1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    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!

Adjoint of a Commutator

  1. Sep 5, 2016 #1
    1. The problem statement, all variables and given/known data
    Show that
    [tex] \left [ A,B \right ]^{\dagger}=-\left [A,B \right ] [/tex]
    2. Relevant equations
    [tex] \left [ A,B \right ] = AB-BA [/tex][tex] \left (AB \right)^{\dagger}= B^{\dagger}A^{\dagger} [/tex]
    3. The attempt at a solution
    [tex] \left [ A,B \right ]^{\dagger}=\left (AB-BA \right )^{\dagger} [/tex][tex]=\left (AB \right )^{\dagger}-\left (BA \right )^{\dagger} [/tex][tex]=B^{\dagger}A^{\dagger}-A^{\dagger}B^{\dagger}[/tex][tex]=-\left (A^{\dagger}B^{\dagger}-B^{\dagger}A^{\dagger} \right )[/tex][tex]=-\left [ A^{\dagger},B^{\dagger} \right ][/tex]
    I can only see this working if the operators are Hermitian but the question did not specify it as such.
  2. jcsd
  3. Sep 5, 2016 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    A quick way to solve the problem of being given a potentially incorrectly specified question is to look for a counterexample. Keep it as simple as possible. In this case, you can choose two 2 x 2 real matrices that are non-symmetric - hence non-Hermitian. Then do the calc and see if the result holds.

    I used the following R code to test this (the 't' function performs a transpose and %*% does matrix mult)

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

Have something to add?
Draft saved Draft deleted