We had the following exam question in our quantum theory undergrads course:

Solve the time independent Schroedinger equation for the following Hamiltonian:

$\hat{H} = \frac{\hat{p}^2}{2m}$ for $x \in [-\frac{L}{2},\frac{L}{2}]$ with the boundary conditions $\frac{d\psi}{dx}|_{\pm\frac{L}{2}} = 0$.

Does sombedy know any application of this type of boundary conditions, or know where they can be used?

  • $\begingroup$ Is 𝜓′ the derivative of the wavefunction evaluated at the boundary? or is it the wavefunction itself? It's not clear. $\endgroup$
    – JQK
    Commented Aug 10, 2023 at 15:22
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    $\begingroup$ Essentially duplicates of physics.stackexchange.com/q/178090/25301, physics.stackexchange.com/q/30374/25301, possibly others? $\endgroup$
    – Kyle Kanos
    Commented Aug 10, 2023 at 15:25
  • $\begingroup$ These describe the simplest case of Neumann boundary conditions. Essentially you solve it like usual to get your sines and cosines then "enforce the condition" at the end. $\endgroup$
    – QPhysl
    Commented Aug 10, 2023 at 15:55
  • $\begingroup$ This question is just the particle in an infinite one dimensional well problem, which is covered thoroughly in every quantum textbook. It has also been covered extensively on this site, so you should look at one of those solutions. $\endgroup$ Commented Aug 10, 2023 at 18:15
  • $\begingroup$ Does this answer your question? Is particle in 1D box, finite & infinite well same case? $\endgroup$ Commented Aug 10, 2023 at 18:17