1. Not finding help here? Sign up for a free 30min 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!

Commutator trouble

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

    Hi...
    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]
    where:

    [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:

    [tex]
    \begin{align}
    & \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] \\
    \end{align}
    [/tex]

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

    Thanks in advance.


    Regards
     
  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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Commutator trouble
  1. Commutator algebra (Replies: 5)

  2. Commutator relations (Replies: 3)

  3. Commutator calculus (Replies: 5)

  4. Commutator relations (Replies: 1)

Loading...