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Homework Help: Commutator trouble

  1. May 25, 2010 #1
    1. The problem statement, all variables and given/known data

    I'm having something about the Interaction/Dirac picture.
    The equation of motion, for an observable [itex]A[/itex] that doesn't depend on time in the Schrödinger picture, is given by:

    [tex]\[i\hbar \frac{d{{A}_{I}}}{dt}=\left[ {{A}_{I}},{{H}_{0}} \right]\][/tex]

    [tex]\[{{\hat{H}}_{0}}=\hbar \omega {{\hat{a}}^{\dagger }}\hat{a}+\frac{\hbar {{\omega }_{0}}{{{\hat{\sigma }}}_{z}}}{2}\][/tex]

    From this I have to commutate with [itex]\[{{\hat{\sigma }}_{+}}\][/itex], [itex]\[{{\hat{\sigma }}_{-}}\][/itex] and [itex]\[{{\hat{\sigma }}_{z}}\][/itex], where [itex]\[{{\hat{\sigma }}_{z}}\][/itex] is the last of the Pauli matrices, and [itex]\[{{\hat{\sigma }}_{\pm }}=\frac{\left( {{{\hat{\sigma }}}_{x}}\pm i{{{\hat{\sigma }}}_{y}} \right)}{2}\][/itex].

    2. Relevant equations


    3. The attempt at a solution
    Is it just as always ? By inserting, and then just take the normal commutator, and get:

    & \left[ {{\sigma }_{z}},{{H}_{0}} \right]=...=0 \\
    & \left[ {{\sigma }_{+}},{{H}_{0}} \right]=...=-\hbar {{\omega }_{0}}\left[ \begin{matrix}
    0 & 1 \\
    0 & 0 \\
    \end{matrix} \right] \\
    & \left[ {{\sigma }_{-}},{{H}_{0}} \right]=...=\hbar {{\omega }_{0}}\left[ \begin{matrix}
    0 & 0 \\
    1 & 0 \\
    \end{matrix} \right] \\

    Or am I way off ?
    I'm kinda stuck, so a hint would be helpfull :)

    Thanks in advance.

  2. jcsd
  3. May 25, 2010 #2
    Are you supposed to answer it in terms of matrices. I haven't checked your answer but it can be simply done just by knowing the commutation relations between the pauli matrices.
  4. May 26, 2010 #3
    I think so...
    There is given a hint that I should look how to commutate the spin matrices, where fx. [Sx, Sy] = ihSz (i = complex number, h = h-bar) - if I remember correctly.
  5. May 26, 2010 #4
    You can use that commutation you just wrote out, along with the permutations of it to solve the problem in terms of spin matrices alone.
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