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

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Timestamps

Real spherical harmonics are defined to remove the complex terms – 0:00
Complex spherical harmonics are chosen using the basis solutions rather than the linear combinations – 2:24
Real spherical harmonics in their proportional forms, ignoring the normalization constants – 3:49
Complex conjugate in polar form flips the sign of the exponential power via Euler's formula – 4:43
The Associated Legendre equation depends on m², so -m gives a solution that is proportional to that using +m, via the Rodrigues' formula – 11:23
Case where m is greater than 0 – 18:23
Case where m = 0 – 26:39
- Note that this yields just the ordinary Legendre polynomial – 30:20
Case where m is less than 0 – 32:20
Putting it all together to get our real spherical harmonics in proportional form – 45:29
- Writing it in more compact form – 47:53