How to solve the Schrödinger equation.

[grayson]

Hello, I have tried many, many times to learn the Schrödinger equation. The only problem is that it seems you need to know the wave equation to solve for it. Can anyone help me?

I meant the wave function, not wave equation btw

[joigus]

Depends very much on the potential energy.

Free Schrödinger equation: plane waves, (harmonic)x(plane waves), etc.

Schrödinger equation in potential wells, barriers etc: Plane waves with discrete harmonics, reflected and transmitted waves, etc.

Schrödinger equation with inverse-square-distance potential: Hydrogen eigenfunctions

Schrödinger equation with linear potential: Airy functions

Schrödinger equation with harmonic-oscillator potential: Hermite polynomials,

etc.

So when people talk about "the" Schrödinger equation, they actually mean "the infinitely many" Schrödinger equations associated to infinitely many possible potential energy where the electron can be captured.

Edited December 15, 2023 by joigus
minor correction