Planck's Constant as Quantum of Energy via FractalWoman's Modified Unit Analysis https://youtu.be/UFg4wzaQGT8

In this video I compare the conventional view of the Planck's constant as being merely a proportionality constant to solve the blackbody radiation problem vs FractalWoman's unique insight that Planck's constant may actually represent the minimum energy unit of ALL electromagnetic frequencies.

#science #STEM #physics #QuantumPhysics #education

Spherical Harmonics: Atomic Orbitals, Laplacian in Polar and Spherical Coordinates, Laplace Equation - 3Speak - YouTube - Summary - Notes - Playlist

In this video, I go over a deep dive into the famous spherical harmonics, which are the solutions to Laplace’s equation in spherical coordinates and are very prevalent in physics, especially in the quantized configurations of atomic orbitals or the electron charge distribution around the atom's nucleus.

#math #sphericalharmonics #calculus #quantumphysics #science

Solid Spherical Harmonics – Solutions to Laplace's Equation in Spherical Coordinates - 3Speak - YouTube - Summary - Notes - Playlist

In this video, I go over the derivation of the solid spherical harmonics, which are solutions to the Laplace equation in spherical harmonics. They are referred to as "solid" because they include the radial term as well. I use separation of variables to define a function that is a multiple of 3 separated functions, and then selected separation constants to obtain a periodic, repeating solution. I solve the function corresponding to the azimuthal angle ɸ via the derivative of an exponential function. The function containing the radial r term is the Euler-Cauchy equation, and I solve it via substitution. The middle function is the associated Legendre equation, whose solution is beyond the scope of this video, so I just plugged in the corresponding function. Combining all of these obtains our solid spherical harmonics!

#math #sphericalharmonics #calculus #quantumphysics #laplace

Spherical Harmonics – Solutions to the Angular Laplacian - 3Speak - YouTube - Summary - Notes - Playlist

In this video, I go over the actual spherical harmonics equations, which exclude the radial terms from the solid spherical harmonics I had derived earlier. The spherical harmonics can be combined into one function that contains all of the angular terms, thus essentially treating the sphere's radius as a constant. I also show how the spherical harmonics can be considered the solution of a modified Laplace equation, and hence a modified Laplace operator termed the angular Laplacian.

#math #sphericalharmonics #calculus #quantumphysics #laplace

Real Spherical Harmonics – Removing the Imaginary Terms - 3Speak - YouTube - Summary - Notes - Playlist

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