1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Verify kinetic energy operator is Hermitian.

  1. Oct 19, 2016 #1
    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 :oldsmile:
  2. jcsd
  3. Oct 19, 2016 #2


    User Avatar

    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.
  4. Oct 19, 2016 #3
    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 :)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted