Hilbert Space commutators help

[mooeypoo]

Hey guys, I got a homework assignment I'm having a lot of trouble with. This is post #1, I'll post a few more regarding the other problems in the homework. I'm not looking for an answer, I'm looking for assistance in how to proceed.

 

This is new to me and very confusing, and my confidence in QM is very low after last semester's course (we all failed, if you remember... had a makeup test that also didn't go well, but I ended up getting a C+, which I usually consider as failure, but in this case, at least it's *passing*... err)

 

Anyways, problem 1:

 

If A,B are two linear operators on a Hilbert space, show that

 

a^(A) B e^(-A) = B + [A,B] + 1/(2!) [A,[A,B]] + ... + 1/(n!) [A, [A, [A, ... [A,B] ... ]]] + ...

 

where the n-th term involves n successive commutators with A's.

 

(for some reason, this doesn't render in latex, there seems to be a problem with the factorial... if you guys want, I am leaving the tex syntax as-is, in case it helps:

e^{A}B e^{-A} = B + \left[ A, B \right] +1/2 \left[ A, \left[ A, B \right] \right] + \ldots + 1/n \left[ A, \left[ A, \ldots \left[ A, B \right] \ldots \right] \right] \right] + \ldots

 

Argh.

 

I don't even know where to START.. I looked up online rules of commutators (and we dealt with commutators last semester, so I know the rules), and I know that the above is defined as an identity, but I can't see how to start with the proof for it.

 

The only thing that popped to my mind is, perhaps, using the identity:

[math][A^{n},B] = nA^{n-1}[A,B][/math]

But I'm not sure how that helps me any here...

 

Anyone? Help..

[timo]

Just to be sure here: you know what the exponential of a matrix is, right?

 

your tex problem was caused by an extra \right] at the very end:

[math]e^{A}B e^{-A} = B + \left[ A, B \right] +1/2 \left[ A, \left[ A, B \right] \right] + \ldots + 1/n \left[ A, \left[ A, \ldots \left[ A, B \right] \ldots \right] \right] + \ldots [/math]

 

EDIT: Oh, and I'm not sure if "Hilbert space commutators" is technically correct. The Hilbert space is the vector space of the states. What you commute here is operators on that space, not elements of the space itself.

Edited September 25, 2010 by timo

[mooeypoo]

That's how the question was phrased. Honestly, I'm very confused, but now I feel like an idiot. Of course I know the exponential of a matrix, we learned it in math two semesters ago. Bah. I think I panicked a bit by the fact I'm looking at something new (commutators) and couldn't see how to start.

 

Exponential of a matrix:

 

[math]A^{b}= \sum \frac{1}{n!} b^{n}[/math]

(gah! it's not rendering again.. here's the latex in plaintext:

A^{b}= \sum \frac{1}{n!} b^{n}

 

Anyways, I'll work on it that way and post my result.

 

Thanks for the help -- and the patience!

 

~moo