Schroedinger equation with derivative boundary conditions

Question

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?

Is ′ the derivative of the wavefunction evaluated at the boundary? or is it the wavefunction itself? It's not clear. — JQK, Aug 10 '23 at 15:22
Essentially duplicates of https://physics.stackexchange.com/q/178090/25301, https://physics.stackexchange.com/q/30374/25301, possibly others? — Kyle Kanos, Aug 10 '23 at 15:25
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. — QPhysl, Aug 10 '23 at 15:55
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. — Matt Hanson, Aug 10 '23 at 18:15
Does this answer your question? Is particle in 1D box, finite & infinite well same case? — Matt Hanson, Aug 10 '23 at 18:17
@MattHanson OP to this question is asking about the Neumann boundary conditions, which I do not see covered in the answers you linked. — Kyle Kanos, Aug 10 '23 at 18:40
They come up essentially automatically when you cover the particle in a box or infinite potential well. I suppose you could have used Dirichlet conditions, but I’ve always seen the Neumann conditions, even when they are not called that. — Matt Hanson, Aug 10 '23 at 18:44

0 Answers0