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Encyclopedia > Spherical harmonics

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 in the computation of atomic electron configurations, the representation of the gravitational field, geoid, and magnetic field of planetary bodies, as well as characterization of the cosmic microwave background radiation. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... In mathematics, Laplaces equation is a partial differential equation named after its discoverer Pierre-Simon Laplace. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... Electron atomic and molecular orbitals In atomic physics and quantum chemistry, the electron configuration is the arrangement of electrons in an atom, molecule, or other physical structure (eg, a crystal). ... The gravitational field is a field (physics), generated by massive objects, that determines the magnitude and direction of gravitation experienced by other massive objects. ... The GOCE project will measure high-accuracy gravity gradients and provide an accurate geoid model based on the Earths gravity field. ... To meet Wikipedias quality standards, this article may require cleanup. ... WMAP image of the CMB anisotropy,Cosmic microwave background radiation(June 2003) The cosmic microwave background radiation (CMB) is a form of electromagnetic radiation that fills the whole of the universe. ...

Contents

Introduction

Real spherical harmonics Ylm, for l=0 to 4 (top to bottom) and m=0 to 4 (left to right). The negative order harmonics Yl-m are rotated about the z axis by 90/m degrees with respect to the positve order ones.

Laplace's equation in spherical coordinates is: Image File history File links Rotating_spherical_harmonics. ...

nabla^2 f = {1 over r^2}{partial over partial r}left(r^2 {partial f over partial r}right) + {1 over r^2sintheta}{partial over partial theta}left(sintheta {partial f over partial theta}right) + {1 over r^2sin^2theta}{partial^2 f over partial varphi^2} = 0

(see also nabla in cylindrical and spherical coordinates). For f(r,θ,φ) = R(r)Θ(θ)Φ(φ), the angular portion of Laplace's equation satisfies This is a list of some vector calculus formulae of general use in working with standard coordinate systems. ...

{Phi(varphi) over sintheta}{d over dtheta}left(sintheta {dTheta over dtheta} right) + {Theta(theta) over sin^2 theta}{d^2Phi over dvarphi^2} + l(l+1)Theta(theta)Phi(varphi) = 0.

Using the technique of separation of variables, the angular solutions can be shown to be a products of trigonometric functions and associated Legendre functions: In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to re-write an equation so that each of two variables occurs on a different side of the equation. ... All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. In mathematics, the trigonometric functions are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other applications. ... In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation or This equation has solutions that are nonsingular on [−1, 1] only if and are integers with , or with trivially equivalent negative values. ...

Y_ell^m (theta, varphi ) = N , e^{i m varphi } , P_ell^m (cos{theta} ),

where Y_ell^m is a called a spherical harmonic function of degree ell and order m, P_ell^m is an associated Legendre function, N is a normalization constant, and θ and varphi represent colatitude and longitude, respectively. The spherical coordinates used in this article are consistent those used by physicists, but differ from those employed by mathematicians (see spherical coordinates). In particular, the colatitude θ, or polar angle, ranges from 0leqthetaleqpi and the longitude varphi, or azimuth, ranges from 0leqvarphi<2pi. Thus, θ is 0 at the North Pole, π / 2 at the Equator, and π at the South Pole. In mathematics, the associated Legendre polynomials, named after Adrien-Marie Legendre, are defined by: These differ from the Legendre polynomials. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... In spherical coordinates, colatitude is the complementary angle of the latitude. ... Longitude, sometimes denoted by the Greek letter λ (lambda),[1][2] describes the location of a place on Earth east or west of a north-south line called the Prime Meridian. ... Azimuth is the horizontal component of a direction (compass direction), measured around the horizon toward the East, i. ...


When Laplace's equation is solved on the surface of the sphere, the periodic boundary conditions in varphi, as well as regularity conditions at both the north and south poles, ensure that the degree ell and order m are integers that satisfy ell ge 0 and |m| le ell. In contrast, if the function f were only to have been defined for theta le theta_0, then the resulting spherical cap harmonics would have been defined for integer order, but non-integer degree. The general solution to Laplace's equation is a linear combination of the spherical harmonic functions multplied by the solutions of R(r):

f(r, theta, varphi) = sum_{ell=0}^infty sum_{m=-ell}^ell r^{-1-ell} , f_ell^m , Y_ell^m (theta, varphi ) + sum_{ell=0}^infty sum_{m=-ell}^ell r^ell , f_ell^{m'} , Y_ell^m (theta, varphi ),

where f_ell^m and f_ell^{m'} are constants. The terms in the first summation approach zero as r goes to infinity, whereas the terms in the second summation approach zero at the origin.


Normalizations

Several different normalizations are in common use for the spherical harmonic functions. In physics and seismology, these functions are generally defined as

Y_ell^m( theta , varphi ) = sqrt{{(2ell+1)over 4pi}{(ell-m)!over (ell+m)!}} , P_ell^m ( cos{theta} ) , e^{i m varphi }

which are orthonormal

int_{theta=0}^piint_{varphi=0}^{2pi}Y_ell^m , Y_{ell'}^{m'*} , dOmega=delta_{ellell'}, delta_{mm'},

where δaa = 1, δab = 0 if a ≠ b, (see Kroenecker delta) and dOmega=sintheta,dvarphi,dtheta. The disciplines of geodesy and spectral analysis use In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

Y_ell^m( theta , varphi ) = sqrt{{(2ell+1) }{(ell-m)!over (ell+m)!}} , P_ell^m ( cos{theta} ), e^{i m varphi }

which possess unit power

{1 over 4 pi} int_{theta=0}^piint_{varphi=0}^{2pi}Y_ell^m , Y_{ell'}^{m'*} dOmega=delta_{ellell'}, delta_{mm'}.

The magnetics community, in contrast, uses Schmidt semi-normalized harmonics

Y_ell^m( theta , varphi ) = sqrt{{(ell-m)!over (ell+m)!}} , P_ell^m ( cos{theta} ) , e^{i m varphi }

which have the normalization

int_{theta=0}^piint_{varphi=0}^{2pi}Y_ell^m , Y_{ell'}^{m'*}dOmega={4 pi over (2 ell + 1)}delta_{ellell'}, delta_{mm'}.

Using the identity (see associated Legendre functions) In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation or This equation has solutions that are nonsingular on [−1, 1] only if and are integers with , or with trivially equivalent negative values. ...

P_ell ^{-m} = (-1)^m frac{(ell-m)!}{(ell+m)!} P_ell ^{m}

it can be shown that all of the above normalized spherical harmonic functions satisfy

Y_ell^{m*} (theta, varphi) = (-1)^m Y_ell^{-m} (theta, varphi),

where the superscript * denotes complex conjugation.


Condon-Shortley Phase

One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of (-1)^m,, commonly referred to as the Condon-Shortley phase in the quantum mechanical literature. In the quantum mechanics community, it is common practice to either include this phase factor in the definition of the associated Legendre functions, or to append it to the definition of the spherical harmonic functions. It should be noted that there is no requirement to use the Condon-Shortley phase in the definition of the spherical harmonic functions, but including it can simplify some quantum mechnical operations. It should be noted that the geodesy and magnetics communities never include the Condon-Shortley phase factor in their definitions of the spherical harmonic functions. In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation or This equation has solutions that are nonsingular on [−1, 1] only if and are integers with , or with trivially equivalent negative values. ...


Spherical harmonics expansion

The spherical harmonics form a complete set of orthonormal functions and thus form a vector space analogue to unit basis vectors. On the unit sphere, any square-integrable function can thus be expanded as a linear combination of these:

f(theta,varphi)=sum_{ell=0}^{infty} sum_{m=-ell}^ell f_ell^m , Y_ell^m(theta,varphi).

This expansion is exact as long as ell goes to infinity. Truncation errors will arise when limiting the sum over ell to a finite bandwidth L. The expansion coefficients can be obtained by multiplying the above equation by the complex conjugate of a spherical harmonic, integrating over the solid angle Omega!,, and utilizing the above orthogonality relationships. For the case of orthonormalized harmonics, this gives

f_ell^m=int_{Omega} f(theta,varphi), Y_ell^{m*}(theta,varphi)dOmega = int_0^{2pi}dvarphiint_0^{pi}dthetasintheta f(theta,varphi)Y_ell^{m*} (theta,varphi).

An alternative set of spherical harmonics for real functions may be obtained by taking the set

Y_{ell m} = begin{cases} Y_ell^0 qquadqquadqquadqquadqquadqquadqquadqquadqquadqquadquadquadmbox{ if } m=0 {1oversqrt2}left(Y_ell^m+(-1)^m , Y_ell^{-m}right) = sqrt{2} N P_ell^m(theta) cos mvarphi qquadquadquad mbox{if } m>0 {1over isqrt2}left(Y_ell^{|m|}-(-1)^{|m|}, Y_ell^{-|m|}right) = sqrt{2} N P_ell^{|m|}(theta) sin |m|varphi quadmbox{ if } m<0. end{cases}

These functions have the same normalization properties as the complex ones above. In this notation, a real square-integrable function can be expressed as an infinite sum of real spherical harmonics as

f(theta, varphi) = sum_{ell=0}^infty sum_{m=-ell}^ell f_{lm} , Y_{lm}(theta, varphi).

