Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Is this operator Hermitian?

  1. Oct 20, 2010 #1
    1. Let G be an operator on H (Hilbert Space). Show that:
    (a) H = 1/2 (G + G[tex]^{\dagger}[/tex]) is Hermitian.
    (b) K = -1/2 (G - G[tex]^{\dagger}[/tex]) is Hermitian.
    (c) G = H + iK

    2. Relevant equations

    3. The attempt at a solution:

    (a) Since the adjoint of the sum of two operators does not change their position (addition of operators is commutative), it is very straight forward.

    (b) This is where I run into trouble, and I think it is because the problem is wrong. The operator given appears to be anti-hermitian (skew-hermitian), however I would like confirmation. This result makes (c) very difficult, as it uses an incorrect answer. My suspicion is that the intended question includes an i (as in, H = -i/2 (...)):

    K[tex]^{\dagger}[/tex] = -1/2 (G[tex]^{\dagger}[/tex] - G) = 1/2 (G - G[tex]^{\dagger}[/tex]) = -K.

    (c) Well if I am correct about (b), (c) is wrong. I was just hoping someone could confirm I am right, or show me how K is Hermitian. I can take it from there.
  2. jcsd
  3. Oct 20, 2010 #2
    If the problem should be as stated this will give a condition for G that allows to solve all equations.
    If G is arbitrary and the equations should be satisfied for general G there has to be a change in definition.
    K= -i/2(...)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook