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Prove the Hamiltonian Operator is Hermitian

  1. Nov 5, 2011 #1
    1. The problem statement, all variables and given/known data

    Show that the Hamiltonian operator ([itex]\hat{H}[/itex])=-(([itex]\hbar[/itex]/2m) d2/dx2 + V(x)) is hermitian. Assume V(x) is real

    2. Relevant equations

    A Hermitian operator [itex]\hat{O}[/itex], satisfies the equation

    <[itex]\hat{O}[/itex]>=<[itex]\hat{O}[/itex]>*

    or

    ∫[itex]\Psi[/itex]*(x,t)[itex]\hat{O}[/itex][itex]\Psi[/itex](x,t)dx = ∫[itex]\Psi[/itex](x,t)[itex]\hat{O}[/itex]*[itex]\Psi[/itex]*(x,t)dx between -∞ and +∞


    3. The attempt at a solution

    This is my first time using LaTex and I'm having trouble inputting what I want, but basically, I've just substituted the Hamiltonian operator for [itex]\hat{O}[/itex], into the second expression above. This is where I am stuck. I'm essentially left with two parts, one thats "telling" me to prove that the Kinetic Energy portion of H is Hermitian and another telling me to prove the Potential Energy portion is hermitian. However I can't seem to see how to do it.

    Thanks

    P.s. can you help with a solutio that does not use Bra-Ket notation, this question is in the section that precedes Dirac notation
     
  2. jcsd
  3. Nov 6, 2011 #2

    Simon Bridge

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  4. Nov 6, 2011 #3

    dextercioby

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    Working in the Hilbert space L2(R) one proceeds like this:

    a) finds the domain of H.
    b) checks if domain is dense everywhere in H.
    c) finds the domain of [itex] H^{\dagger} [/itex]
    d) checks that the domain of H is included in the domain of its adjoint.
    e) finally checks that the ranges of the 2 operators are equal for all vectors in the common domain (the domain of H).
     
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