Spectrum Analysis

The total power of a function f is defined in the signal processing literature as the integral of the function squared, divided by the area it spans. Using the orthonormality properties of the real unit-power spherical harmonic functions, it is straightforward to verify that the total power of a function defined on the unit sphere is related to its spectral coefficients by a generalization of Parseval's theorem: In mathematics, Parsevals theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. ...

frac{1}{4 , pi} int_Omega f(Omega)^2, dOmega = sum_{l=0}^infty S_{f!f}(l),

where

S_{f!f}(l) = sum_{m=-l}^l f_{lm}^2

is defined as the angular power spectrum. In a similiar manner, one can define the cross-power of two functions as

frac{1}{4 , pi} int_Omega f(Omega) , g(Omega) , dOmega = sum_{l=0}^infty S_{fg}(l),

where

S_{fg}(l) = sum_{m=-l}^l f_{lm} g_{lm}

is defined as the cross-power spectrum. If the functions f and g have a zero mean (i.e., the spectral coefficients f00 and g00 are zero), then S_{f!f}(l) and Sfg(l) represent the contributions to the function's variance and covariance for degree ell, respectively. It is common that the (cross-)power spectrum is well approximated by a power law of the form

S_{f!f}(l) = C , ell^{beta}.

When β = 0, the spectrum is "white" as each degree possesses equal power. When β < 0, the spectrum is termed "red" as there is more power at the low degrees with long wavelengths than higher degrees. Finally, when β > 0, the spectrum is termed "blue".


Addition theorem

A mathematical result of considerable interest and use is called the addition theorem for spherical harmonics. Two vectors r and r', with spherical coordinates (r,theta,varphi) and (r ',theta ',varphi '),respectively, have an angle γ between them given by

cosgamma=costhetacostheta'+sinthetasintheta'cos(varphi-varphi').

The addition theorem expresses a Legendre polynomial of order l in the angle γ in terms of products of two spherical harmonics with angular coordinates (theta,varphi) and (theta',varphi'): Note: The term Legendre polynomials is sometimes used (wrongly) to indicate the associated Legendre polynomials. ...


P_l( cos gamma ) = frac{4pi}{2l+1}sum_{m=-l}^l Y_{lm}^*(theta',phi') , Y_{lm}(theta,varphi).


This expression is valid for both real and complex harmonics. However, it should be emphasized that the quoted form above is valid only for the orthonormalized spherical harmonics. For unit power harmonics it is only necessary to remove the factor of 4π.


Visualization of the spherical harmonics

Schematic representation of Ylm on the unit sphere. Ylm is equal to 0 along m great circles passing through the poles, and along l-m circles of equal latitude. The function changes sign each time it crosses one of these lines.
Schematic representation of Ylm on the unit sphere. Ylm is equal to 0 along m great circles passing through the poles, and along l-m circles of equal latitude. The function changes sign each time it crosses one of these lines.

The spherical harmonics are easily visualized by counting the number of zero crossings they possess in both the latitudinal and longitudinal directions. For the latitudinal direction, the associated Legendre functions possess l − | m | zeros, whereas for the longitudinal direction, the trigonomentric sin and cos functions possess 2 | m | zeros. Image File history File links Download high resolution version (668x777, 41 KB) Représentation dune harmonique sphérique cpar écart à une sphère Representation of a spherical harmonic by discrepancy to a sphere Auteur/author : Christophe Dang Ngoc Chan (Cdang) File links The following pages link to this file... Image File history File links Download high resolution version (668x777, 41 KB) Représentation dune harmonique sphérique cpar écart à une sphère Representation of a spherical harmonic by discrepancy to a sphere Auteur/author : Christophe Dang Ngoc Chan (Cdang) File links The following pages link to this file...


When the spherical harmonic order m is zero, the spherical harmonic functions do not depend upon longitude, and are referred to as zonal. When l = | m | , there are no zero crossings in latitude, and the functions are referred to as sectoral. For the other cases, the functions checker the sphere, and they are referred to as tesseral.


