1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Commutator Identity

  1. Oct 5, 2010 #1


    User Avatar
    Gold Member

    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]
  2. jcsd
  3. Oct 5, 2010 #2


    User Avatar
    Homework Helper
    Gold Member

    Neither is correct.


    You won't have anything involving commutators until you multiply by both exponentials and collect terms in powers of the parameter [itex]x[/itex].
  4. Oct 5, 2010 #3


    User Avatar
    Gold Member

    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

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Commutator Identity
  1. Commutator algebra (Replies: 5)

  2. Commutator relations (Replies: 3)

  3. Commutator calculus (Replies: 5)

  4. Commutator relations (Replies: 1)