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Creation/Anhilation Operator Commutation Relation

  1. Feb 14, 2015 #1
    1. The problem statement, all variables and given/known data
    Simplify the following commutator involving the creation and annihilation operators.

    [tex][a^{\dagger}a,a \sqrt{a^\dagger a} ][/tex]

    2. Relevant equations
    I know that [tex] [a,a^\dagger] = 1[/tex].

    3. The attempt at a solution
    I think I should be trying to put the creation operators to the left (normal ordering). I have also worked out
    [tex][a^{\dagger}a,a]=a[/tex], but can't seem to figure out what to do in this case.
     
  2. jcsd
  3. Feb 15, 2015 #2
    There's no need to try too hard with normal ordering - notice that you have an operator and its square root in the commutation relation! Just break up the commutator using the identity [itex]\left[A, BC\right] = \left[A, B\right]C + B\left[A, C\right] [/itex]
     
  4. Feb 15, 2015 #3
    I see. Then the result is just:
    [tex][a^{\dagger}a,a \sqrt{a^\dagger a} ]=a\sqrt{a^\dagger a} + (a^\dagger a)^{3/2}-(a^\dagger a)^{3/2}=a\sqrt{a^\dagger a}[/tex]
     
  5. Feb 15, 2015 #4
    Hmm...I think you are off by a minus sign. [itex][a^{\dagger}a,a] = - a [/itex]
     
  6. Feb 15, 2015 #5
    Yep, you're right. My original post above is off by a negative too.
     
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