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, representation of gravitational fields, geoids, and the magnetic fields of planetary bodies, and characterization of the cosmic microwave background radiation. In 3D computer graphics, spherical harmonics plays a special role in a wide variety of topics including indirect lighting (ambient occlusion, global illumination, precomputed radiance transfer, etc.) and in recognition of 3D shapes.

Introduction

Laplace's equation in spherical coordinates is:

: abla^2 f = {1 over r^2}{partial over partial r}left(r^2 {partial f over partial r} ight) + {1 over r^2sin heta}{partial over partial heta}left(sin heta {partial f over partial heta} ight) + {1 over r^2sin^2 heta}{partial^2 f over partial varphi^2} = 0

(see also del in cylindrical and spherical coordinates). For "f"("r",θ,φ)="R"("r")Θ(θ)Φ(φ), the angular portion of Laplace's equation satisfies

:{Phi(varphi) over sin heta}{d over d heta}left(sin heta {dTheta over d heta} ight) + {Theta( heta) over sin^2 heta}{d^2Phi over dvarphi^2} + l(l+1)Theta( heta)Phi(varphi) = 0.

Using the technique of separation of variables, two differential equations result:

:frac{1}{Phi(varphi)} frac{d^2 Phi(varphi)}{dvarphi^2} = -m^2

:l(l+1)sin ^2( heta) + frac{sin( heta)}{Theta( heta)} frac{d}{d heta} left [ sin( heta) frac{dTheta}{d heta} ight ] = m^2

for some "m" and "l". Hence, the angular solutions can be shown to be a products of trigonometric functions and associated Legendre functions:

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

where Y_ell^m is 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 φ represent colatitude and longitude, respectively. The spherical coordinates used in this article are consistent with those used by physicists, but differ from those employed by mathematicians (see spherical coordinates). In particular, the colatitude θ, or polar angle, ranges from 0 ≤ θ ≤ π and the longitude φ, or azimuth, ranges from 0 ≤ φ ≤ 2π. Thus, θ is 0 at the North Pole, π/2 at the Equator, and π at the South Pole.

When Laplace's equation is solved on the surface of the sphere, the periodic boundary conditions in φ, 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 ≥ 0 and |"m"| ≤ ell. In contrast, if the function "f" were only to have been defined for θ ≤ θ0, 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 multiplied by the solutions of "R"("r"):

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

Orthogonality and normalization

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

which are orthonormal

:int_{ heta=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 Kronecker delta) and "d"Ω = sinθ "d"φ "d"θ. The disciplines of geodesy and spectral analysis use

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

which possess unit power

:{1 over 4 pi} int_{ heta=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( heta , varphi ) = sqrt(ell-m)!over (ell+m)! , P_ell^m ( cos{ heta} ) , e^{i m varphi }

which have the normalization

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

In quantum mechanics this normalization is often used, too, and is there named Racah's normalization after Giulio Racah.

Using the identity (see associated Legendre functions)

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

where the superscript * denotes complex conjugation. Alternatively, this equation follows from the relation of the spherical harmonic functions with the Wigner D-matrix.

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. 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 mechanical operations, especially the application of raising and lowering operators. The geodesy and magnetics communities never include the Condon-Shortley phase factor in their definitions of the spherical harmonic functions.

pherical harmonics expansion

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

f( heta,varphi)=sum_{ell=0}^{infty} sum_{m=-ell}^ell f_ell^m , Y_ell^m( heta,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( heta,varphi), Y_ell^{m*}( heta,varphi)dOmega = int_0^{2pi}dvarphiint_0^{pi}d hetasin heta f( heta,varphi)Y_ell^{m*} ( heta,varphi).

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

Y_{ell m} = egin{cases}Y_ell^0 qquadqquadqquadqquadqquadqquadqquadqquadqquadqquadquadquadmbox{ if } m=0\{1oversqrt2}left(Y_ell^m+(-1)^m , Y_ell^{-m} ight) = sqrt{2} N_{(l,m)} P_ell^m(cos heta) cos mvarphi qquadquadquad mbox{if } m>0 \{1over isqrt2}left(Y_ell^{-m}-(-1)^{m}, Y_ell^{m} ight) = sqrt{2} N_{(l,m)} P_ell^{-m}(cos heta) sin mvarphi quadmbox{ if } m<0.end{cases}

where N_{(l,m)} denotes the normalization constant as a function of l and m. 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( heta, varphi) = sum_{ell=0}^infty sum_{m=-ell}^ell f_{lm} , Y_{lm}( heta, varphi).

See here for a list of real spherical harmonics up to and including "l = 5". Note, however, that the listed functions differ by the phase (-1)"m" from the phase given in this article.

pectrum 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:

: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 similar 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 "f"00 and "g"00 are zero), then "Sff"("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^{eta}.

When &beta; = 0, the spectrum is "white" as each degree possesses equal power. When &beta; < 0, the spectrum is termed "red" as there is more power at the low degrees with long wavelengths than higher degrees. Finally, when &beta; > 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, heta,varphi) and (r ', heta ',varphi '),respectively, have an angle ,!gamma between them given by

:cosgamma=cos hetacos heta'+sin hetasin heta'cos(varphi-varphi').

The addition theorem expresses a Legendre polynomial of order l in the angle ,!gamma in terms of products of two spherical harmonics with angular coordinates ( heta,varphi) and ( heta',varphi'):

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

Visualization of the spherical harmonics

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.

