Commutator Identity


by kreil
Tags: commutator, identity
kreil
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#1
Oct5-10, 04:11 PM
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P: 518
1. The problem statement, all variables and given/known data


Prove the following identity:

[tex]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!}+...[/tex]

where A and B are operators and x is some parameter.

2. Relevant equations
[tex] e^x = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...[/tex]
[tex] e^{-x} = 1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+...[/tex]

3. The attempt at a solution

[tex] \hat B e^{-x \hat A} = \hat B - [\hat B, \hat A]x + \frac{[[\hat B, \hat A], \hat A] x^2}{2!}+...[/tex]

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. [tex] \hat B (\hat A x)^2 = [\hat B, \hat A \hat A] x^2....??[/tex]

or [tex] \hat B (\hat A x)^2 = [[\hat B, \hat A], \hat A] x^2...??[/tex]
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gabbagabbahey
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#2
Oct5-10, 06:21 PM
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Quote Quote by kreil View Post
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. [tex] \hat B (\hat A x)^2 = [\hat B, \hat A \hat A] x^2....??[/tex]

or [tex] \hat B (\hat A x)^2 = [[\hat B, \hat A], \hat A] x^2...??[/tex]
Neither is correct.

[tex]\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 [itex]x[/itex].
kreil
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#3
Oct5-10, 07:23 PM
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P: 518
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 ) [/tex]

So I multiply this out, collect terms in powers of x, and simplify to the commutator relations

Thanks


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