Spherical Harmonics in Atomic Orbitals and Magnetic Fields

mes (67)in DTube • yesterday

In this video, I go over an overview of how spherical harmonics show up in the equations for atomic orbitals and magnetic fields. The Schrödinger equation is a partial differential equation that governs the wave function that mathematically describes a quantum state. Solutions to the Schrödinger equation are standing waves called stationary states or energy eigenstates or "atomic orbitals". I go over the Schrödinger equation for the electron in a hydrogen atom, which is also applicable to hydrogen-like atoms (any atom or ion with a single electron), and the corresponding solution that involves spherical harmonics. Similarly, I show that the spherical harmonics arise also in the equation of magnetic fields, which are derived from the mathematical scalar potential function. The complete magnetic fields (and similarly atomic orbitals in general quantum states) are superpositions of multiple such basis functions/eigenstates.

#math #sphericalharmonics #calculus #atomicphysics #magnet

Timestamps:

Atomic orbitals contain the radial term in the spherical harmonics solutions to the Schrödinger equation for an electron in a hydrogen (or hydrogen-like) atom – 0:00
Grok AI summary of terms in Schrödinger equation and their physical meaning – 4:10
Solution to Schrödinger equation for the hydrogen or hydrogen-like atom (any atom or ion with a single electron) via separation of variables. Multi-electron atoms require approximate solutions – 7:36
Visualization of the hydrogen wave function – 10:20
3D views of some hydrogen-like atomic orbitals – 12:07
Table of hydrogen-like real-valued wave functions – 12:36
Each orbital is an eigenstate, although an actual electron is a superposition of states – 17:26
Equations for the real-valued wave function – 19:52
Animation and figures showing superpositions (lobes) and true eigenstates (rainbow colors): https://grok.com/share/c2hhcmQtMg_3122e883-13a6-42d8-abe1-cc9c72d1de4f – 22:39
Magnetic fields can be expressed as spherical harmonics – 25:10
- Scalar potential outside sources is a function describing magnetic fields away from a source of current, such as away from a permanent magnet – 25:53
- Magnetic field is the negative gradient of the scalar potential – 28:06
- Producing lobes and nodes around a magnetic via a Hall sensor: The Project Effect – 28:42
- Any complex magnetic field can be expressed as the sum of weighted spherical harmonics of different orders. Figure from MRI website – 30:32