Prove the Hamiltonian Operator is Hermitian

[atay5510]

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 that's "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

 

[Simon Bridge]

$$\int A^* \phi^*(x) \psi(x) \, \mathrm{d}x = \int \phi^*(x) A \psi(x) \, \mathrm{d}x$$

partial integration

see:
https://www.physicsforums.com/showthread.php?t=180611

 

[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).