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!

Linear Operator formulation

  1. Sep 7, 2014 #1
    1. The problem statement, all variables and given/known data
    Show that any linear operator [itex]\hat{L}[/itex] can be written as [itex]\hat{L} = \hat{A} + i\hat{B}[/itex], where [itex]\hat{A}[/itex] and [itex]\hat{B}[/itex] are Hermitian operators.


    2. Relevant equations
    The properties of hermitian operators.


    3. The attempt at a solution
    I am not sure where to start with this one. For example, we know that if an operator, A is hermitian, then [itex]\langle g\mid A f \rangle = \langle f\mid A g\rangle^*[/itex]. But I do not see how to break up L into any combination of other operators. Any help would be appreciated, perhaps a nudge in the right direction.
     
  2. jcsd
  3. Sep 7, 2014 #2

    strangerep

    User Avatar
    Science Advisor

    What would ##\hat L^\dagger## look like?
     
  4. Sep 7, 2014 #3

    Matterwave

    User Avatar
    Science Advisor
    Gold Member

    If ##\hat{A},\hat{B}## are Hermitian then ##i\hat{B}## is anti-Hermitian. So the problem is really just asking you to prove that any operator is a sum of a Hermitian part and an anti-Hermitian part. This is very similar that any real linear operator is a sum of a symmetric part and an anti-symmetric part. Do you know how that property is proved?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Linear Operator formulation
  1. Linear Operators? (Replies: 7)

  2. Linear Operators (Replies: 9)

Loading...