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I'm having trouble figuring out the following commutator relation problem:
Suppose A and B commute with their commutator, i.e., [B,[A,B]]=[A,[A,B]]=0. Show that
[A,B^{n}]=nB^{n-1}[A,B]
I have
[A,B^{n}] = AB^{n} - B^{n}A
and also
[A,B^{n}] = AB^{n} - B^{n}A = ABB^{n-1} - BB^{n-1}A
I don't know where to go from here. I'm not positive the above relation is correct either.
Suppose A and B commute with their commutator, i.e., [B,[A,B]]=[A,[A,B]]=0. Show that
[A,B^{n}]=nB^{n-1}[A,B]
I have
[A,B^{n}] = AB^{n} - B^{n}A
and also
[A,B^{n}] = AB^{n} - B^{n}A = ABB^{n-1} - BB^{n-1}A
I don't know where to go from here. I'm not positive the above relation is correct either.