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Canonical static concept record

Spherical harmonic

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Summary

A simultaneous eigenfunction of the spherical Laplace–Beltrami operator (equivalently $\mathbf L^2$ ) and a chosen component $L_z$ ; the $Y_{\ell m}$ form an orthonormal basis of $L^2(S^2)$ .

Record metadata

Canonical name: Spherical harmonic
Concept ID: spherical_harmonic
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Analysis
Domains: Analysis
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Special-function basis

Carrier(s)

unit sphere $S^2$
complex-valued square-integrable functions

Data

functions $Y_{\ell m}$

Axioms / constraints

$-\Delta_{S^2}Y_{\ell m}=\ell(\ell+1)Y_{\ell m}$
$L_zY_{\ell m}=m\hbar Y_{\ell m}$ under the physics normalization

Canonically induces

orthonormal decomposition of $L^2(S^2)$

Notes

Normalization and phase conventions vary.

Concept sources

Wikipedia — Spherical harmonics — Spherical harmonics · encyclopedic reference · source ID wp-spherical_harmonic
PDG — Review of Particle Physics 2026: Mathematical Tools — Review of Particle Physics 2026: Mathematical Tools · Particle Data Group reference · source ID pdg-2026-mathematical-tools
NIST — Atomic Spectroscopy: An Introduction — Atomic Spectroscopy: An Introduction · NIST technical reference · source ID nist-atomic-spectroscopy

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Laplace operator Spherical harmonic

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: spherical_from_hilbert
Relation type: Framework specialization framework-specialization
Direction: source → target
Endpoint roles: source: Specializes to; target: Special case or refinement of
Authored annotation: eigenfunctions on $S^2$

Authored explanation

Spherical harmonics are eigenfunctions of the Laplace–Beltrami operator on $S^2$ and form an orthonormal basis of $L^2(S^2)$ .

How to interpret this relation type

A more specific theory, model, entity class, or regime is obtained by restricting or extending the scope of a broader framework.

Relation sources

Wikipedia — Hilbert space — Hilbert space · expository · source ID wp-hilbert
nLab — Hilbert space — Hilbert space · reference · source ID nlab-hilbert
Princeton — Functional Analysis Lecture Notes — Functional Analysis Lecture Notes · course notes · source ID princeton-functional-analysis
Wikipedia — Spherical harmonics — Spherical harmonics · encyclopedic reference · source ID wp-spherical_harmonic
PDG — Review of Particle Physics 2026: Mathematical Tools — Review of Particle Physics 2026: Mathematical Tools · Particle Data Group reference · source ID pdg-2026-mathematical-tools
NIST — Atomic Spectroscopy: An Introduction — Atomic Spectroscopy: An Introduction · NIST technical reference · source ID nist-atomic-spectroscopy

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.