First few spherical harmonics

Analytic expressions for the first few orthonormalized spherical harmonics that use the Condon-Shortley phase convention:

Y_{0}^{0}(theta,varphi)={1over 2}sqrt{1over pi}
Y_{1}^{-1}(theta,varphi)={1over 2}sqrt{3over 2pi} , sintheta , e^{-ivarphi}
Y_{1}^{0}(theta,varphi)={1over 2}sqrt{3over pi}, costheta
Y_{1}^{1}(theta,varphi)={-1over 2}sqrt{3over 2pi}, sintheta, e^{ivarphi}
Y_{2}^{-2}(theta,varphi)={1over 4}sqrt{15over 2pi} , sin^{2}theta , e^{-2ivarphi}
Y_{2}^{-1}(theta,varphi)={1over 2}sqrt{15over 2pi}, sintheta, costheta, e^{-ivarphi}
Y_{2}^{0}(theta,varphi)={1over 4}sqrt{5over pi}, (3cos^{2}theta-1)
Y_{2}^{1}(theta,varphi)={-1over 2}sqrt{15over 2pi}, sintheta,costheta, e^{ivarphi}
Y_{2}^{2}(theta,varphi)={1over 4}sqrt{15over 2pi}, sin^{2}theta , e^{2ivarphi}
Y_{3}^{0}(theta,varphi)={1over 4}sqrt{7over pi}, (5cos^{3}theta-3costheta)
More spherical harmonics up to Y10

This is a table of spherical harmonics up to l=10. ...

Generalizations

The spherical harmonics map can be seen as representations of the symmetry group of rotations around a point (SO(3)) and its double-cover SU(2). As such they capture the symmetry of the two-dimensional sphere (or two-sphere). Each set of spherical harmonics with a given value for the l-parameter map onto a different irreducible representation of SO(3). In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ... In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ... A sphere (< Greek σφαίρα) is a perfectly symmetrical geometrical object. ... A sphere is a perfectly symmetrical geometrical object. ... The factual accuracy of this article is disputed. ... In mathematics, the term irreducible is used in several ways. ... Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ...


In addition, the two-sphere is equivalent to the Riemann sphere. The complete set of symmetries of the Riemann sphere are described by the Mobius transformation group PSL(2,C), which is isomorphic as a real Lie group to the Lorentz group. The analog of the spherical harmonics for the Lorentz group are given by the hypergeometric series; indeed, the spherical harmonics can be re-expressed in terms of the hypergeometric series, as SO(3) is a subgroup of PSL(2,C). A sphere is a perfectly symmetrical geometrical object. ... A rendering of the Riemann Sphere. ... Möbius transformations should not be confused with the Möbius transform. ... The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ... In mathematics, a hypergeometric series is a power series in which the ratios of successive coefficients k is a rational function of k. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...


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[1][2][3][4] In mathematics, the term symmetric space has several different meanings. ... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ...


See also

This article may be too technical for most readers to understand. ... In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R (where U is an open subset of Rn) which satisfies Laplaces equation, i. ... In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ... In mathematics and its applications, a classical Sturm-Liouville equation, named after Jacques Charles François Sturm (1803-1855) and Joseph Liouville (1809-1882), is a real second-order linear differential equation of the form where the functions p(x), q(x), and w(x) are specified at the outset...

References

Cited references

  1. ^ N. Vilenkin, Special Functions and the Theory of Group Representations, Am. Math. Soc. Transl., vol. 22, (1968).
  2. ^ J. D. Talman, Special Functions, A Group Theoretic Approach, (based on lectures by E.P. Wigner), W. A. Benjamin, New York (1968).
  3. ^ W. Miller, Symmetry and Separation of Variables, Addison-Wesley, Reading (1977).
  4. ^ A. Wawrzyńczyk, Group Representations and Special Functions, Polish Scientific Publishers. Warszawa (1984).

General 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 0-521-09209-4, See chapter 3.
  • J.D. Jackson, Classical Electrodynamics, ISBN 0-471-30932-X
  • Albert Messiah, Quantum Mechanics, volume II. (2000) Dover. ISBN 0-486-40924-4.
  • D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii Quantum Theory of Angular Momentum,(1988) World Scientific Publishing Co., Singapore, ISBN 9971-5-0107-4

Web resources

  • Spherical harmonics on Mathworld
  • Spherical Harmonic Models of Planetary Topography
  • Spherical harmonics generator in OpenGL
  • General Solution to LaPlace's Equation in Spherical Harmonics (Spherical Harmonic Analysis). Solid Earth Geophysics.

Software


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