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Name of this relation, and struggle proving it.

  1. Mar 11, 2014 #1
    [a, a[itex]^{+(n)}[/itex]] = na[itex]^{+(n-1)}[/itex]

    1) What's the name of this relation if it has any?

    2) I tried to prove this by induction, I started by saying that for n=1, this holds since [a, a[itex]^{+}[/itex]] = 1 (as we all know and as we can all prove)

    then I assumed it true for (n-1), but I didn't go too far afterwards. Can someone give me a hint concerning its proof.

  2. jcsd
  3. Mar 11, 2014 #2


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    Is it homework?
  4. Mar 11, 2014 #3


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    Induction works just fine.

    There's an identity for working with commutators that helps:

    [itex][A, B C] = [A, B] C + B [A, C][/itex]

    Apply to the case [itex]A = a[/itex], [itex]B = (a^\dagger)^{n-1}[/itex], [itex]C=a^\dagger[/itex].
    Then it should be easy to prove it by induction.
    Last edited: Mar 11, 2014
  5. Mar 11, 2014 #4
    Oh thank you!!! I am now convinced, I tried proving it several ways, and didn't use what you proposed, and the procedures kept turning me down! Thank you, again!

    Bill_K,this wasn't a homework, I was trying to convince myself that its true. I asked my professor and he told me to prove it by induction and didn't give further hints.

    Thanks guys.
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