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Commutating Annihilators with a beamsplitter

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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_adj such that [a,a_adj] = 1 and another annihilator b with creator b_adj, I see that the argument in a proof claims the following:

Let G = a_adj*b - a*b_adj. 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!

This post has been migrated from (A51.SE)
asked Apr 18, 2012 in Theoretical Physics by Abe Asfaw (0 points) [ no revision ]
This question is not research-level. It should be migrated to physics.se

This post has been migrated from (A51.SE)
It works! Thank you!

This post has been migrated from (A51.SE)
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.

This post has been migrated from (A51.SE)

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