schrodinger equation mode solving

[simbilod]

beginning some code to mode solve for simple quantum mechanical problem

@HelgeGehring am I implementing the below correctly? (worrying about signs and units, I need to make the potential unrealistically large to see the right spectrum. In finite-difference there is a division by the grid spacing on the K0 derivative term that makes the units work out)

Weak form:

adapted from your example and https://physics.stackexchange.com/questions/618172/weak-solution-of-schr%C3%B6dinger-equation#:~:text=%CF%88m(x)%3D%E2%88%9A,n%CF%80x%2FL)., adding potential term

[HelgeGehring]

I've adjusted the numbers to https://young.physics.ucsc.edu/115/quantumwell.pdf and switched it to a dense solver

the dense solver is quite nice for development (but sadly too slow for most stuff), as it avoids some problems of sparse solvers, no need to guess the eigenvalue and it also shows the full spectrum

[simbilod]

I've had good luck using shift-invert mode on sparse eigensolvers, and shift-invert targeting a number lower than my smallest eigenvalue (e.g. energy at the bottom of the well). The spectrum of the 1D time-independent Shrodinger Equation is simpler than the Maxwell wave equation

[HelgeGehring]

Yeah, we should try the sparse now :)
Using the dense solver is just easier for getting the equations right :)