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I have to find out the following commutator $$ [a^\dagger a^\dagger a, a^\dagger a a] $$ and after expanding it with $[A,B]=AB-BA$ $$ [a^\dagger a^\dagger a, a^\dagger a a] = a^\dagger a^\dagger a a^\dagger a a - a^\dagger a a a^\dagger a^\dagger a $$ I begun to wonder - is there some kind of easier method to calculate such expressions or I just have to use the relation $ [a, a^\dagger] = a a^\dagger - a^\dagger a = 1$?

Qmechanic
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michelson
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1 Answers1

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You can start by using the fact that $[a,a] = [a^{\dagger},a^{\dagger}] = 0$ together with $[A,BC]=B[A,C]+[A,B]C$ and hence \begin{eqnarray} [a^{\dagger} a^{\dagger} a, a^{\dagger} a a] &=& a^{\dagger}[a^{\dagger} a^{\dagger} a, a a]+[a^{\dagger} a^{\dagger} a, a^{\dagger} ]a a \\ &=& a^{\dagger}[a^{\dagger} a^{\dagger}, a a]a + a^{\dagger} a^{\dagger} [ a, a^{\dagger} ]a a \\ &=& a^{\dagger}[a^{\dagger} a^{\dagger}, a a]a + a^{\dagger} a^{\dagger} a a \ . \end{eqnarray} Now what remains is simpler.

secavara
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