] H=\frac{P^2}{2\mu} -\frac{Ze^2}{r}+ \frac{\alpha}{r^3} L.S[/itex] (which we can use [itex] L.S=\frac{1}{2} (J^2-L^2-S^2)[/itex]in the third term) how to realize that the third term,[itex]\frac{\alpha}{r^3} L.S[/itex], commutes with sum of the first two terms,[itex] \frac{P^2}{2\mu} -\frac{Ze^2}{r}[/itex], and then conclude that the set of the operators [itex] J^2, J_z, L^2, S^2[/itex] are the best to work with? ([itex] \alpha, Z, e, \mu[/itex] are constants)
     
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  3. Jan 14, 2014 #2

    DrClaude

    User Avatar

    Staff: Mentor

    Usually, you arrive at it by steps. You first have the Hamiltonian in the absence of spin, from which you find that you have quantum numbers ##n##, ##l##, and ##m_l##, the latter usually chosen to be the projection along z. When you add spin, you find that the Hamiltonian is independent of spin, so you describe the eigenstates with
    $$
    \left| n, l, m_l, s, m_s \right\rangle
    $$

    When you add spin-orbit coupling, you realize that ##m_l## and ##m_s## are no longer good quantum numbers (conserved quantities), and you try to figure out what is still conserved. the total angular momentum ##j## and its projection ##m_j## are, so you switch to
    $$
    \left| n, l, s, j, m_j \right\rangle
    $$

    Note that ##j## and ##m_j## were already good quantum numbers, but since they are redundant with ##l##, ##m_l##, ##s##, and ##m_s## in the absence of spin-orbit coupling, they are not explicitely mentionned.
     
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