Commutating Annihilators with a beamsplitter

Question

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!

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Comments

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.

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It works! Thank you!

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This question is not research-level. It should be migrated to physics.se

This post has been migrated from (A51.SE)