- #1
sa1988
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- 23
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