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## Homework Statement

Show that the Hamiltonian operator ([itex]\hat{H}[/itex])=-(([itex]\hbar[/itex]/2m) d

^{2}/dx

^{2}+ V(x)) is hermitian. Assume V(x) is real

## Homework 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 +∞

## 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