# Commutating Annihilators with a beamsplitter

I am reading Nielsen and Chuang on P. 291, for anyone interested in the origin of my question.

Given an annihilator $a$ and its corresponding creator $a^\dagger$ such that $[a,a^\dagger] = 1$ and another annihilator $b$ with creator $b^\dagger$, an argument in a proof claims the following:

Let $G = a^\dagger b - ab^\dagger$. Then, $[G,a] = -b$ and $[G,b] = a$.

I don't see how these two relations hold. Can someone please point me in the right direction or prove them?

Thank you SOCommunity!

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## migrated from theoreticalphysics.stackexchange.comApr 20 '12 at 13:19

This question came from our site for scientific theorists and academic scholars interested in theoretical, research-level physics.

Hi, just use $[XY,Z]=XYZ-ZXY = XYZ-XZY+XZY-ZXY = X[Y,Z]+[X,Z]Y$ and the basic commutators $[a,a^\dagger]=1$ and similarly for $b$ while other commutators vanish. You will see that from the right hand side, only one term survives and it gives you what you need. –  Luboš Motl Apr 18 '12 at 16:48
It works! Thank you! –  Abe Asfaw Apr 18 '12 at 16:53
This question is not research-level. It should be migrated to physics.se –  Frédéric Grosshans Apr 20 '12 at 8:59