Homework Help: Verify kinetic energy operator is Hermitian.

1. Oct 19, 2016

sa1988

1. The problem statement, all variables and given/known data

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

2. Relevant equations

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

3. 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

2. Oct 19, 2016

Staff: Mentor

The second time you are doing the integration by parts, you are making the wrong choice for $u$ and $v'$.

Also, don't forget that this is a definite integral.

3. Oct 19, 2016

sa1988

Thanks for this. Relieving to see it was a fairly simple arithmetic error, but worrying that my eyes continually didn't pick up on it...

Pretty late in the evening now and I don't have any pen or paper handy for going over it all, but I'll check it out tomorrow.

Cheers :)