[![](https://images.hive.blog/DQmXx43erFBUAcR9qXMNihbH76kwPqejKRkpK55fyRrKgV7/441371d016fa13ea3f0ac69f70c1bdf3)](https://3speak.tv/watch?v=mes/220d6572) ▶️ [Watch on 3Speak](https://3speak.tv/watch?v=mes/220d6572) - [Odysee](https://odysee.com/Spherical-Harmonics:b411814bbf4b88a22d2b136854be4c867f77591b) - [BitChute](https://www.bitchute.com/video/gFSCuLXGejxk) - [Rumble](https://rumble.com/v76qf26-spherical-harmonics-atomic-orbitals-laplacian-in-polar-and-spherical-coordi.html) - [Telegram](https://youtu.be/R7SAhbuZPig) - [YouTube](https://youtu.be/p5SdobFPiKY) - [Summary](https://inleo.io/threads/view/mes/re-leothreads-x5wsh2ax) - [PDF Notes](https://1drv.ms/b/c/88862ef47bcaf6cd/IQCuF1y7tt2IR6fssJkQvz4VAUXIXkwKoyrNEqqKQKsGDOc) - [Hive Notes](https://peakd.com/hive-128780/@mes/220d6572) - [Sections playlist](https://www.youtube.com/playlist?list=PLai3U8-WIK0FHh2kZzebadnPFwO6Yi5qu) - [MES Links](https://mes.fm/links) --- 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 #atomicorbitals #science ![Spherical Harmonics.jpeg](https://files.peakd.com/file/peakd-hive/mes/23ydRCbjj1UKSrutn1S2JKNZsySufGoRk4g1qWUFekLJd3CFYjzktYWmdVi2fasLZ7G2D.jpeg) ## 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 --- ### View Video Notes Below! --- >[Become a MES Super Fan](https://www.youtube.com/channel/UCUUBq1GPBvvGNz7dpgO14Ow/join) - [Donate](https://mes.fm/donate) - [Subscribe via email](http://mes.fm/subscribe) - [MES merchandise](https://mes.fm/store ) >- [MES Links webpage](https://mes.fm/links) - [MES Links Telegram](https://t.me/meslinks) - [MES Truth](https://mes.fm/truth) > >**Reuse of my videos:** >- Feel free to make use of / re-upload / monetize my videos as long as you provide a link to the original video. >>**Fight back against censorship:** >- Bookmark sites/channels/accounts and check periodically. >- Remember to always archive website pages in case they get deleted/changed. > >**Recommended Books:** ["Where Did the Towers Go?"](https://mes.fm/judywoodbook) by Dr. Judy Wood > >**Join my forums:** [Hive community](https://peakd.com/c/hive-128780) - [Reddit](https://reddit.com/r/AMAZINGMathStuff) - [Discord](https://mes.fm/chatroom) > >**Follow along my epic video series:** [MES Science](https://mes.fm/science-playlist) - [MES Experiments](https://peakd.com/mesexperiments/@mes/list) - [Anti-Gravity](https://peakd.com/antigravity/@mes/series) ([MES Duality](https://peakd.com/antigravity/@mes/antigravity-part-6-video-1-objects-in-rotation-defy-mainstream-physics-mes-duality-concept)) - [Free Energy](https://mes.fm/freeenergy-playlist) - [PG](https://peakd.com/pg/@mes/videos) > >--- > >**NOTE 1:** If you don't have time to watch this whole video: > >- Skip to the end for Summary and Conclusions (if available) >- Play this video at a faster speed. >-- TOP SECRET LIFE HACK: Your brain gets used to faster speed! >-- [MES tutorial](https://peakd.com/video/@mes/play-videos-at-faster-or-slower-speeds-on-any-website) >- Download and read video notes. >- Read notes on the Hive blockchain $HIVE >- Watch the video in parts. >-- Timestamps of all parts are in the description. > >**Browser extension recommendations:** [Increase video speed](https://mes.fm/videospeed-extension) - [Increase video audio](https://mes.fm/volume-extension) - [Text to speech](https://mes.fm/speech-extension) ([Android app](https://mes.fm/speech-android)) – [Archive webpages](https://chrome.google.com/webstore/detail/archive-page/gcaimhkfmliahedmeklebabdgagipbia) --- ## Topics to Cover - Introduction to Spherical Harmonics - Visualization of Real Spherical Harmonics - Atomic Orbitals - Magnetic Fields - Mathematical Review - Laplace Operator in Polar Coordinates - Laplace Operator in Spherical Coordinates - Solid Spherical Harmonics: Solutions to Laplace Equation in Spherical Coordinates - Associated Legendre Equation - Spherical Harmonics: Solution to the Angular Laplacian - Deriving the Real Spherical Harmonics - Graphing Real Spherical Harmonics in Desmos Calculator --- # Introduction to Spherical Harmonics [Spherical harmonics](https://en.wikipedia.org/wiki/Spherical_harmonics) are special functions defined on the surface of a sphere, and have many applications in physics. [From Wikipedia](https://en.wikipedia.org/wiki/Spherical_harmonics): >Spherical harmonics originate from solving [Laplace's equation](https://en.wikipedia.org/wiki/Laplace%27s_equation) in the spherical domains. Functions that are solutions to Laplace's equation are called [harmonics](https://en.wikipedia.org/wiki/Harmonic_function). The Laplace Equation in [rectangular coordinates](https://en.wikipedia.org/wiki/Three-dimensional_space) (x, y, z) is shown below. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23uRLJc4mySmtY7ACnBy6kcsFAkPyvW8uhefA7GV3oEuJgxvNXpyLHtqg2ntspwm1uhU4.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23u6YguacRrD3R66GYW3DxB581uUDjwbHvLLED6RdmU61CB6DmLtCKfA1448UnaLm8NDM.png) Note that the above equation describes steady-state conditions (no sources or sinks), i.e. where things are in "harmony", hence solutions are called "harmonic functions". The above equation can be rewritten to obtain the Laplace Equation in [spherical coordinates](https://en.wikipedia.org/wiki/Spherical_coordinate_system) (r, θ, ɸ). ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tHb6kphfZJ5nVQgCo6L1PNE8yk4JqNFfHdNovYuqHziFJB2SBKFwSABQ5BxZtFmV8op.png) >The ***physics convention***. Spherical coordinates (r, θ, ɸ) as commonly used: radial distance r ([slant distance](https://en.wikipedia.org/wiki/Slant_distance) to origin), polar angle θ ([theta](https://en.wikipedia.org/wiki/Theta)) (angle with respect to positive polar axis), and azimuthal angle ɸ ([phi](https://en.wikipedia.org/wiki/Phi)) (angle of rotation from the initial meridian plane). ![image.png](https://files.peakd.com/file/peakd-hive/mes/23vsqxzwFziucqdL7zPxwKB5J6qYoDyqPVSRYwBavntrZeshz9NcEBvsUgWSCLDSxBdTc.png) Solving the above equation using separation of variables [obtains the Solid Spherical Harmonics](https://grok.com/share/c2hhcmQtMg_efc76689-e3d9-4ef6-8834-38352c655356): ![image.png](https://files.peakd.com/file/peakd-hive/mes/23x1aTvxqNwdhUKGyps7c5nscQLTnYypiXjkZHUT9vRrbf2C9quaq8cynEdSpB6DzXfbi.