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Commutators of Pauli Matrices

  1. Aug 5, 2013 #1
    1. The problem statement, all variables and given/known data

    Express the product

    where σy and σz are the other two Pauli matrices defined above.

    commutatorpauli1.png


    2. Relevant equations



    3. The attempt at a solution

    I'm not sure if this is a trick question, because right away both exponentials combine to give 1, where the result is simply σx

    commutatorpauli2.png
     
  2. jcsd
  3. Aug 5, 2013 #2
    That's the whole point of the exercise, to see that the exponentials do not combine to give 1. To do such a thing, you would have to pass [itex]exp(i\alpha\sigma^z)[/itex] to the other side of [itex]\sigma^x[/itex]. But [itex]\left[\sigma^z,\sigma^x\right]\neq0[/itex] so that you can't simply commute them.

    Hint: express the exponential as a series.
     
  4. Aug 7, 2013 #3
    Ah, I see what you mean, as the sum is a matrix and not a number. Silly me.
     
    Last edited: Aug 7, 2013
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