# Prove the Hamiltonian Operator is Hermitian

## Homework Statement

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

## Homework 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 +∞

## 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

dextercioby
c) finds the domain of $H^{\dagger}$