# Prove the Hamiltonian Operator is Hermitian

1. Nov 5, 2011

### atay5510

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

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

2. Relevant equations

A Hermitian operator $\hat{O}$, satisfies the equation

<$\hat{O}$>=<$\hat{O}$>*

or

∫$\Psi$*(x,t)$\hat{O}$$\Psi$(x,t)dx = ∫$\Psi$(x,t)$\hat{O}$*$\Psi$*(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 $\hat{O}$, 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. Nov 6, 2011

3. Nov 6, 2011

### dextercioby

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 $H^{\dagger}$
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).