Spherical harmonic
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In mathematics, the spherical harmonics are an orthogonal set of solutions to Laplace's equation represented in a system of spherical coordinates. The solutions are generally expressed in terms of trigonometric functions and Legendre polynomials. This form comes from separation of variables once the Laplacian is written in the spherical coordinate system. Note that the spherical coordinates <math>\theta<math> and <math>\varphi<math> in this article are used in the physicist's way, as opposed to the mathematician's definition of spherical coordinates. That is, <math>\theta<math> is the colatitude or polar angle, ranging from <math>0\leq\theta\leq\pi<math> and <math>\phi<math> the azimuth or longitude, ranging from <math>0\leq\varphi<2\pi<math>.
The spherical harmonic with parameters l, m can be written as:
- <math> Y_{\ell,m}( \theta , \varphi ) = e^{i m \varphi } \cdot P_\ell^m ( \cos{\theta} ) <math>
where <math>P_\ell^m<math> are the associated Legendre polynomials.
Spherical harmonics are important in many theoretical and practical applications, particularly the computation of atomic electron configurations, and the approximation of the Earth's gravitational field and the geoid.
| Y1 | Missing image Legendre_Y1_xy.png | Missing image Legendre_Y1_polaire.png |
| Y2 | Missing image Legendre_Y2_xy.png | Missing image Legendre_Y2_polaire.png |
| Y3 | Missing image Legendre_Y3_xy.png | Missing image Legendre_Y3_polaire.png |
In space
Harmoniques_spheriques_positif_negatif.png
then the representative surface looks like a "battered" sphere;
Ylm is equal to 0 along circles (the representative surface intersects the ρ = ρ0 sphere at these circles). Ylm is alternatively positive and negative between two circles.
Traces_harmonique_spherique.png
See also
References
- A.R. Edmonds, Angular Momentum in Quantum Mechanics, (1957) Princeton University Press, ISBN 0-691-07912-9.
- E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, (1970) Cambridge at the University Press, ISBN 521-09209-4 See chapter 3.
- Albert Messiah, Quantum Mechanics, volume II. (2000) Dover. ISBN 0486409244.
- "General Solution to LaPlace's Equation in Spherical Harmonics (http://solid_earth.ou.edu/notes/harmonic/harmonic.html)" (Spherical Harmonic Analysis). Solid Earth Geophysics.
- Spherical harmonics on Physicsworld (http://mathworld.wolfram.com/SphericalHarmonic.html)