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Determine whether or not is a Hermitian operator

  1. Dec 5, 2011 #1

    dje

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    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

    dextercioby

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    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

    dje

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    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
     
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