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## Homework Statement

Not actually a homework question but is an exercise in my lecture notes.

## Homework Equations

I'm following this which demonstrates that the momentum operator is Hermitian:

## The Attempt at a Solution

$$KE_{mn} = (\frac{-\hbar^2}{2m}) \int\Psi_{m}^{*} \Psi_{n}^{''} dx $$

$$ by parts: \int uv' = uv - \int vu' $$

$$ KE_{mn} = (\frac{-\hbar^2}{2m}) \Big( \Psi_{m}^* \Psi_{n}^{'} - \int \Psi_{n}^{'} \Psi_{m}^{'*} dx \Big) $$

$$ KE_{mn} = (\frac{-\hbar^2}{2m}) \Big( \Psi_{m}^* \Psi_{n}^{'} - (\Psi_{n}^{'} \Psi_{m}^{*} - \int \Psi_{m}^{*}\Psi_{n}^{''} dx) \Big) $$

$$ KE_{mn} = (\frac{-\hbar^2}{2m}) \int \Psi_{m}^{*}\Psi_{n}^{''} dx $$

$$KE_{mn} = KE_{mn}$$

Can anyone see the gaping error in my working?

Thanks