Spherical harmonics

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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. 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.

Introduction

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Top to bottom: l=0 to 4
Left to right: m=0 to ±4 (non-imaginary harmonics)
The two non-imaginary spherical harmonics that are linear combinations of Yl,m and Yl,-m are equivalent to each other but rotated 90 degrees relative to the z axis relative to each other.

Laplace's equation in spherical coordinates is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {1 \over r^2}{\partial \over \partial r}\left(r^2 {\partial f \over \partial r}\right) + {1 \over r^2\sin\theta}{\partial \over \partial \theta}\left(\sin\theta {\partial f \over \partial \theta}\right) + {1 \over r^2\sin^2\theta}{\partial^2 f \over \partial \varphi^2} = 0}

(see nabla in cylindrical and spherical coordinates).

Separation of variables leads to solutions expressed in terms of trigonometric functions and Legendre polynomials. Note that the spherical coordinates Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta\!\,} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi\!\,} in this article are used in the physicist's way, as opposed to the mathematician's definition of spherical coordinates. That is, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta\!\,} is the colatitude or polar angle, ranging from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\leq\theta\leq\pi} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi\!\,} the azimuth or longitude, ranging from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\leq\varphi<2\pi} .

The general solution which remains finite towards infinity is a linear combination of functions of the form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r^{-1-\ell} \cos (m \varphi) P_\ell^m (\cos{\theta} ) }

and

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where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_\ell^m} are the associated Legendre polynomials, and with integer parameters Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell \ge 0} and m from 0 to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell} .

Put in another way, the solutions with integer parameters Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell \ge 0} and m from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - \ell} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell} , can be written as linear combinations of:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{\ell,m}(r,\theta , \varphi ) = r^{-1-\ell} Y_\ell^m( \theta , \varphi )}

where the functions Y are the spherical harmonics with parameters l, m, which can be written as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_\ell^m( \theta , \varphi ) = \sqrt{{(2\ell+1)\over 4\pi}{(\ell-m)!\over (\ell+m)!}} \cdot e^{i m \varphi } \cdot P_\ell^m ( \cos{\theta} ) }

The spherical harmonics obey the normalisation condition (δaa = 1 and δab = 0 if a ≠ b)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\theta=0}^\pi\int_{\varphi=0}^{2\pi}Y_\ell^mY_{\ell'}^{m'*}\,d\Omega=\delta_{\ell\ell'}\delta_{mm'}\quad\quad d\Omega=\sin\theta\,d\varphi\,d\theta}
Y1
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Y2
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Y3
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An alternative set of spherical harmonics with no imaginary component may be obtained by taking the set

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_\ell^0\quad\quad for\ 0\le\ell\le\infin}

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {1\over\sqrt2}\left((-1)^mY_\ell^m+Y_\ell^{-m}\right)\quad\quad \mbox{ for } \ 0\le\ell\le\infin,\ 1\le m\le \ell}

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {1\over i\sqrt2}\left((-1)^mY_\ell^m-Y_\ell^{-m}\right)\quad\quad \mbox{ for } \ 0\le\ell\le\infin,\ 1\le m\le \ell}

The spherical harmonics in cartesian coordinates may be obtained by substituting

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos\theta={z\over r},\qquad e^{\pm ni\varphi}\cdot\sin^n\theta={(x\pm iy)^n\over r^n},\qquad r=\sqrt{x^2+y^2+z^2}} .

First few spherical harmonics

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Representation as ρ = ρ0 + ρ1·Ylm(θ,φ)
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 alternately positive and negative between two circles.
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the Y32 with four sections

These are the first few spherical harmonics:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_{0}^{0}(\theta,\varphi)={1\over 2}\sqrt{1\over \pi}}


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_{1}^{-1}(x)={1\over 2}\sqrt{3\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\quad={1\over 2}\sqrt{3\over 2\pi}\cdot{(x-iy)\over r}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_{1}^{0}(x)={1\over 2}\sqrt{3\over \pi}\cdot\cos\theta\quad={1\over 2}\sqrt{3\over \pi}\cdot{z\over r}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_{1}^{1}(x)={-1\over 2}\sqrt{3\over 2\pi}\cdot e^{i\varphi}\cdot\sin\theta\quad={-1\over 2}\sqrt{3\over 2\pi}\cdot{(x+iy)\over r}}


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_{2}^{-2}(\theta,\varphi)={1\over 4}\sqrt{15\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_{2}^{-1}(\theta,\varphi)={1\over 2}\sqrt{15\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot\cos\theta}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_{2}^{0}(\theta,\varphi)={1\over 4}\sqrt{5\over \pi}\cdot(3\cos^{2}\theta-1)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_{2}^{1}(\theta,\varphi)={-1\over 2}\sqrt{15\over 2\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot\cos\theta}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_{2}^{2}(\theta,\varphi)={1\over 4}\sqrt{15\over 2\pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta}


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_{3}^{0}(\theta,\varphi)={1\over 4}\sqrt{7\over \pi}\cdot(5\cos^{3}\theta-3\cos\theta)}
More spherical harmonics up to Y10

Generalizations

The spherical harmonics in a certain sense capture the symmetry properties of the two-sphere. The symmetry properties of the two-sphere are given by the Lie groups SO(3) and its double-cover SU(2). The spherical harmonic transform under the integer-spin representations of these groups; they are a part of the representation theory of these groups. However, the two-sphere can also be understood to be the Riemann sphere. The complete set of symmetries of the Riemann sphere can be understood to be described by the Mobius transformation group SL(2,C), of which the Lorentz group is but a representation. The analog of the spherical harmonics for the Lorentz group are given by the hypergeometric series; indeed, the spherical harmonics are easily re-expressed in terms of the hypergeometric series, as SO(3) is a subgroup of SL(2,C).

More generally, hypergeometric series can be generalized to describe the symmetries of any symmetric space; in particular, hypergeometric series can be developed for any Lie group.

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.
  • D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii Quantum Theory of Angular Momentum,(1988) World Scientific Publishing Co., Singapore, ISBN 9971-50-107-4
  • "General Solution to LaPlace's Equation in Spherical Harmonics" (Spherical Harmonic Analysis). Solid Earth Geophysics.
  • Spherical harmonics on Mathworld
  • Spherical harmonics generator in OpenGL