Real Spherical Harmonics – Removing the Imaginary Terms

In this video, I derive the real spherical harmonics by excluding the imaginary complex i terms from the complex spherical harmonics. The real spherical harmonics are what are typically plotted when graphing atomic orbitals or magnetic fields, since they are easier to visualize in 3D and still inherit the important properties of complex spherical harmonics. I derive the proportion forms (thus excluding the proportionality constants) of the real spherical harmonics from their typical definition, and make use of complex conjugation and Euler's formula to obtain a compact formula for the cases where m is positive or negative.

#math #sphericalharmonics #calculus #quantumphysics #3d