Uploaded March 2026 | Updated September 2026, 3 weeks ago
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. Laplace's equation is the sum of the 2nd partial derivatives of a function with respect to the x, y, z terms (in rectangular coordinates) and is equated to zero. Converting this equation into spherical coordinates (r, θ, φ) (which I do by first obtaining the corresponding Laplace operator or Laplacian in polar coordinates), and then solving it produces solutions known as solid spherical harmonics. Focusing only on the angular terms θ and φ obtains the spherical harmonics, which, when plotting only the real terms (ignoring the imaginary terms), and distorting radial terms proportional to the magnitude of values on the sphere, we obtain the distinct 3D lobes!
#math #sphericalharmonics #calculus #quantumphysics #science
Timestamps:
- Intro – 0:00
- Topics to cover – 0:47
- Introduction to Spherical Harmonics – 1:40
- Visualization of Real Spherical Harmonics – 25:58
- Atomic Orbitals – 31:24
- Magnetic Fields – 56:40
- Mathematical Review – 1:04:00
- Laplace Operator in Polar Coordinates – 1:14:07
- Laplace Operator in Spherical Coordinates – 2:09:00
- Solid Spherical Harmonics: Solutions to Laplace Equation in Spherical Coordinates – 2:50:38
- Associated Legendre Equation – 3:50:00
- Spherical Harmonics: Solution to the Angular Laplacian – 4:05:39
- Deriving the Real Spherical Harmonics – 4:16:44
- Graphing Real Spherical Harmonics in Desmos Calculator – 5:07:41
- Outro – 5:19:55
Notes and playlists:
- Summary: inleo.io/threads/view/mes/re-leothreads-x5wsh2ax
- Notes: peakd.com/hive-128780/@mes/220d6572
- PDF Notes: https://1drv.ms/b/c/88862ef47bcaf6cd/IQCuF1y7tt2IR6fssJkQvz4VAUXIXkwKoyrNEqqKQKsGDOc
- Sections playlist: youtube.com/playlist?list=PLai3U8-WIK0FHh2kZzebadnPFwO6Yi5qu .
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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. Laplace's equation is the sum of the 2nd partial derivatives of a function with respect to the x, y, z terms (in rectangular coordinates) and is equated to zero. Converting this equation into spherical coordinates (r, θ, φ) (which I do by first obtaining the corresponding Laplace operator or Laplacian in polar coordinates), and then solving it produces solutions known as solid spherical harmonics. Focusing only on the angular terms θ and φ obtains the spherical harmonics, which, when plotting only the real terms (ignoring the imaginary terms), and distorting radial terms proportional to the magnitude of values on the sphere, we obtain the distinct 3D lobes!
#math #sphericalharmonics #calculus #quantumphysics #science
Timestamps:
- Intro – 0:00
- Topics to cover – 0:47
- Introduction to Spherical Harmonics – 1:40
- Visualization of Real Spherical Harmonics – 25:58
- Atomic Orbitals – 31:24
- Magnetic Fields – 56:40
- Mathematical Review – 1:04:00
- Laplace Operator in Polar Coordinates – 1:14:07
- Laplace Operator in Spherical Coordinates – 2:09:00
- Solid Spherical Harmonics: Solutions to Laplace Equation in Spherical Coordinates – 2:50:38
- Associated Legendre Equation – 3:50:00
- Spherical Harmonics: Solution to the Angular Laplacian – 4:05:39
- Deriving the Real Spherical Harmonics – 4:16:44
- Graphing Real Spherical Harmonics in Desmos Calculator – 5:07:41
- Outro – 5:19:55
Notes and playlists:
- Summary: inleo.io/threads/view/mes/re-leothreads-x5wsh2ax
- Notes: peakd.com/hive-128780/@mes/220d6572
- PDF Notes: https://1drv.ms/b/c/88862ef47bcaf6cd/IQCuF1y7tt2IR6fssJkQvz4VAUXIXkwKoyrNEqqKQKsGDOc
- Sections playlist: youtube.com/playlist?list=PLai3U8-WIK0FHh2kZzebadnPFwO6Yi5qu .
------------------------------------------------------
Become a MES Super Fan! youtube.com/channel/UCUUBq1GPBvvGNz7dpgO14Ow/join
DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate
SUBSCRIBE via EMAIL: https://mes.fm/subscribe
MES Links: https://mes.fm/links
MES Truth: https://mes.fm/truth
Official Website: https://MES.fm
Hive: peakd.com/@mes
Email me: contact@mes.fm
Free Calculators: https://mes.fm/calculators
BMI Calculator: https://bmicalculator.mes.fm
Grade Calculator: https://gradecalculator.mes.fm
Mortgage Calculator: https://mortgagecalculator.mes.fm
Percentage Calculator: https://percentagecalculator.mes.fm
Free Online Tools: https://mes.fm/tools
iPhone and Android Apps: https://mes.fm/mobile-apps