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/48RtbrWkqoK2junxdgDEmqk2MK8CKtMCbabmFTYKTswGubeoki1RZohXSgYhdF31EG.png) >Where: > >n = 0, 1, 2, 3, … (non-negative integer) > >m = -n, -n+1, …, n (integer) > >P_n^m(t) is the [Associated Legendre function](https://en.wikipedia.org/wiki/Associated_Legendre_polynomials) with t = cos θ (so t ∈ [−1, 1]): ![image.png](https://files.peakd.com/file/peakd-hive/mes/23w3EdNWmNTwBidLc58eCd7HxEbyBsCnYPn967SvRUcRGJj2UYHGFEdrj3kzrCErU6zj4.png) >P_n(t) is the Ordinary Legendre Polynomials ([Rodrigues' formula](https://en.wikipedia.org/wiki/Rodrigues%27_formula)): ![image.png](https://files.peakd.com/file/peakd-hive/mes/23uRL1eje7fJobopDB2LzvYbGy5wyD8pJVeH73dLpmpRw6b15LyWi8hQobWGjCmbz19uk.png) >Or compacted together: ![image.png](https://files.peakd.com/file/peakd-hive/mes/EogUXX25VGe5b5w73SYHajcfpxTquAsnUJEgZAn8eZMPdj1oaSNGvFb1xfib5unfek7.png) The general solutions to the Laplace Equation are called [Solid Spherical Harmonics](https://en.wikipedia.org/wiki/Solid_harmonics) as they contain both the radial term (r) and angular terms (θ, ɸ). If instead we remove the radial term so that we obtain only the angular terms, then such solutions are termed [Spherical Harmonics](https://en.wikipedia.org/wiki/Laplace_operator). ![image.png](https://files.peakd.com/file/peakd-hive/mes/23viRcjJQra5KyBNMfqZAfM8Kyi3n8S3WFeCwZTbDGV8TpMwa9XzxwHMfyWKz4a1dq8gH.png) >Where ∇² (or Δ) is the [Laplace operator or Laplacian](https://en.wikipedia.org/wiki/Laplace_operator): Multiplying by r² we get: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23uRKsyid8dRWcYG9u1jcwcBuZjtTVceSeQqDF4JpNY2snDA2rSGqmhYTbbAgxmECo46k.png) >or ![image.png](https://files.peakd.com/file/peakd-hive/mes/EocDksNvFgPz89QYSkVqAJSrhSLzEUxtw9JFySCGqgSk7YnEeTDN4G8Uvb23bp4zj2P.png) >Where Λ² is the [angular Laplacian](https://grok.com/share/c2hhcmQtMg_61a88cec-b61e-4ca7-a74b-7549f15af961) (or [Laplace-Beltrami operator](https://en.wikipedia.org/wiki/Laplace%E2%80%93Beltrami_operator) on the sphere): ![image.png](https://files.peakd.com/file/peakd-hive/mes/23viTAxLiqjLTA4PLNfsjHpa6nxEn2ge53QBX4FshEap9tLhkqkkhBJ9ZQ5t9AsKY99uC.png) Rewriting f: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23xAwa5W2gkeyhBXAqkKvGVRwTcbWwpjtaUL6J1C2EDdd4C7Dhy2YfT6EtH2QbzdX3tzt.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23xLKhBsFFVzxKqCdSGxEocEEhbm13RnUJppfDJPWBD7upYwpcGQJ2pKi18g5awD5HcmM.png) Solving Y we obtain the complex Spherical Harmonics: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23wXQcVFK3VykgY8tHsy6BWVQQNvjYF6mk111Gq8boYnZZL4B7bAUgwqySAnfUHdB1Lq8.png) >Where N is a normalization constant, usually: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23uFwMHA88orVpjWqKUfm8mi5sJJxEg7VRzy3btEZAuL2mGUmJM21JCWYsRCVDYeY8ZSe.png) To ensure the [following integral](https://en.wikipedia.org/wiki/Spherical_harmonics) (used in Quantum Mechanics for probability) is equal to 1: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23vsqxzwFqLP1G92PTYM4D4mvyBfGYsmBKSuuLMAMQtCYG7m4s1H4aHHvnakFESrJEm3S.png) >or ![image.png](https://files.peakd.com/file/peakd-hive/mes/23wgoGrppohEob1bfGQrBDo1hHjQ2r5Lq7eZbGScjotf9Ses5Rb5iZZGTC7a8w8UBy8LV.png) The **real Spherical Harmonics** are defined such that the functions are real-valued everywhere, with no complex term i. [The usual definition](https://grok.com/share/c2hhcmQtMg_279828ef-748b-44aa-8dce-ad48866477c5) is given below, and is almost always used when displaying Spherical Harmonics visually, since they [exclude the complex Phase factor](https://grok.com/share/c2hhcmQtMg_3af8d7dd-5260-44c6-820a-a9e8ac8f33bc) e^imφ (which is the rotating about the z-axis). ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tHbg12Bcpx1sj5h1oYzSiGvGSU73ro8VbigZ72gXnj3ijae4WSHk3FdDvhZY931dFvA.png) [Plugging in all the values into the above definition](https://grok.com/share/c2hhcmQtMg_a08d5833-e537-4e3d-88a0-497029993abe) obtains the simplified form below, in which the proportionality symbol ∝ is used to just show the functional shape, rather than the normalization constant. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tSz8J2EzZz3GKcUpaMF2tYQhAjYvDEuWZDEruz4QqVvxQ576atX5NnwS1t3nTGK8fhz.png) or ![image.png](https://files.peakd.com/file/peakd-hive/mes/EocDj8Z7Qxr5M5bzS8Ks6qj4o1PshU6JhREmXTaiactVsmBHurRa9ATAmvAr8aP3X3g.png) The complete, correctly normalized real spherical harmonics would be something like below. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tcNhnr5GNn7W2xXj4PvNBbAyGwjRKUmqDVuwyNRjMT5em9isJ6xjeMqHUusFaP4do9a.png) >Where the exact value of the normalization constant is usually: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23uFwMHA8Rz8rXrN3rnBjCRCDowzS7YA2gBW9ySCGK2suMTZaQzCot69FUGXXT5scvvgp.png) Thus the [full general solution to Laplace's equation](https://grok.com/share/c2hhcmQtMg_ec846ed6-098b-4171-a1bb-a74b810f2d31) in spherical coordinates is a sum (or series) over all possible n and m: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23viTPW7RSvKDieFAHDZgYndfFfckzGrhx5eaQMcDdjoGREWQ4BJKnmXRxc4npP2eUEVK.png) >Where: > >A_nm and B_nm are constants determined by boundry conditions (e.g., values of f. on a sphere or at infinity). The Spherical Harmonics describe angular patterns that fit the sphere's geometry, and are "natural modes" or "basis functions" for solving problems on or around spheres, such as waves on a balloon or electron clouds in atoms. --- ## Visualization of Real Spherical Harmonics Visualizations of the first few real [spherical harmonics](https://en.wikipedia.org/wiki/Spherical_harmonics) are shown below. ![image.png](https://files.peakd.com/file/peakd-hive/mes/AKGddBKAAHXHwB6dCNtGuaeMFSW4r1T51wj2bYC5yoYnRiU2Cy7rcX5vdpErEJk.png) >[Animation](https://en.wikipedia.org/wiki/Spherical_harmonics#/media/File:Rotating_spherical_harmonics.gif): Real (Laplace) spherical harmonics for (top to bottom) and (left to right) Note that the spherical harmonics can be visualized by [distorting the sphere radially proportional to the magnitude of the values on the sphere](https://grok.com/share/c2hhcmQtMg_b901793e-ea40-4a7d-b5da-9135600505d3), thus obtaining the familiar "lobes". ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tmjgGNFjcqcNPCp351BQszNoT9hfyZTBReXtBTjpRkZKBjvpkiwaDNBQQEsK7ApQfYm.png) >[Alternative picture](https://en.wikipedia.org/wiki/Spherical_harmonics#/media/File:Sphericalfunctions.svg) for the real spherical harmonics Y_ℓm. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23t7eqdtCGPHH1Cc6JtpKKnDo9WJdW3jDZSGjuqa5zjDtG6AXaG29kyNt8DVyPc3jUfn5.png) >[Figure](https://en.wikipedia.org/wiki/Spherical_harmonics#/media/File:Spherical_Harmonics.png): Blue portions represent regions where the function is positive, and yellow for when it is negative. The distance of the surface from the origin indicates the absolute value of Yℓm(θ,φ) in angular direction (θ,φ). ![image.png](https://files.peakd.com/file/peakd-hive/mes/23xpVqUgHV2kW47U6ES6VUhXEFRr6AtAaodDgLfRACZmCMAvxQegHBgG3MX9b4swoBzYi.png) >[Instagram post](https://www.instagram.com/p/DTk7l8zDGPg/?e=b4d42d14-1156-432b-afbe-08a0bd4aa1af&g=5) --- ## Atomic Orbitals Interestingly, spherical harmonics resemble [atomic orbitals](https://en.wikipedia.org/wiki/Atomic_orbital), which describe the location and wave-like behavior of an electron in an atom (where ℓ is often used instead of n as I do in this video). Note that atomic orbitals are not simply radial distortions as in pure spherical harmonics, but they [actually contain the radial term in the solutions](https://grok.com/share/c2hhcmQtMg_b901793e-ea40-4a7d-b5da-9135600505d3). Atomic orbitals are solutions to the [Schrödinger equation](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation), which is shown for the electron in a hydrogen atom (or a hydrogen-like atom) as well as its solution below. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23uRLJc4nAZFpCs4pqLD9osbMwhbZV95N8y2y5k8itZLeAVf1sjB3UNYBBrd2YehBuw6i.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tmjWHT3mDc4Dh9JrLnZtdazQSaPhCDvMEJWaSpfpjHvgjcpBjfAesC3TrdK3dr8hVcC.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tw8Af2WmjY9BBmzeCTEkZJpCg5ubQRJaMCfXxQkNvJXT5gic1gis4T3RS29DNF4AiDV.png) Summary of terms [via Grok AI](https://grok.com/share/c2hhcmQtMg_9a1876d5-4d9b-447f-b93b-91f08d406bd2) and their physical meaning. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23sxovdWzzvRp2YUnnRYbYNR3y9SikfAxsQ8xsrFvPZc3r8dbZxiDDv7RKv8LN86hrtxG.png) Using separation of variables in spherical coordinates: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23xf97RGtRihWq2mkmXGm2em2xNrWfNW6B6Vcy9EDUo5hnyvCKpwAZPbCeUrSPAkpaCMT.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/EoKC9wvf84GUvgaJXFvvUzTVx6sbeN7H9ZNCUcAfbjzsjwhYH3ngnHdwCMoicyXMiwd.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23viTPW7RfHi2RsrPHpJ9fxg6qHVVdZXm2vXG7tZbixQCUPnn4zjrVBwoaKi6KfSi5jcH.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23vsr8HUGSABDMsYMLiUWtsZwrHQvyTdNQtUBzxB9sCFJ7x9Nsiuvi76CPkriHcy5GS6w.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tvYJJUDSHK47DPnu8wyF93vj3AYQupZZPTSJKxgpuJuXV2Pcvcfa4TH7t67w4L8haZo.png) >[Figure](https://en.wikipedia.org/wiki/Atomic_orbital#/media/File:Hydrogen_Density_Plots.png): Cross-sections of atomic orbitals of the electron in a hydrogen atom at different energy levels. The probability of finding the electron is given by the color, as shown in the key at upper right (the - / + indicate the phase of the Wave Function). ![image.png](https://files.peakd.com/file/peakd-hive/mes/23xf9Ky3ZA1pR9NpbqjTYUQC2ptrUEA89pcpADRpLFGPhWktUCPpo2JNPGx6948zwkpf4.png) >[Figure](https://en.wikipedia.org/wiki/Atomic_orbital#/media/File:Atomic-orbital-clouds_spdf_m0.png): 3D views of some [hydrogen-like](https://en.wikipedia.org/wiki/Hydrogen-like_atom) (any atom or ion with a single electron) atomic orbitals showing probability density and phase (g orbitals and higher not shown) The table below shows the real hydrogen-like wave functions for all atomic orbitals up to 7s, and therefore covers the occupied orbitals in the ground state of all elements in the periodic table up to [radium](https://en.wikipedia.org/wiki/Radium). ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tw5xn9bpPs3e9kSuYtBSebmFcuU97byhxUZy6du8ds6SfADB8GueNEfydnJD8UWjcwE.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8AjTfzgsWwcZiX5UTtMNozMzR2QVZXKtxWMEcgGSdoe5hA8Tcz2en9r3fK9dxzkxpk.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tSxMXDFGM2USdyZ5Q7FwtnUwHGWxQmb8dkP8ncUYZgmzCqh9agPe93cUSTqdBPm8GEJ.png) >"One should remember that these orbital 'states', as described here, are merely [eigenstates](https://en.wikipedia.org/wiki/Eigenstates) of an electron in its orbit. An actual electron exists in a superposition of states, which is like a [weighted average](https://en.wikipedia.org/wiki/Weighted_average), but with [complex number](https://en.wikipedia.org/wiki/Complex_number) weights." >"The [quantum state](https://en.wikipedia.org/wiki/Quantum_state#Eigenstates_and_pure_states) after a measurement is in an **eigenstate** corresponding to that measurement and the value measured." >"Numerical or analytic solutions in quantum mechanics can be expressed as pure states. These solution states, called **eigenstates**, are labeled with quantized values, typically quantum numbers. For example, when dealing with the energy spectrum of the electron in a hydrogen atom, the relevant pure states are identified by the principal quantum number n, the angular momentum quantum number ℓ, the magnetic quantum number m, and the spin z-component s_z." Instead of the complex orbitals described earlier, it is common, especially in the chemistry literature, to use real atomic orbitals. These real orbitals arise from simple linear combinations of complex orbitals. Real orbitals are related to complex orbitals in the same way that the real spherical harmonics are related to complex spherical harmonics. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tHbkj8BGBprPwNxmAPzse2Z1w8T3nm84fEbbFhsPvvRANz7coPeo497s5DyYLMpc3bE.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23u6WukGn8msqMRyp4gfQfz3whGPkU8nHrvwJyxrLgyQyeATeqgrm4W5Cdv5BuqZR1w1R.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/EpGUbrYo5vUZJzNnksR7s79po2iUmWGzJTfsfzmkAQWGifjpuY9fCtAHDefjUozdewJ.png) >[Animation](https://en.wikipedia.org/wiki/Atomic_orbital#/media/File:Orbital_p1-px_animation.gif): Continuously varying superpositions between the p₁ and the p_x orbitals. [Note that the superpositions are the separated](https://grok.com/share/c2hhcmQtMg_673c088d-ad69-4f92-8799-6b06952d7db7) lobes (chosen so the complex terms are eliminated) and not the rainbow eigenstates (or "pure" fundamental solutions. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23wCbbr2wyjoQPPKu7kj6LdQs28pq5nuxcCzLkzAe3S8dc9cUcgN2XRnyrhoAt978eY7w.png) >[Figure](https://en.wikipedia.org/wiki/Atomic_orbital#/media/File:Atomic_orbitals_spdf_m-eigenstates_and_superpositions.png): Atomic orbitals spdf m-eigenstates (right) and superpositions (left) Fascinating stuff and a bit beyond the scope of this video and my head (for now)! --- ## Magnetic Fields Likewise, magnetic fields can be expressed as spherical harmonics, as expressed in [Lori Gardi aka FractalWoman's paper](https://peakd.com/phisics/@fractalwoman/self-similar-structures-in-physical-fields-an-analogy-between-magnetic-projections-and-electron-orbitals). For a magnetic multipole, the [scalar potential outside sources](https://inleo.io/threads/view/askrafiki/re-mes-mm0zar8p) (mathematical function describing magnetic fields [away from a source of current](https://inleo.io/threads/view/askrafiki/re-mes-mm0zgxab)) can be expressed as: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23viTTuZXgmUmuw5wZqUjgMfWZDf95JCY5iTLv5wrodhYfTEnmGhgyuRcJReYQG3G6dNk.png) The magnetic field is then obtained as the negative gradient: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23xVitpf4D9BQmSWLq7FLuyXkkypiFoSwjZcGX5XnGu5Cmf3YzvdBezJRmtiUzpjK1qSm.png) Measuring a single component of **B** (via a [Hall sensor](https://youtu.be/ANNyArOO1kQ)) gives a scalar projection that depends on the orientation, producing lobes and nodes analogous to orbital probability patterns. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23uRDdyqXGqPUV8Y42JjBSzccdZKVHAm3WTdc7UTogLZqSYxdDs41PLX1iSrXFS6HD9Hr.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23r23txHrvag1CeMPs8AMJKr3q5spjQKTAyjaS5eaxqYAeQzxhL4trvtdvMR8Nze9sHk6.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/241tLjP8W8JpNzLrE7Ji6n5mm3Bj7FLH89rmDwcJfKCUP1FUdTMHB2ya8v3xLtLJqNCRG.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/EoARy34YymTxoC1q52EBv4wUzCE7FyRxCpicUj1HGnww3aMSagfxS3Gojp4HqYdFra9.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tRpGSWxMyZHuRBFVkwWzG24SocnUZAeZV8Nw1FwHxrFMVtW9jkGKqw8vBDyy99tKPTz.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23t6zDfaApfBFSn9pCfiKzgf18DWUtQxdbppJsB1haULwjtnDgyxxrhFSKLmUo2DAagBj.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/EoH5CEJEVddeDrSDJHfwVSph7DvVbsNeV8SfD2HZgtbDvjWX8mHwVj7F5jN6GUXh19Z.png) >[Figure](https://mriquestions.com/how-to-map-bo.html): Any complex magnetic field can be expressed in terms of a sum of weighted spherical harmonics of different orders. Order 0 is a constant value. Order 1 contains simple linear harmonics, here principally directed and labeled in the Cartesian coordinate system as Z, Y, and X respectively. The 2nd Order has five harmonics, called Z², ZX, X²-Y², ZY, and XY. Order 3 has seven harmonics, a few of which are illustrated. As the order increases so do both the number and complexity of the harmonics. --- # Mathematical Review [Laplace's Equation](https://en.wikipedia.org/wiki/Laplace%27s_equation) is a second-order **partial differential equation** named after Pierre-Simon Laplace who first studied it in 1786. A [partial differential equation (PDE)](https://en.wikipedia.org/wiki/Partial_differential_equation) is an equation which involves a **multivariable function** and one or more of its **partial derivatives**. A [multivariable function](https://en.wikipedia.org/wiki/Function_of_several_real_variables) is a function with more than two arguments, such as the volume of a cone. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23wN27PsatxdiT6vmXyK27PVMHrpZeqCi8kniUKgp1gsFPBLmeuP5Qa8eeyHFZ9QKMcXA.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23uRKjLsbpX5VSc4WXp6jvrwxrbWXdP3tkcmdPApnLLv7cPAoJxjZALAXaqWM62ZiyaWV.png) A [partial derivative](https://en.wikipedia.org/wiki/Partial_derivative) is the derivative of a multivariable function with respect to one of those variables, with the others held constant. The partial derivative of a function f(x, y, …) with respect to the x-direction is denoted by any of the following ways: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23vsr8H93AifuHsApVz2eVccHP9vBZsN1zxc9aAvEU1bSZ6EVRHqAYPurRvofxT3FhGkg.png) >It can be thought of as the rate of change of the function in the x-direction. For a function f(x, y, z) of three variables, Laplace's equation is ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tSyukFYFvtQKhYMAXL7kxCJKaeFNETTgFXTM5D8f6gnaCDCarSSVsMrRq3rjK8dRijB.png) >A function that obeys this equation is called a **harmonic function**. The Laplace equation can be written using the [Laplace operator or Laplacian](https://en.wikipedia.org/wiki/Laplace_operator) (Δ or ∇²) or the divergence operator (∇ ·) of the [gradient](https://en.wikipedia.org/wiki/Gradient) (∇f). ![image.png](https://files.peakd.com/file/peakd-hive/mes/23xAwZGLegy69yn3nCvGUgu3QwVh5MqVLkTquabSijxx2sY4yACveQcFZAk48FAsPAY8k.png) The [nabla symbol](https://en.wikipedia.org/wiki/Nabla_symbol) ∇, written as an upside-down triangle and pronounced "del", denotes the [vector differential operator](https://en.wikipedia.org/wiki/Del). ![image.png](https://files.peakd.com/file/peakd-hive/mes/23twAKVLETWpgaymMQMVVtyJfgYk3xaqkwmAhQ7LS55bnUqrDL4zprbXYsSzN27c6Jpkt.png) The [gradient](https://en.wikipedia.org/wiki/Gradient) is the vector field (or [vector function](https://peakd.com/hive-128780/@mes/omhbcxkw)) ∇f whose value at a given point gives the direction and rate of fastest increase. >Note that F_x, F_y, F_z are [vector components of F and not partial derivatives](https://grok.com/share/c2hhcmQtMg_998aa7b7-fe08-49c2-8753-314bab601e49). In coordinate-free terms, the gradient of a function f(**r**) may be defined by: ![image.png](https://files.peakd.com/file/peakd-hive/mes/EouExREncbAH8ZnqERVo4q75wsDov5H6MnSFvgUH2fyctZujBBFoTQ9orz8uPBeeu9y.png) >where df is the total infinitesimal change in f for an infinitesimal displacement d**r**, and is seen to be maximal when d**r** is in the direction of the gradient ∇f. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23yJRCfw22y1tWE3fMaRwfSSEytehBhijoBdTAXZi3BYvtfCFDjcGAcmGhbA3sVefx6Zk.png) >[Figure](https://en.wikipedia.org/wiki/Gradient#/media/File:Gradient2.svg): The gradient, represented by the blue arrows, denotes the direction of greatest change of a scalar function. The values of the function are represented in greyscale and increase in value from white (low) to dark (high). In 3D coordinates, the [divergence](https://en.wikipedia.org/wiki/Divergence) is just the sum of partial derivatives and uses the [dot product](https://peakd.com/hive-128780/@mes/vectors-and-the-geometry-of-space-the-dot-product) notation as a [mnemonic device](https://en.wikipedia.org/wiki/Mnemonic) to aid in remembering. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23w3EneiZPx6qrWbu4e1x5x3rpmeQsKLpwi67qF9AcEYirQDfoQrYhnygPvZ8bBxaYUDR.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tcLqusWoh6ZxzHh1PgP7knqJuAsZuQosi8YgNLpv539Xz2EwfRmEJ1RHRAzNzwATtmY.png) >[Image](https://en.wikipedia.org/wiki/Divergence#/media/File:Divergence_(captions).svg): The divergence of different vector fields in 2 dimensions, and represents the [flux (creation / source or sink) at a point](https://grok.com/share/c2hhcmQtMg_fc5fb68b-adf0-49d5-80e8-1059c21ad420). --- # Laplace Operator in Polar Coordinates I will generally follow the derivations shown in videos by Michael Dabkowski from the Mike, the Mathematician YouTube channel for: 1. Converting the [Laplacian in Polar Coordinates](https://youtu.be/wlbZxa66ncg) 2. Converting the [Laplacian into Spherical Coordinates](https://youtu.be/fPidnMY5yD0) 3. Solving the [Laplacian to obtain Spherical Harmonics](https://youtu.be/36ymXf-Ziho). I have also used some help from [Grok AI](https://grok.com/share/c2hhcmQtMg_7bfaf3b9-bf7c-4fcd-8298-b4ab607dade7) for when the above videos are lacking. To help in obtaining the Laplace Operator or Laplacian in Spherical Coordinates, [lets first obtain it](https://youtu.be/wlbZxa66ncg) for the simpler, but related, [polar coordinates](https://peakd.com/@mes/video-notes-polar-coordinates). ![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8CfKPvfgJksPLEWKyXw3HVPaDG3AhVLncAgbRcdHjLo2omJBZj3ZhA2AUN6zdPT4FX.png) Let's write out the first partial derivative of f in terms of x and y by using the [Chain Rule for Partial Derivatives](https://en.wikipedia.org/wiki/Chain_rule) and its associated [Tree Structure](http://www2.math.ou.edu/~jjackson/teaching/1405slides.pdf). ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tcP5UHCWxsHLCSj2uD8Ex5jYdJ5bAMxcuyH5fNKHDsheK1n3cWZkvk4tDcPx7xYi7WJ.png) I will cover partial derivatives more closely in future videos, but the chain rule is similar to that for normal derivatives from [my earlier video](https://peakd.com/hive-128780/@mes/derivative-examples-chain-rule). ![image.png](https://files.peakd.com/file/peakd-hive/mes/Eo4Jo4pzbMHyvTFvMhH37oqs4pntxhq9h6HtLAdZdcUb919LgSkpgDQGBJ13eX4nQWq.png) We can solve the first partial derivative of f in terms of x as follows from our polar coordinates formulas. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tHb6kpi7DTJ4XPyWrrx8whoGwPohgcaApQcpSjnpn94kS1dJDhdAuEa5QxAzYUTiUVS.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/Eo2BM1YUZ3eZRYKeAxdyQtZsoc8spTdJ4b6LKiKiG1HTbGci5mEiacS4KHoqkDeUipp.png) Recall from my [earlier video](https://peakd.com/hive-128780/@mes/derivative-of-arctanx-or-inverse-tanx) the derivative of inverse tangent. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tSym852DMWBE3poKLBfKtwwp3c4Hznht1KU5X3hYAahWx1ruJdKSE3q1Py718bQ72ay.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tHbYLE2ibRUkyg4RvDj7UdEtJTL33oL9VaWxmbUXRb349gLkDtsuR14jGn4GKanmWDQ.png) Similarly, for the partial derivative of f in terms of y we have: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tHbkj8BK8fKYU8hcL1Y7ho2nG9SvMm5KkiuESgYKiozQfGSBmHY2Sh7qWq1W9GFWxd7.png) Now let's find the second partial derivative of f in terms of x: ![image.png](https://files.peakd.com/file/peakd-hive/mes/48GjPY3wkUpmcN8m8FMq9wsRducm1BB9fQDqyutqUWC8wd6doPWSrHbojWtx39zRr2.png) Likewise, we have the second partial derivative of f in terms of y: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tSym8QDRkYZht1NgAqMmn8qrdzgerZMBQbsuDxsYzSCYbcZksvGiTcb3WyjpE6mTbNj.png) Note that, by [symmetry of second derivatives](https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives) (also called the Schwarz integrability condition or Clairaut's theorem), we have: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23u6YmnLJ3M3HhUp1MDAMQ6y4jUdWoTwqqTDGnk9qubu2hZr13Ed82CDEALZ8F15Wdq8Z.png) >Note that we can actually cancel out terms below without using the symmetry of second derivatives! Now we can put it all together to obtain the Laplacian in polar coordinates. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tSzLNbhdLVHxN8gNAZJ1G7iPJJfCgZWAvNYLjySfqAGnbejoXyT3CuYTGfEy9jzLZNj.png) [Double-checking with Wikipedia](https://en.wikipedia.org/wiki/Laplace_operator), we obtain the correct answer. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tmmT3BGYuMEZwyBwqRuwASGTZofjNuSrTPggUeLVV1PtsjoiwLjZkHtJoR8WN52Ha8g.png) --- # Laplace Operator in Spherical Coordinates Now let's determine the [Laplacian in Spherical Coordinates](https://youtu.be/fPidnMY5yD0) by first converting rectangular coordinates into [spherical coordinates](https://en.wikipedia.org/wiki/Spherical_coordinate_system). ![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8CSPFCQM2hoYWsbR1pgkHYK2G86ri8Fp7a7Hd7HXmWsnmR5gwZn6MLT9RAHjrnMnbX.png) Recall the [distance formula in 3D (and 2D)](https://www.youtube.com/watch?v=Y9-7SjOxwZw&list=PLai3U8-WIK0GiAJeLMzhOsrMliibfce5m&index=6): ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tcNnfFqYQ9ekxYn72GxEwZpRSjLgokcKdBSfwwCBiTLGMJUvA3f9yDzRnvy7VJccVm5.png) We will now apply the Laplacian in Polar Coordinates **twice** to find the Laplacian in Spherical Coordinates: once for the x-y terms, and once for the z-s terms. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8CfKPvh2Xjqed37AKQVKuUDzSgN82hZiFovroH9A8QAXk9kTtp1UwajgXtzqpBHTGh.png) Likewise for the z-s terms: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tHbKgySZio8sMcouiPfUBpi2bUwM6k5QhK9Zj1PsW3hLZ1LbVC1b1WzogZHns7BK4W3.png) Adding these together we get: ![image.png](https://files.peakd.com/file/peakd-hive/mes/Eo2BJ5b2bdSxTKMnQkHRDBgxT5b8a9VErBtTMk7LH2BGG7hoXVrXGeHnJsoohww15Lu.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tcNeS8UAdy3w4c2UPjWdnM3dz892dHpDyHSLHs61f3UxTtQBmvoL7J4Mz44swEuUGPs.png) Thus our Laplacian in spherical coordinates becomes: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tSzLNbh8MUDgrPcDQwYAiPr4jZYMVfjh3cboqiEYGbaXWM21hpyEvjyncDqPkmU9fHG.png) [Double-checking with Wikipedia](https://en.wikipedia.org/wiki/Laplace_operator), we obtain the correct answer, which lines up with the third variation of the formula! ![image.png](https://files.peakd.com/file/peakd-hive/mes/EoeMBAfxGjZKNaQLgxKjMvEQ1yVA2J1JhnxD9DMLbN9pD32zEmBVz1y9c6fWe27LxNp.