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Recently, I have been researching the Schrodinger Equation. I am now attempting to derive the solution to the equation for the Hydrogen Atoms potential energy function. What I am wondering is that whether it is easier to do a coordinate change in the Schrodinger equation to make it Spherical. I am specifically working with this version: $${ih}\frac{\partial\psi}{\partial t}=[\frac{-h^2}{2m}\nabla^2+V(\mathbf r, t)]\psi$$ Additionally, I am wondering if there are any Spherical Harmonics you could pull out of this. Thanks in advance.

ExtantCanvas966
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    Yes, it's much easier in spherical coordinates. One of the outcomes is that a stationary solution will have constant angular momentum. That implies that the solutions will be spherical harmonics multiplied by a radial function. I am not sure if anyone published a solution derived in Cartesian coordinates. – Andrei Apr 05 '19 at 19:17
  • Thanks, this is really helpful. I was actually thinking along the same lines, but I got spooked by how messy spherical coordinates can get. – ExtantCanvas966 Apr 05 '19 at 22:02
  • For all central potentials just use spherical coordinates. It will help separate the wave function into spherical harmonics and a radial function – Andrei Apr 06 '19 at 01:27
  • https://chemistry.stackexchange.com/questions/39680/why-is-there-a-need-of-polar-coordinates-to-solve-the-schr%C3%B6dinger-wave-equation – Cheong Sik Feng Jan 01 '20 at 04:06

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