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Prove Commutator Exponentials Algebra

  • #1
901
3

Homework Statement



Prove the following for operators A and B.

e^A B e^-A = B + [A,B] + (1/2!) * [A,[A,B]] + (1/3!) * [A,[A,[A,B]]] + ...

Homework Equations



e^A = 1 + A + (1/2!)A^2 + (1/3!)A^3 + ...

The Attempt at a Solution



I have no clue how to start.

For the highly special case of [A,B] = constant and [A,B] commutes with A and B, we can prove that e^A B e^-A = B + [A,B]

through the following:

Take the given identity [A,F(B)] = [A,B] dF/dB and set F = e^B

[A,e^B] = Ae^B - e^B*A = e^B * [A,B]

multiply both sides by e^-B to left.

e^-B A e^B - A = [A,B]

e^-B A e^B = A + [A,B]

This is the first 2 terms of the series and if we take [A,B] = c, then the other terms are zero. How do I prove it for general [A,B] then?
 

Answers and Replies

  • #2
901
3
any help with this? i am seriously stuck and everything i've looked up involves Lie algebras or some abstract math stuff i've never learned in physics before =(
 

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