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nrqed
#2
Jun22-06, 01:31 PM
Sci Advisor
HW Helper
P: 2,953
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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., [tex][B,[A,B]]=[A,[A,B]]=0[/tex]. Show that

[tex][A,B^{n}]=nB^{n-1}[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^{n-1} - BB^{n-1}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^{n-1} [/itex].
, that is use [itex] [A,B^n] = B[A,B^{n-1}] + [A,B] B^{n-1} [/itex].
Now, repeat this again on the first term using now [itex] C= B^{n-2} [/itex]. You will get a recursion formula that will give you the proof easily.