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/EoeMA4cQFLQqcqnqhLZkWJWGhtuLSuFJDeswm6AK48Ff6gxaHArQciEauMB1rSRUgr8.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23viTAxLiqjL4ui4ex8uxTymwhCF66UCwfc9diBcf7a41CsX3yCEU1cTF6oJm4zRS128n.png) --- # Solid Spherical Harmonics: Solutions to the Laplace Equation in Spherical Coordinates [Solid spherical harmonics](https://en.wikipedia.org/wiki/Spherical_harmonics) are the solutions to the Laplace Equation in spherical coordinates. Laplace's Equation is just the Laplace operator or Laplacian set to equal zero. From the Laplacian in spherical coordinates (r, θ, ɸ), [we can write Laplace's Equation as follows](https://youtu.be/36ymXf-Ziho): ![image.png](https://files.peakd.com/file/peakd-hive/mes/Eo4JhGpWc1y6YuPD7pbCsJLkvh7RDVEHabi6TweJVhGgLSVqTP429262JCiEUSCood6.png) We can apply [separation of variables](https://en.wikipedia.org/wiki/Separation_of_variables#pde) to obtain. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8D6MYfN9KjpanAMGxzycURDchYj4Z4Hi4iKBByrnZTz9NLHLMPxHpnrqj1w9oj2ym4.png) We now set the equations equal to their [corresponding separation constants](https://grok.com/share/c2hhcmQtMg_038acc46-fb1d-42b2-88ca-88bfe1b7dce8), to ensure a periodic solution, with the middle function to be set as the difference between the constants, thus ensuring the overall equation sums to zero. ![image.png](https://files.peakd.com/file/peakd-hive/mes/48FmUS4BL5A5EdFKUmyrN5yPQNCiZHDVz3oh9GUetuT7j1WoqLxovmjhTxkEaGBbaP.png) The first constant -m² (where m is an integer) is selected such that we have a periodic solution: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8CfKPvTxqEvpsn5CgWCsKJVm2QhPbmaCDSyFoHmeMQd7t7gMuDK839GGnsxDRunq43.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/Eo8ZeGF6MReSc1yrsqrfgdqfzeo8CY6oNEYfNZEZVaMLvL7MSG6PDbnfffXhay5GUGi.png) Note that the [general solution allows for writing every possible solution as a linear combination](https://grok.com/share/c2hhcmQtMg_2a062298-7689-4b26-ad74-2704f692bc1f) of two basis functions. This complex exponential function are equivalent to trigonometric equations via [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula), as shown in my [earlier video](https://youtu.be/aZkd2HKvCdU). ![image.png](https://files.peakd.com/file/peakd-hive/mes/23twABpCt4y1Lp1t8N9NRg6ReJN9ZA2rjYBq9CcEPJVtTXKfUHx3xnkKLEk9BQAWf2szC.png) Thus the periodicity for our solution is: ![image.png](https://files.peakd.com/file/peakd-hive/mes/Eo4JhGpWc1yS6n7QvkvBM8nrBjDW7ArNZGaZuoMfmPR7Dfd6nbjP9qQRKTTatkc6vyg.png) The selection of the negative sign (-) in -m² allows for our specific complex solution and the [squared ensures we get m to be integer values and thus our solution is periodic](https://grok.com/share/c2hhcmQtMg_6140ed59-45e2-458c-954d-ae9e7bcbe3b4) about 2π or 360 degrees. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8CSPFC3qt61CoeMZoHKnHvisaFHCaPQQAC6PLa8dB2o8a3PVomcAe6XLJHR5hvSbdF.png) Let's examine the second separated function and its chosen constant n(n+1) (where n is a positive integer), which yields the [Euler-Cauchy equation](https://en.wikipedia.org/wiki/Cauchy%E2%80%93Euler_equation), which we can solve via substitution. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8CfKPvYt89Jfn1h1ARZWQ1EVc7Nu8hztj43yce1CYri67d7QigGeEvbAmEJM4y5RoP.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8D6MYfatFGaKWK9f9fXJURvnuKsoo4dMxut4QjXu5FBg1B1EJxgNa6LZ6fuA9xbJHm.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/48HDP36TaL1SmdNiqS6ruN3KK5EqTpmLh1DnQVzsmYZHo86uF89fdNpUm4eFYqzk9L.png) Again, note the [general solution is the linear combination](https://grok.com/share/c2hhcmQtMg_a444a881-47af-4366-9a3b-dbb735980b19) of the two basis functions. Recall my [earlier video](https://youtu.be/H46ILjs10_c) on the Quadratic Formula. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tw7oNjTynsjyB9SdiLehjfFwEkux8jHYJjpyJpbwVX3x89ZUEAbZ8GkMKJUABmkrjZx.png) Let's now solve the middle term in the Laplace equation, this term is key to Spherical Harmonics. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23twAQDSE6b3S3ybsqwnmGgt7DMWU9wx29WyNwMc2PwqUQB11AJs2SM63w51SKvBN7tiq.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tHbkj8AzaMhhNHEb38XJriYVW19f3G6CJkYVrRZD2enEmJrmvYNW9GAbXsgLWvQYfHP.png) --- ## Associated Legendre Equation This is the [Associated Legendre Equation](https://en.wikipedia.org/wiki/Associated_Legendre_polynomials): ![image.png](https://files.peakd.com/file/peakd-hive/mes/23wCdT2J55ZSd3DgQ3dJrc5PkfsJEzbs31Y6GxZTZ5mxf87MX4vVBdj5RFX2oVKiBU128.png) >or ![image.png](https://files.peakd.com/file/peakd-hive/mes/23w3EneiZDA2YWwqzT4b5DJF6X9w1Sv6Jtxph6WiuJB4ZitmK6N8xQKim5zE6qPrvYEwE.png) >where ℓ and m are integers and refer to the degree and order (not a power) of the associated Legendre polynomial, respectively. The above equation has nonzero solutions that are [non-singular](https://grok.com/share/c2hhcmQtMg_62bbe7e6-78a2-4e69-9c42-16bbd3f72a9e) (or regular and means P remains finite, bounded, continuous, and differentiable) on [-1, 1] only if ℓ and m are integers with 0 ≤ m ≤ ℓ, or with trivially equivalent negative values (replacing m with - m). The solution is given in terms of the [Legendre polynomials](https://en.wikipedia.org/wiki/Legendre_polynomials) Pℓ(x) (m ≥ 0). ![image.png](https://files.peakd.com/file/peakd-hive/mes/23wCdahRRYhCCieyG36Sr8sYXb4ZLmuohq5Ebk66ghPBQmsYNMYwmibXwHXD47sq7fgby.png) >Note the factor (-1)^m is known as the [Condon-Shortley phase](https://en.wikipedia.org/wiki/Spherical_harmonics#Condon%E2%80%93Shortley_phase) but some authors omit this phase factor (which just rotates / shifts the phase or angle). Where P_ℓ(x) is the solution of the Legendre equation: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23wXQhMzzHTzxrupLsSuoyxNToyzMyw4XvWGxJwdxoqy6dMdYpLbUPWGEGDidz1TyeSje.png) By [Rodrigues' formula](https://en.wikipedia.org/wiki/Rodrigues%27_formula) we can obtain Pℓ(x) as: ![image.png](https://files.peakd.com/file/peakd-hive/mes/EoKCdAJd25ZyMJxzqzSUWAicoPGBZBTt65shF71XBxXUVhh9FD45zZUHyitaLLLmYp6.png) Then P_m^ℓ(x) can be expressed as: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23uRLJc4n7cRwxrXXQEs3DTXSzyUdz1ba6URR1rB7DUFerKNnBQuwQPkFxBTgCZqPEGF3.png) The derivation of the Legendre polynomials are outside the scope of this video but I may cover them in a future video, see [Mike, the Mathematician's playlist](https://www.youtube.com/playlist?list=PLLXsDm32L84jhw_OFGVYuKHgLB9yuPTnA) for now. Now plugging all of this together, we obtain our solution for the Solid Spherical Harmonics: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23viTAxLiqjL4ui4ex8uxTymwhCF66UCwfc9diBcf7a41CsX3yCEU1cTF6oJm4zRS128n.