Spherical harmonic

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.

The spherical harmonic with parameters l, m can be written as:

where 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
Y2
Y3

In space

See also

• Clebsch-Gordan coefficients
• Harmonic function
• Rotation group
• Sturm-Liouville theory

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" (Spherical Harmonic Analysis). Solid Earth Geophysics.
• Spherical harmonics on Physicsworld