1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Determine whether or not is a Hermitian operator

  1. Dec 5, 2011 #1


    User Avatar

    1. The problem statement, all variables and given/known data

    The operator F is defined by Fψ(x)=ψ(x+a) + ψ(x-a), where a is a nonzero constant. Determine whether or not F is a Hermitian operator.

    2. Relevant equations

    ∫(x+a)d/dx + (x-a)d/dxψ

    3. The attempt at a solution

    f = (1=ax) + (1-ax)ψ

    What are the steps I need to do to figure this out. Thanks.
  2. jcsd
  3. Dec 6, 2011 #2


    User Avatar
    Science Advisor
    Homework Helper

    No matter the value of a, one can show that F is bounded, so the adjoint of it exists. Then all you need is to check is the hermiticity condition in integral form:

    [tex] \int dx \psi^{*}(x) F\psi(x) = ? [/tex]

    Try to get the psi with exchanged argument under the complex conjugate sign.
  4. Dec 6, 2011 #3


    User Avatar

    I am not sure of these steps but I will try. Can you show me if I am still not understanding this. thanks.

    Fψ(x)=Fψ(x+a) + ψ(x-a) Fτ= F to be Hermitian
    Fψ (x+a) + (x-a) = F dt/dx? (x+a) + (x-a)

    = F dt/dx (x + a) + (x-a)

    ∫(x+a)d/dx + (x-a)d/dx ψ

    F τ= (1+ax)ψ + (1-ax)ψ

    KEY= * below/symbol I am wanting here is circle with vertical line through it.
    (θ*/ψ) = (ψ/θ*)
    θ* (x) ψ(x+a) + ψ(x-a) dt/dx dx
    = ψ(x +a) + ψ (x-a) dθ*/dx dx

    Solution- F is Hermitian operator Fτ= F
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook