In mathematics, the spherical harmonics are the angular portion of 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, the approximation of the Earth's gravitational field and the geoid, and the parameter fitting for the anisotropy map of the cosmic microwave background.

Introduction

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Laplace's equation in spherical coordinates is:

$\{1\; \backslash over\; r^2\}\{\backslash partial\; \backslash over\; \backslash partial\; r\}\backslash left(r^2\; \{\backslash partial\; f\; \backslash over\; \backslash partial\; r\}\backslash right)$

+ {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).

For $f(\backslash theta,\backslash phi)=\backslash Theta(\backslash theta)\backslash Phi(\backslash phi)\backslash !\backslash ,$ that satisfies the angular portion of the Laplace's equation,

$\{\backslash Phi(\backslash phi)\; \backslash over\; sin\backslash theta\}\{d\; \backslash over\; d\backslash theta\}\backslash left(sin\backslash theta\; \{d\backslash Theta\; \backslash over\; d\backslash theta\}\; \backslash right)\; +\; \{\backslash Theta(\backslash theta)\; \backslash over\; sin^2\; \backslash theta\}\{d^2\backslash Phi\; \backslash over\; d\backslash phi^2\}\; +\; l(l+1)\backslash Theta(\backslash theta)\backslash Phi(\backslash phi)\; =\; 0$

Separation of variables leads to solutions expressed in terms of trigonometric functions and Legendre polynomials. Note that the spherical coordinates $\backslash theta\backslash !\backslash ,$ and $\backslash varphi\backslash !\backslash ,$ in this article are used in the physicist's way, as opposed to the mathematician's definition of spherical coordinates. That is, $\backslash theta\backslash !\backslash ,$ is the colatitude or polar angle, ranging from $0\backslash leq\backslash theta\backslash leq\backslash pi$ and $\backslash varphi\backslash !\backslash ,$ the azimuth or longitude, ranging from .

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

$r^\{-1-\backslash ell\}\; \backslash cos\; (m\; \backslash varphi)\; P\_\backslash ell^m\; (\backslash cos\{\backslash theta\}\; )$

and

$r^\{-1-\backslash ell\}\; \backslash sin\; (m\; \backslash varphi)\; P\_\backslash ell^m\; (\backslash cos\{\backslash theta\}\; )$

where $P\_\backslash ell^m$ are the associated Legendre polynomials, and with integer parameters $\backslash ell\; \backslash ge\; 0$ and m from 0 to $\backslash ell$.

Put in another way, the solutions with integer parameters $\backslash ell\; \backslash ge\; 0$ and m from $-\; \backslash ell$ to $\backslash ell$, can be written as linear combinations of:

$U\_\{\backslash ell,m\}(r,\backslash theta\; ,\; \backslash varphi\; )\; =\; r^\{-1-\backslash ell\}\; Y\_\backslash ell^m(\; \backslash theta\; ,\; \backslash varphi\; )$

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

$Y\_\backslash ell^m(\; \backslash theta\; ,\; \backslash varphi\; )\; =\; \backslash sqrt\; \backslash cdot\; e^\{i\; m\; \backslash varphi\; \}\; \backslash cdot\; P\_\backslash ell^m\; (\; \backslash cos\{\backslash theta\}\; )$

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

$\backslash int\_\{\backslash theta=0\}^\backslash pi\backslash int\_\{\backslash varphi=0\}^\{2\backslash pi\}Y\_\backslash ell^mY\_\{\backslash ell\text{'}\}^\{m\text{'}*\}\backslash ,d\backslash Omega=\backslash delta\_\{\backslash ell\backslash ell\text{'}\}\backslash delta\_\{mm\text{'}\}\backslash quad\backslash quad\; d\backslash Omega=\backslash sin\backslash theta\backslash ,d\backslash varphi\backslash ,d\backslash theta$

|| Legendre Y1 polaire.png || Legendre Y2 polaire.png || Legendre Y3 polaire.png

An alternative set of spherical harmonics with no imaginary component may be obtained by taking the set

$Y\_\backslash ell^0\backslash quad\backslash quad\; \backslash mbox\{\; for\; \}\backslash \; 0\backslash le\backslash ell\backslash le\backslash infin$

and

$\{1\backslash over\backslash sqrt2\}\backslash left((-1)^mY\_\backslash ell^m+Y\_\backslash ell^\{-m\}\backslash right)\backslash quad\backslash quad$

\mbox{ for } \ 0\le\ell\le\infin,\ 1\le m\le \ell

and

$\{1\backslash over\; i\backslash sqrt2\}\backslash left((-1)^mY\_\backslash ell^m-Y\_\backslash ell^\{-m\}\backslash right)\backslash quad\backslash quad$

\mbox{ for } \ 0\le\ell\le\infin,\ 1\le m\le \ell

The spherical harmonics in cartesian coordinates may be obtained by substituting

$\backslash cos\backslash theta=\{z\backslash over\; r\},\backslash qquad\; e^\{\backslash pm\; ni\backslash varphi\}\backslash cdot\backslash sin^n\backslash theta=\{(x\backslash pm\; iy)^n\backslash over\; r^n\},\backslash qquad\; r=\backslash sqrt\{x^2+y^2+z^2\}$.

Spherical harmonics form a complete set of orthonormal functions and thus form a vector space analogue to unit basis vectors. Any (square-integrable) function of $\backslash theta\backslash !\backslash ,$ and $\backslash varphi\backslash !\backslash ,$ can be expanded as a linear combination of spherical harmonics.

