Quote by dimensionless
I'm having trouble figuring out the following commutator relation problem:
Suppose A and B commute with their commutator, i.e., [tex][B,[A,B]]=[A,[A,B]]=0[/tex]. Show that
[tex][A,B^{n}]=nB^{n1}[A,B][/tex]
I have
[tex][A,B^{n}] = AB^{n}  B^{n}A[/tex]
and also
[tex][A,B^{n}] = AB^{n}  B^{n}A = ABB^{n1}  BB^{n1}A[/tex]
I don't know where to go from here. I'm not positive the above relation is correct either.

Do you know the relation
[A,BC] = B[A,C] + [A,B] C
?
It's easy to prove. Just expand out.
Now, use with [itex] C= B^{n1} [/itex].
, that is use [itex] [A,B^n] = B[A,B^{n1}] + [A,B] B^{n1} [/itex].
Now, repeat this again on the first term using now [itex] C= B^{n2} [/itex]. You will get a recursion formula that will give you the proof easily.