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: 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]\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.


    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

    User Avatar
    Science Advisor
    Homework Helper

  4. Nov 6, 2011 #3


    User Avatar
    Science Advisor
    Homework Helper

    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).
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook