# Commutator Identity

by kreil
Tags: commutator, identity
 P: 516 1. The problem statement, all variables and given/known data Prove the following identity: $$e^{x \hat A} \hat B e^{-x \hat A} = \hat B + [\hat A, \hat B]x + \frac{[\hat A, [\hat A, \hat B]]x^2}{2!}+\frac{[\hat A,[\hat A, [\hat A, \hat B]]]x^3}{3!}+...$$ where A and B are operators and x is some parameter. 2. Relevant equations $$e^x = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...$$ $$e^{-x} = 1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+...$$ 3. The attempt at a solution $$\hat B e^{-x \hat A} = \hat B - [\hat B, \hat A]x + \frac{[[\hat B, \hat A], \hat A] x^2}{2!}+...$$ It seems after I rearrange the commutation orders, the signs all become positive and this is the required result, so I know I must be doing something wrong. I think it has to do with how I'm multiplying out B into the series.. i.e. $$\hat B (\hat A x)^2 = [\hat B, \hat A \hat A] x^2....??$$ or $$\hat B (\hat A x)^2 = [[\hat B, \hat A], \hat A] x^2...??$$
HW Helper
P: 5,004
 Quote by kreil It seems after I rearrange the commutation orders, the signs all become positive and this is the required result, so I know I must be doing something wrong. I think it has to do with how I'm multiplying out B into the series.. i.e. $$\hat B (\hat A x)^2 = [\hat B, \hat A \hat A] x^2....??$$ or $$\hat B (\hat A x)^2 = [[\hat B, \hat A], \hat A] x^2...??$$
Neither is correct.

$$\hat{B}(x\hat{A})^2=BA^2x^2[/itex] You won't have anything involving commutators until you multiply by both exponentials and collect terms in powers of the parameter $x$.  P: 516 ahh I see it now I think.. [tex]e^{x \hat A} \hat B e^{-x \hat A} = \left ( 1+\hat A x + \frac{1}{2} \hat A^2 x^2 + \frac{1}{6} \hat A^3 x^3+... \right ) \left ( \hat B - \hat B \hat A x + \frac{1}{2} \hat B \hat A^2 x^2 - \frac{1}{6}\hat B \hat A^3 x^3+... \right )$$ So I multiply this out, collect terms in powers of x, and simplify to the commutator relations Thanks

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