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Commutation relation of operators

  1. Aug 1, 2010 #1
    Im reading in a quatum mechanics book and need help to show the following relationship, (please show all the steps):

    If A,B,C are operators:

    [A,BC] = B[A,C] + [A,B]C
     
  2. jcsd
  3. Aug 1, 2010 #2
    Just expand out the commutation operators based on the definition, i.e. [X,Y]=XY-YX.

    When you see how easy this is, you will laugh and might even be embarrassed that you posted such an easy question :smile: - no offense intended ... I've done much worse.
     
  4. Aug 1, 2010 #3
    I've often done much worse too. :redface:
    Moreover, there are some similar formulas, which can be proven easily and similarly:
    [tex] [A,BC] = \{A,B\} C - B\{ A , C \} [/tex]
    [tex] \{ A , BC \} = \{A,B\}C - B[A,C] = [A,B]C + B\{A,C\}[/tex]
    where the square bracket denotes the commutator and the curly bracket denotes the anti-commutator.

    Probably mathfilip wanted to specify the point that the algebra must be "associative", or these formulas are not valid.
     
    Last edited: Aug 1, 2010
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