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tHbkj8BGBprPvUecPKgodoStH7M3rATaJobRJ6chAWZo1jWbBNEPBKpiVYYtoKVWgPw.png) --- # Spherical Harmonics: Solution to the Angular Laplacian The solid spherical harmonics includes the radial terms r, while the spherical harmonics, in general, typically are just in terms of the angular terms θ and φ. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23viTAxLjHPVfSBMLWzcXmYpN2WjrzPoUb5bQdn9Y74HFQsR8y5rttKXK9kcAZ3KZyYst.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23wrAFLM1AN7umik6StG7rXPA63sLr7CPqTZoborUJi7ix9MryxURMHkTEydn1X69LxZc.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23wrBoZPKckMmLSGrcWfnEuZHJ3LL2PqwK7b31m1rUnMNQbtde3CBFeTCFCqMovVYje8m.png) We can combine the radial terms as a function Y(θ,φ): ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tSzChULjofYomj1K51ba14EqLQVFeAX3jBcSUfSMWd5YLnsb63jvHACK3GtYab9Vbj6.png) This Y is typically considered as the solution to the [angular Laplacian](https://grok.com/share/c2hhcmQtMg_61a88cec-b61e-4ca7-a74b-7549f15af961) (or [Laplace-Beltrami operator](https://en.wikipedia.org/wiki/Laplace%E2%80%93Beltrami_operator) on the sphere), which can be denoted as Λ² (or [denoted as Δs²](https://en.wikipedia.org/wiki/Laplace_operator)). ![image.png](https://files.peakd.com/file/peakd-hive/mes/23sxpRyy9Khy6bRyLoS3fdid2KZd4EfR8JNYy3JF4R7CNk6D3Yomoj7KDWghzPy2GyTZk.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/Eo6SF51cKUNrRNcb5vvTxjM9GZBKt4GvLXnmDkVrDSx7dEUk5fxnC7wmsTiciVSzMHH.png) Y is a solution to the above equation involving the angular Laplacian: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23w3EvKqum7vyauzUEzT6n3jkmrJjvYKhpMgBKaorzHjiBxCtjo4bwv84G8NKcendnZeh.png) --- # Deriving the Real Spherical Harmonics The **real Spherical Harmonics** are defined such that the functions are real-valued everywhere, with no complex term i, with the typical definition shown as follows: ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tHb6kphfZJ5nWzNFiqNSmah7Sk67SpvNapVxsCtjxHTbhDARbYMjVmwV6soGm8K7iRH.png) Note that the complex spherical harmonics are [chosen using the pure complex exponentials](https://grok.com/share/c2hhcmQtMg_266bfa1e-ac33-4fee-bb04-899b775857b8) (the basis solutions), rather than the linear combinations, and the Y_n^-m term is defined via the following [complex conjugate](https://en.wikipedia.org/wiki/Complex_conjugate) form (flip the sign of the complex rotation angle term) because this makes important formulas and operations in quantum mechanics [cleaner, simpler, and more elegant](https://grok.com/share/c2hhcmQtMg_2191c6fe-2468-4476-a880-20d3076b8782). And using the proportional forms (ignoring the Normalization constant discussed earlier), we obtain. https://img.leopedia.io/DQmY1GTr5bgGxBWHpmDTP8zHXa9zG12n4FcAgyR9SUmTmgy/image.png Note the [complex conjugation](https://en.wikipedia.org/wiki/Complex_conjugate) in the above definition. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tSzLNbhdLVpkfkaGk6pfQ8AtxFvPe1Az6owyhz4BJBi2oZoANZqEFhkNXezTsvkxH5c.png) For the Y_n^-m term, we use P_n^m instead of P_n^-m because the Associated Legendre equation depends on m², and not m. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tcP5UHCp9CSiTHw4781SLCzhv2VRxJ2MyJKzm2dpmZiJfHbGHMuLK6mj2nGTcLFgizV.png) Note that this exact relation is obtained from Rodrigues' formula for the Associated Legendre Function in which we [have the following identity](https://en.wikipedia.org/wiki/Associated_Legendre_polynomials), which I may derive in a future video. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23vsrFxGPbr3jcPGPCz548p4MNK5gaU6BrxXKLcZ6sUYxMELBQsfGJ9f2ugCFDEt1o7hk.png) **For m > 0, the definition is:** ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tcNRVyjeyzJjXxHCWUMxHgC1YzkP3o5QtLNRfzF1DWAkgqWdmskVhzqa9EtoCtuq195.png) **For m = 0, the definition is:** ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tcP5UHCi3kx2dtDia21JG74eXzHmJkyLHcjg4o4sUc97P9WzXLVAXm5bGogZxKrfrZN.png) Note that P_n⁰ = P_n and is just the ordinary Legendre polynomial. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tHbg12Bgz5HpNFyjQGpzFoyRzkx5Dmxjm575pDLFomExHJEq9mAhN6Eh8NHcyAZq3Bz.png) **For m < 0, the definition is:** ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tHbKgySQBjyhttFbpLtc6eXPWwZKKsSoVdsbiCNLaEKcdx4TneQ7gQSxvmXzHZpAAzo.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tcP5UHCNV3ESRD3UnMVAzJABzp6mY7XXqZCNTftRfyo9EtbeL747FNfHTpp6qijbU2U.png) Thus, [putting it all together](https://grok.com/share/c2hhcmQtMg_a08d5833-e537-4e3d-88a0-497029993abe), we obtain the real spherical harmonics in proportional form, excluding the normalization constant. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tSzR6hgcHkeSMgij34bH7Cjny28ZZ9q1cppgKydHkTNK3pmS4MxHwAV5oAn9HzJttiP.png) --- ## Graphing Real Spherical Harmonics in Desmos Calculator Here is an amazing Real Spherical Harmonics graph in Desmos Calculator that someone made! >[https://www.desmos.com/calculator/nsfrepxacm](https://www.desmos.com/calculator/nsfrepxacm) Note the normalization constants used, as well as the same formulas we obtained above! ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tmmjqriPM3s5WFqsC8LS4xpMjgb3Y8P7GxEEubP6xSCgEdaguJEcaUMoM6Dih7hu4bu.png) Note the Associated Legendre Polynomials and its giant derivative function. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tHbkj8AudoYbJ6Wgjdx3Tuqsg9DWfs62rLHBwt8FhrHzJ7TcJArY3DomhG7GGg6bp2V.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/Eo8ZeGF6LNgB5AvKSY2V7VuGAcgstN1qByDAqnmVKNYkkv5MVYNcdqBLWi6XYmBpnaq.png) Since the Spherical Harmonics are graphed on the sphere, the radial disance needs to be stretched proportional to the magnitude of its value on the sphere. This gives the distinctive lobes. ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tcNzkBEXVo2s8k3j9tqt5SnscuNnAez9JPQUjefE55idxNncARKQzTzXqyNvdM8nvZE.png) Note the relationships between the principle, orbital angular momentum, and magnetic quantum numbers. The principle sets the maximum orbital angular momentum which then sets positive and negative limits of the magnetic. ![image.png](https://files.peakd.com/file/peakd-hive/mes/Enz43VKA9fa96JcGA2YU7CFUdEGmnKShmJthCied2iFaru4YtHzzCdbATRLxkQJJN52.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/23tRp4zmPC7BoGWREX3BhwJNFrvEi4Uf14SWtNPCS1ryRi61sJmMci6a35nxueW5yeWfj.png) Here are the Real Spherical Harmonics for various values of the quantum numbers. ![image.png](https://files.peakd.com/file/peakd-hive/mes/246av6YqkwKQjkXrTBmBgEfTJUAv82QVEGr5tVdLdJjWpAp5gVNfMuHkwtaz2gwhrKJXf.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/246av6ZU4DJoSPfbnbdxHc79A4CfdqShZWnzgTQebvadKHwevfnudzEWQxLmcR7D5Wv5G.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/246avK1NHjwDrjRegtuYKEsoYMgsuD7LpSZV1gz2SkSMXpbpceEU3FFiZ6m6Wa8WZCgRr.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/246avK1NHjwDrjRegtuYKEsoYMgvZ6XU7GHyPhcAF8bvxrwKmDh7pbSSse7Q4mnm77vfP.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/246av6ZU4DJoSPfbnbdxHc79A4CfdqSkWV6f1wRhVdS4jytXb1J9WWwzQZew3KmGS5eRs.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/246av6ZU4DJoSPfbnbdzEMHNc646K4aa6C1EdwP3SBz3NaGCbBLnEXkNejo5Gfy69sDvP.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/246avK1hTwBxtPpoprx1qzqQcrJxmJ9c3WdLnurju8gpUkSnMouhWH1bqeTXgM1CC3uYn.png) ![image.png](https://files.peakd.com/file/peakd-hive/mes/246azgW7bEArX3QLsxp1MbXnm4VYafeKNb7fQNSb9apVV2U39yAFxJq91L4oRnGCm1BqD.png) Fascinating stuff!! --- # Sections Playlist and Thumbnails [Sections playlist](https://www.youtube.com/playlist?list=PLai3U8-WIK0FHh2kZzebadnPFwO6Yi5qu) ## Spherical Harmonics: Atomic Orbitals, Laplacian in Polar and Spherical Coordinates, Laplace Equation [3Speak](https://3speak.tv/watch?v=mes/220d6572) - [Odysee](https://odysee.com/Spherical-Harmonics:b411814bbf4b88a22d2b136854be4c867f77591b) - [BitChute](https://www.bitchute.com/video/gFSCuLXGejxk) - [Rumble](https://rumble.com/v76qf26-spherical-harmonics-atomic-orbitals-laplacian-in-polar-and-spherical-coordi.html) - [Telegram](https://youtu.be/R7SAhbuZPig) - [YouTube](https://youtu.be/p5SdobFPiKY) - [PDF Notes](https://1drv.ms/b/c/88862ef47bcaf6cd/IQCuF1y7tt2IR6fssJkQvz4VAUXIXkwKoyrNEqqKQKsGDOc) - [Hive Notes](https://peakd.com/hive-128780/@mes/220d6572) - [Summary](https://inleo.io/threads/view/mes/re-leothreads-x5wsh2ax) ![Spherical Harmonics.jpeg](https://files.peakd.com/file/peakd-hive/mes/23ydRCbjj1UKSrutn1S2JKNZsySufGoRk4g1qWUFekLJd3CFYjzktYWmdVi2fasLZ7G2D.jpeg) ## Introduction to Spherical Harmonics ![1 Introduction.jpeg](https://files.peakd.com/file/peakd-hive/mes/23tmjWGnRcS2gEpCygwo5o3SW5dTfoJfCK24YAZQ3nNtGxeR658s5vAR3TNUUW8qm4Hxv.jpeg) [3Speak](https://3speak.tv/watch?v=mes/cac50999) - [Telegram](https://t.me/meslinks/33963) - [YouTube](https://youtu.be/R6A3KtYQCyw) - [Summary](https://inleo.io/threads/view/mes/re-leothreads-t477w8we) ## Spherical Harmonics in Atomic Orbitals and Magnetic Fields ![2 Orbitals.jpeg](https://files.peakd.com/file/peakd-hive/mes/23xAipGS5kUwJ1HrYgXja8xtpEoupMkrWMeRa2hsNSxVPp2eFj2eME6dxgNvXAtNQb5iy.jpeg) [3Speak](https://3speak.tv/watch?v=mes/27ef8fad) - [Telegram](https://t.me/meslinks/34009) - [YouTube](https://youtu.be/6Ct0zWqK_JU) - [Summary](https://inleo.io/threads/view/mes/re-leothreads-31jxnwwwg) ## Mathematical Review: Partial Differential Equations (PDE), Laplace's Equation, Gradient, Divergence ![3 Review.png](https://files.peakd.com/file/peakd-hive/mes/23t8B1xtpkeApFNvkEwsCNUnYKAGHrR1JxRCR8BVLE7gr187Kwk3Ag3jbMUNpLcuhvUmT.png) [3Speak](https://3speak.tv/watch?v=mes/a305d74f) - [Telegram](https://t.me/meslinks/34038) - [YouTube](https://youtu.be/u8QT6RD5JAo) - [Summary](https://inleo.io/threads/view/mes/re-leothreads-p6dy9vxe) ## Laplace Operator (Laplacian) in Polar Coordinates – PROOF ![4 Laplacian Polar.png](https://files.peakd.com/file/peakd-hive/mes/23tSxBXxmewxwSNGFmYAyAjzySphWdWKwJH2w2dWT7TETLsvhcpU3y5D6PVJaKEBA46us.png) [3Speak](https://3speak.tv/watch?v=mes/294e58ab) - [Telegram](https://t.me/meslinks/34077) - [YouTube](https://youtu.be/k8udQSNEi7E) - [Summary](https://inleo.io/threads/view/mes/re-leothreads-tvmwb2zs) ## Laplace Operator (Laplacian) in Spherical Coordinates – PROOF ![5 Laplacian Spherical.png](https://files.peakd.com/file/peakd-hive/mes/23tmjb17kzsiPAp4ckEpfKQdKJdU5ojKDo6BwwSh4JcyzAYHVUmKZHYeC3XuEWYFPzuxk.png) [3Speak](https://3speak.tv/watch?v=mes/034f3436) - [Telegram](https://t.me/meslinks/34095) - [YouTube](https://youtu.be/hJtUt6n_Ya4) - [Summary](https://inleo.io/threads/view/mes/re-leothreads-2q5fx2jfy) ## Solid Spherical Harmonics – Solutions to Laplace's Equation in Spherical Coordinates ![6 Solid Spherical Harmonics.png](https://files.peakd.com/file/peakd-hive/mes/23tSxGFxiyWACCTMuTMFitF7Kx3KCsQLWwzerEsvBHNNSq5bKnTabWL9Xqm3Xo2ut6B5h.png) [3Speak](https://3speak.tv/watch?v=mes/9d41089b) - [Telegram](https://t.me/meslinks/34126) - [YouTube](https://youtu.be/gAQddWT0_7g) - [Summary](https://inleo.io/threads/view/mes/re-leothreads-2tekjj63p) ## Complex Spherical Harmonics – Solutions to the Angular Laplacian ![7 Spherical Harmonics.png](https://files.peakd.com/file/peakd-hive/mes/23tSxBXxmhXVFzHRQ7769ZuwmBFbfvkzekcH2S8X8XmbmALfCq59xjDC5Q8SfXTpSUoAG.png) [3Speak](https://3speak.tv/watch?v=mes/65445edf) - [Telegram](https://t.me/meslinks/34160) - [YouTube](https://youtu.be/5avQYYEYnVY) - [Summary](https://inleo.io/threads/view/mes/re-leothreads-gahhcp2z) ## Real Spherical Harmonics – Removing the Imaginary Terms ![7 Spherical Harmonics.png](https://files.peakd.com/file/peakd-hive/mes/23tSxBXxmhXVFzHRQ7769ZuwmBFbfvkzekcH2S8X8XmbmALfCq59xjDC5Q8SfXTpSUoAG.png) [3Speak](https://3speak.tv/watch?v=mes/2d1e95ad) - [Telegram](https://t.me/meslinks/34212) - [YouTube](https://youtu.be/udTyDJ_HS60) - [Summary](https://inleo.io/threads/view/mes/re-leothreads-blfpmusf) ## Graphing Real Spherical Harmonics in Desmos Calculator ![9 Graphing Real Spherical Harmonics.jpeg](https://files.peakd.com/file/peakd-hive/mes/23viTAwLxNubQz9aUe89e6Rdv57fRmarG1gHapERac9gT1qwo2LRTAGGKf5DUotsegY2u.jpeg) [3Speak](https://3speak.tv/watch?v=mes/a058e2ca) - [Telegram](https://t.me/meslinks/34238) - [YouTube](https://youtu.be/aVJj9EmK8vA) - [Summary](https://inleo.io/threads/view/mes/re-leothreads-exowfp4p) ### MES Links: https://mes.fm/links [//]:# (!worldmappin 49.138996 lat -123.108105 long d3scr)

equation resolving