MathLiveSpherical harmonics
Gives the angular basis functions on the surface of a sphere, with degree \ell and order m, normalized so they are orthonormal. They appear whenever a problem with spherical symmetry is solved, such as atomic orbitals, multipole expansions and quantum angular momentum.
Y_\ell^m(\theta, \varphi) = \sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-|m|)!}{(\ell+|m|)!}}\, P_\ell^{|m|}(\cos\theta)\, e^{im\varphi}What the symbols mean
| spherical harmonic of degree \ell and order m, a complex function of polar angle \theta and azimuthal angle \varphi | |
| degree (non-negative integer, \ell \geq 0) | |
| order (integer with |m| \leq \ell) | |
| polar (colatitude) angle, 0 \leq \theta \leq \pi | |
| azimuthal angle, longitude on the sphere | |
| associated Legendre polynomial of degree \ell and order |m| | |
| imaginary unit |
About this formula
- FieldMathematics
- TopicSpecial functions
- LevelUniversity
- Also calledspherical harmonic, spherical harmonic function, angular part of hydrogen wavefunction, spherical harmonics normalization, Laplace spherical harmonics
- SourceWikidata Q877100
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