Spherical harmonics for Schrodinger equation

Spherical harmonics are special mathematical functions defined on the angular coordinates, the azimuthal and polar angles, phi and theta, as shown in the box. Why are spherical harmonics important in quantum mechanics? As we will see, the spherical harmonics make up part of the electron’s wavefunctions in a hydrogen atom. Just like the spatial and time harmonics we have seen earlier are consequences of the spatial and time translation symmetry, the spherical harmonics is a consequence of the rotational symmetry of the hydrogen problem. In this video, we shall review how the spherical harmonics solutions is obtained, and how one can visualize them.
Technical Content: Tony Low
Video Production: Katie Low, Tony Low
Erratum.
In 2.22, the angles \phi and \theta in the equations on the right should be interchanged. Thanks to ‪@cinemaclips4497‬ for pointing out this typo.

[@aravindsai2409]

Thank you. You made it so easy to understand.

[@tgocamargo]

The way you explain the sequence of solutions made understanding much easier. Congratulations on the quality material you have on your channel.

[@professornanoscience]

Glad it was helpful!

[@kisho2679]

While spherical harmonics are the eigenfunctions of the hydrogen atom to describe its state, which are the eigenfunctions for the other 117 chemical elements (helium, etc.)?

[@ManojKumar-cj7oj]

cant belive how easy u made it for me , thank u so much

[@professornanoscience]

Most welcome 😊

[@Zoro-fl2mn]

Can you give a source for the mathematical derivation of the polar solution?

[@cinemaclips4497]

It should be z=rcosθ from the way you labeled your axes.

[@professornanoscience]

thanks! I added an erratum in the video description and point out this typo. Pls check.

[@kisho2679]

how about the spherical harmonicals for the other chemical elements in the periodic table (not just is Hydrogen)?

[@professornanoscience]

Spherical harmonics is just the solutions of the Laplacian with spherical symmetry. It applies to other elements to the extent that the one electron picture is valid, and that is a hairy thing to discuss here.