Spherical harmonics: Difference between revisions
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Carl McBride (talk | contribs) (New page: The spherical harmonics <math>Y_l^m (\theta,\phi)</math> are the angular portion of the solution to Laplace's equation in spherical coordinates. ==See also== *[http://mathworld.wolfram...) |
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The spherical harmonics <math>Y_l^m (\theta,\phi)</math> are the angular portion of the solution to [[Laplace's equation]] in spherical coordinates. | The '''spherical harmonics''' <math>Y_l^m (\theta,\phi)</math> are the angular portion of the solution to [[Laplace's equation]] in spherical coordinates. | ||
They are given by | |||
:<math>Y_l^m (\theta,\phi) = | |||
(-1)^m \sqrt{\frac{2n+1}{4\pi}\frac{(n-m)!}{(n+m)!}} | |||
P^m_n(\cos\theta) e^{i m \phi},</math> | |||
where <math> P^m_n </math> is the [[associated Legendre function]]. | |||
The first few spherical harmonics are given by: | |||
:<math>Y_0^0 (\theta,\phi) = \frac{1}{2} \frac{1}{\sqrt{\pi}}</math> | |||
:<math>Y_1^{-1} (\theta,\phi) = \frac{1}{2} \sqrt{\frac{3}{2\pi}} \sin \theta e^{-i\phi} </math> | |||
:<math>Y_1^0 (\theta,\phi) = \frac{1}{2} \sqrt{\frac{3}{\pi}} \cos \theta </math> | |||
:<math>Y_1^1 (\theta,\phi) = -\frac{1}{2} \sqrt{\frac{3}{2\pi}} \sin \theta e^{i\phi} </math> | |||
==See also== | ==See also== | ||
*[[Wigner D-matrix]] | |||
*[http://mathworld.wolfram.com/SphericalHarmonic.html Spherical Harmonic -- from Wolfram MathWorld] | *[http://mathworld.wolfram.com/SphericalHarmonic.html Spherical Harmonic -- from Wolfram MathWorld] | ||
==References== | |||
*M. E. Rose "Elementary theory of angular momentum", John Wiley & Sons (1967) Appendix III | |||
*[http://dx.doi.org/10.1007/BF01597437 I. Nezbeda, J. Kolafa and S. Labík "The spherical harmonic expansion coefficients and multidimensional integrals in theories of liquids", Czechoslovak Journal of Physics '''39''' pp. 65-79 (1989)] | |||
[[category: mathematics]] | [[category: mathematics]] | ||
Latest revision as of 11:54, 20 June 2008
The spherical harmonics Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_l^m (\theta,\phi)} are the angular portion of the solution to Laplace's equation in spherical coordinates. They are given by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_l^m (\theta,\phi) = (-1)^m \sqrt{\frac{2n+1}{4\pi}\frac{(n-m)!}{(n+m)!}} P^m_n(\cos\theta) e^{i m \phi},}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P^m_n } is the associated Legendre function.
The first few spherical harmonics are given by:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_0^0 (\theta,\phi) = \frac{1}{2} \frac{1}{\sqrt{\pi}}}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_1^{-1} (\theta,\phi) = \frac{1}{2} \sqrt{\frac{3}{2\pi}} \sin \theta e^{-i\phi} }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_1^0 (\theta,\phi) = \frac{1}{2} \sqrt{\frac{3}{\pi}} \cos \theta }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_1^1 (\theta,\phi) = -\frac{1}{2} \sqrt{\frac{3}{2\pi}} \sin \theta e^{i\phi} }
See also[edit]
References[edit]
- M. E. Rose "Elementary theory of angular momentum", John Wiley & Sons (1967) Appendix III
- I. Nezbeda, J. Kolafa and S. Labík "The spherical harmonic expansion coefficients and multidimensional integrals in theories of liquids", Czechoslovak Journal of Physics 39 pp. 65-79 (1989)