$f(\backslash theta,\backslash varphi)=\backslash sum\_\{l=0\}^\{\backslash infty\}\; \backslash sum\_\{m=-l\}^\{l\}\; c\_\{lm\}Y\_\{lm\}(\backslash theta,\backslash varphi)$

This expansion is exact as long as $l\backslash !\backslash ,$ goes to infinity. Errors kick in when limiting the series to a certain order of $l\backslash !\backslash ,$.

The expansion coefficients can be obtained by multiplying the above equation by the complex conjugate of spherical harmonics and integrating over the solid angle $\backslash Omega\backslash !\backslash ,$.

$c\_\{lm\}=\backslash int\_\{S\_2\}f(\backslash theta,\backslash varphi)Y\_\{lm\}^*(\backslash theta,\backslash varphi)d\backslash Omega$

First few spherical harmonics

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These are the first few spherical harmonics:

$Y\_\{0\}^\{0\}(\backslash theta,\backslash varphi)=\{1\backslash over\; 2\}\backslash sqrt\{1\backslash over\; \backslash pi\}$

$Y\_\{1\}^\{-1\}(x)=\{1\backslash over\; 2\}\backslash sqrt\{3\backslash over\; 2\backslash pi\}\backslash cdot\; e^\{-i\backslash varphi\}\backslash cdot\backslash sin\backslash theta\backslash quad=\{1\backslash over\; 2\}\backslash sqrt\{3\backslash over\; 2\backslash pi\}\backslash cdot\{(x-iy)\backslash over\; r\}$

$Y\_\{1\}^\{0\}(x)=\{1\backslash over\; 2\}\backslash sqrt\{3\backslash over\; \backslash pi\}\backslash cdot\backslash cos\backslash theta\backslash quad=\{1\backslash over\; 2\}\backslash sqrt\{3\backslash over\; \backslash pi\}\backslash cdot\{z\backslash over\; r\}$

$Y\_\{1\}^\{1\}(x)=\{-1\backslash over\; 2\}\backslash sqrt\{3\backslash over\; 2\backslash pi\}\backslash cdot\; e^\{i\backslash varphi\}\backslash cdot\backslash sin\backslash theta\backslash quad=\{-1\backslash over\; 2\}\backslash sqrt\{3\backslash over\; 2\backslash pi\}\backslash cdot\{(x+iy)\backslash over\; r\}$

$Y\_\{2\}^\{-2\}(\backslash theta,\backslash varphi)=\{1\backslash over\; 4\}\backslash sqrt\{15\backslash over\; 2\backslash pi\}\backslash cdot\; e^\{-2i\backslash varphi\}\backslash cdot\backslash sin^\{2\}\backslash theta$

$Y\_\{2\}^\{-1\}(\backslash theta,\backslash varphi)=\{1\backslash over\; 2\}\backslash sqrt\{15\backslash over\; 2\backslash pi\}\backslash cdot\; e^\{-i\backslash varphi\}\backslash cdot\backslash sin\backslash theta\backslash cdot\backslash cos\backslash theta$

$Y\_\{2\}^\{0\}(\backslash theta,\backslash varphi)=\{1\backslash over\; 4\}\backslash sqrt\{5\backslash over\; \backslash pi\}\backslash cdot(3\backslash cos^\{2\}\backslash theta-1)$

$Y\_\{2\}^\{1\}(\backslash theta,\backslash varphi)=\{-1\backslash over\; 2\}\backslash sqrt\{15\backslash over\; 2\backslash pi\}\backslash cdot\; e^\{i\backslash varphi\}\backslash cdot\backslash sin\backslash theta\backslash cdot\backslash cos\backslash theta$

$Y\_\{2\}^\{2\}(\backslash theta,\backslash varphi)=\{1\backslash over\; 4\}\backslash sqrt\{15\backslash over\; 2\backslash pi\}\backslash cdot\; e^\{2i\backslash varphi\}\backslash cdot\backslash sin^\{2\}\backslash theta$

$Y\_\{3\}^\{0\}(\backslash theta,\backslash varphi)=\{1\backslash over\; 4\}\backslash sqrt\{7\backslash over\; \backslash pi\}\backslash cdot(5\backslash cos^\{3\}\backslash theta-3\backslash cos\backslash theta)$

More spherical harmonics up to Y₁₀

Addition theorem

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A mathematical result of considerable interest and use is called the addition theorem for spherical harmonics. Two coordinate vectors x and x', with spherical coordinates $(r,\backslash theta,\backslash phi)$ and $(r\; \text{'},\backslash theta\; \text{'},\backslash phi\; \text{'})$,respectively, have an angle $\backslash gamma$ between them. The addition theorem expresses a Legendre polynomial of order $l$ in the angle $\backslash gamma$ in terms of the products of the spherical harmonics of the angles $\backslash theta,\backslash phi$ and $\backslash theta\text{'},\backslash phi\text{'}$:

$P\_l(\; \backslash cos\; \backslash gamma\; )\; =\; \backslash frac\{4\backslash pi\}\{2l+1\}\backslash sum\_\{m=-l\}^l\; Y\_\{lm\}^*(\backslash theta\text{'},\backslash phi\text{'})Y\_\{lm\}(\backslash theta,\backslash phi)$

where $\backslash cos\backslash gamma=\backslash cos\backslash theta\backslash cos\backslash theta\text{'}+\backslash sin\backslash theta\backslash sin\backslash theta\text{'}\backslash cos(\backslash phi-\backslash phi\text{'})\backslash ,$

Generalizations

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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

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

References

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• 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.
• J.D. Jackson, Classical Electrodynamics, ISBN 0-471-30932-X
• 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

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