Creation/Anhilation Operator Commutation Relation

[teroenza]

Homework Statement

Simplify the following commutator involving the creation and annihilation operators.

$$[a^{\dagger}a,a \sqrt{a^\dagger a} ]$$

Homework Equations

I know that $$[a,a^\dagger] = 1$$.

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
$$[a^{\dagger}a,a]=a$$, but can't seem to figure out what to do in this case.

 

[Fightfish]

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 $\left[A, BC\right] = \left[A, B\right]C + B\left[A, C\right]$

 

[teroenza]

I see. Then the result is just:
$$[a^{\dagger}a,a \sqrt{a^\dagger a} ]=a\sqrt{a^\dagger a} + (a^\dagger a)^{3/2}-(a^\dagger a)^{3/2}=a\sqrt{a^\dagger a}$$

 

[Fightfish]

Hmm...I think you are off by a minus sign. $[a^{\dagger}a,a] = - a$

 

[teroenza]

Yep, you're right. My original post above is off by a negative too.