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Operator algebra with commutators

  1. Sep 16, 2012 #1

    chill_factor

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    1. The problem statement, all variables and given/known data

    Consider the operator A and its Hermitian adjoint A*.

    If [A,A*] = 1, evaluate: [A*A,A]

    2. Relevant equations

    standard rules of linear algebra, operator algebra and quantum mechanics

    3. The attempt at a solution

    [A,A*] = AA* - A*A = 1

    A*A = (1+AA*)

    [A*A,A] = AA*A - A*AA = ???????

    How do I even start?
     
    chill_factor, Sep 16, 2012
  2. jcsd
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  3. Sep 16, 2012 #2

    jmcelve

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    Hi chill_factor,

    If we expand [A*A,A], we obtain A*AA-AA*A. Do you see a way to factor this that might allow you to use the [A,A*] commutator?
     
    jmcelve, Sep 16, 2012
  4. Sep 16, 2012 #3

    chill_factor

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    Thank you for answering. I think I see it. There is an A on the right side of both the A*AA and AA*A operators so A*AA - AA*A = (A*A-AA*)A = -[A,A*]A = -A. Is that right?

    I guess my mistake was writing the commutator out incorrectly.
     
    chill_factor, Sep 16, 2012
  5. Sep 16, 2012 #4

    jmcelve

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    Yep. You've got it. Nice work.
     
    jmcelve, Sep 16, 2012
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