How Are 2D Standing Waves Extended to 3D in Quantum Mechanics?

[etotheipi]

I’ve seen the description of electrons around an atom as existing as standing waves with different harmonics corresponding to different energy levels.

The atom is evidently 3-dimensional, and the wave function of an electron must also be in terms if 3 spatial coordinates.

What mathematical extension to 2D standing waves is made to convert them i to a 3D form? I’m almost certainly not nearly good enough to understand the full maths behind this but was just wondering where I could find a more high level description of this.

Thanks

 

[BvU]

Hi $e^\pi$

Google 3D standing waves to get an idea
Or hydrogen orbitals

See e.g. here