Spherical harmonics - Revision history

Dduque: added general formula

2008-06-20T10:54:44Z

added general formula

← Older revision		Revision as of 12:54, 20 June 2008
Line 1:		Line 1:
	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:		The first few spherical harmonics are given by:

Carl McBride: Another slight tidy.

2008-06-20T10:22:40Z

Another slight tidy.

← Older revision		Revision as of 12:22, 20 June 2008
Line 11:		Line 11:
	==See also==		==See also==
	*[[Wigner D-matrix]]		*[[Wigner D-matrix]]
			*[http://mathworld.wolfram.com/SphericalHarmonic.html Spherical Harmonic -- from Wolfram MathWorld]
	==References==		==References==
	*M. E. Rose "Elementary theory of angular momentum", John Wiley & Sons (1967) Appendix III		*M. E. Rose "Elementary theory of angular momentum", John Wiley & Sons (1967) Appendix III
	*[http://mathworld.wolfram.com/SphericalHarmonic.html Spherical Harmonic -- from Wolfram MathWorld]
	*[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)]		*[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]]

Carl McBride: Slight tidy.

2008-06-20T10:22:08Z

Slight tidy.

← Older revision		Revision as of 12:22, 20 June 2008
Line 10:		Line 10:
	:<math>Y_1^1 (\theta,\phi) = -\frac{1}{2} \sqrt{\frac{3}{2\pi}} \sin \theta e^{i\phi} </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]]
			==References==
	*M. E. Rose "Elementary theory of angular momentum", John Wiley & Sons (1967) Appendix III		*M. E. Rose "Elementary theory of angular momentum", John Wiley & Sons (1967) Appendix III
	*[http://mathworld.wolfram.com/SphericalHarmonic.html Spherical Harmonic -- from Wolfram MathWorld]		*[http://mathworld.wolfram.com/SphericalHarmonic.html Spherical Harmonic -- from Wolfram MathWorld]
	*[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)]		*[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]]

Carl McBride: /* See also */ Added a reference

2008-06-18T15:58:28Z

Carl McBride at 13:50, 14 August 2007

2007-08-14T13:50:59Z

← Older revision		Revision as of 15:50, 14 August 2007
Line 1:		Line 1:
	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.
			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==
	*[http://mathworld.wolfram.com/SphericalHarmonic.html Spherical Harmonic -- from Wolfram MathWorld]		*[http://mathworld.wolfram.com/SphericalHarmonic.html Spherical Harmonic -- from Wolfram MathWorld]
	*[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)]		*[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]]

Carl McBride at 13:32, 14 August 2007

2007-08-14T13:32:55Z

← Older revision		Revision as of 15:32, 14 August 2007
Line 2:		Line 2:
	==See also==		==See also==
	*[http://mathworld.wolfram.com/SphericalHarmonic.html Spherical Harmonic -- from Wolfram MathWorld]		*[http://mathworld.wolfram.com/SphericalHarmonic.html Spherical Harmonic -- from Wolfram MathWorld]
			*[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]]

Carl McBride: New page: The spherical harmonics $Y_l^m (\theta,\phi)$ are the angular portion of the solution to Laplace's equation in spherical coordinates. ==See also== *[http://mathworld.wolfram...

2007-05-31T10:21:37Z

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...

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.com/SphericalHarmonic.html Spherical Harmonic -- from Wolfram MathWorld]
[[category: mathematics]]

← Older revision		Revision as of 17:58, 18 June 2008
Line 10:		Line 10:
	:<math>Y_1^1 (\theta,\phi) = -\frac{1}{2} \sqrt{\frac{3}{2\pi}} \sin \theta e^{i\phi} </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==
			*M. E. Rose "Elementary theory of angular momentum", John Wiley & Sons (1967) Appendix III
	*[http://mathworld.wolfram.com/SphericalHarmonic.html Spherical Harmonic -- from Wolfram MathWorld]		*[http://mathworld.wolfram.com/SphericalHarmonic.html Spherical Harmonic -- from Wolfram MathWorld]
	*[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)]		*[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]]