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}( heta,varphi)={1over 2}sqrt{1over pi}

:Y_{1}^{-1}( heta,varphi)={1over 2}sqrt{3over 2pi} , sin heta , e^{-ivarphi}:Y_{1}^{0}( heta,varphi)={1over 2}sqrt{3over pi}, cos heta:Y_{1}^{1}( heta,varphi)={-1over 2}sqrt{3over 2pi}, sin heta, e^{ivarphi}

:Y_{2}^{-2}( heta,varphi)={1over 4}sqrt{15over 2pi} , sin^{2} heta , e^{-2ivarphi}:Y_{2}^{-1}( heta,varphi)={1over 2}sqrt{15over 2pi}, sin heta, cos heta, e^{-ivarphi}:Y_{2}^{0}( heta,varphi)={1over 4}sqrt{5over pi}, (3cos^{2} heta-1):Y_{2}^{1}( heta,varphi)={-1over 2}sqrt{15over 2pi}, sin heta,cos heta, e^{ivarphi}:Y_{2}^{2}( heta,varphi)={1over 4}sqrt{15over 2pi}, sin^{2} heta , e^{2ivarphi}

:Y_{3}^{0}( heta,varphi)={1over 4}sqrt{7over pi}, (5cos^{3} heta-3cos heta)

:More spherical harmonics up to Y10

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 addition, the two-sphere is equivalent to the Riemann sphere. The complete set of symmetries of the Riemann sphere are described by the Möbius 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).

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

later integrals or coefficients

John C. Slater defined the integral of three spherical harmonics as a coefficient c [John C. Slater, Quantum Theory of Atomic Structure, McGraw-Hill (New York, 1960), Volume I]

:c^k(l,m,l,'m')=int d^2Omega Y_l^m(Omega)^* Y_{l'}^{m'}(Omega) Y_k^{m-m'}(Omega)

These integrals are useful and necessary when doing atomic calculations of the Hartree-Fock variety where matrix elements of the Coulomb operator are needed. For an explicit formula, one can use Gaunt's formula under the section on Associated Legendre functions.

Note that the product of two spherical harmonics can be written in terms of these coefficients. By expanding such a product over a spherical harmonic basis, as outlined earlier in this article,:Y_l^m Y_{l'}^{m'} = sum_{l",m"} hat{A}(l,m,l',m',l",m") Y_{l"}^{m"},one may then multiply by Y* and integrate, using the conjugate property and being careful with phases and normalisations::int Y_l^m Y_{l'}^{m'} Y_{L}^{-M} = hat{A}(l,m,l',m',L,M) = c^L(l,m,l',m')

These coefficient obey a number of identities. They include

::egin{align}c^k(l,m,l',m') &=c^k(l,-m,l',-m')\&=(-1)^{m-m'}c^k(l',m',l,m)\&=(-1)^{m-m'}sqrt{frac{2l+1}{2k+1c^l(l',m',k,m'-m)\& = (-1)^{m'}sqrt{frac{2l'+1}{2k+1c^{l'}(k,m-m',l,m)\sum_{m=-l}^{l} c^k(l,m,l,m) &= (2l+1)delta_{k,0}\sum_{m=-l}^l sum_{m'=-l'}^{l'} c^k(l,m,l',m')^2 &= sqrt{(2l+1)(2l'+1)}cdot c^k(l,0,l',0)\sum_{m=-l}^l c^k(l,m,l',m')^2 & = sqrt{frac{2l+1}{2l'+1cdot c^k(l,0,l',0)\sum_{m=-l}^l c^k(l,m,l',m')c^k(l,m, ilde l,m') &= delta_{l', ilde l}cdotsqrt{frac{2l+1}{2l'+1cdot c^k(l,0,l',0)\sum_m c^k(l,m+r,l',m) c^k(l,m+r, ilde l,m) &= delta_{l, ilde l} cdot frac{sqrt{(2l+1)(2l'+1){2k+1}cdot c^k(l,0,l',0)\sum_m c^k(l,m+r,l',m)c^q(l,m+r,l',m) &= delta_{k,q}cdotfrac{sqrt{(2l+1)(2l'+1){2k+1}cdot c^k(l,0,l',0)end{align}

ee also

* Clebsch-Gordan coefficients
* Harmonic function
* Rotation group
* Sturm-Liouville theory
* Atomic orbital
* Solid harmonics
* Vector spherical harmonics
* Table of spherical harmonics

References

Cited referencesGeneral references
* E.W. Hobson, "The Theory of Spherical and Ellipsoidal Harmonics", (1955) Chelsea Pub. Co., ISBN 978-0828401043.
* T.M. MacRobert, "Spherical harmonics: An elementary treatise on harmonic functions, with applications", (1967) Pergamon Press, 349p.
* C. Müller, "Spherical Harmonics", (1966) Springer, Lecture Notes in Mathematics, Vol. 17, ISBN 978-3-540-03600-5.
* 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 9971-5-0107-4Web resources
* [http://mathworld.wolfram.com/SphericalHarmonic.html Spherical harmonics on Mathworld]
* [http://www.ipgp.jussieu.fr/~wieczor/SH/SH.html Spherical Harmonic Models of Planetary Topography]
* [http://adomas.org/shg/ Spherical harmonics generator in OpenGL]
* [http://www.paulsprojects.net/opengl/sh/sh.html OpenGL Spherical harmonics demo]

oftware

* [http://www.ipgp.jussieu.fr/~wieczor/SHTOOLS/SHTOOLS.html SHTOOLS: Fortran 95 software archive]
* [http://healpix.jpl.nasa.gov/ HEALPIX: Fortran 90 and C++ software archive]
* [http://www.cisl.ucar.edu/css/software/spherepack/ SpherePack: Fortran 77 software archive]
* [http://www.cs.dartmouth.edu/~geelong/sphere/ SpharmonicKit: C software archive]
* [http://geoweb.princeton.edu/people/simons/software.html Frederik J Simons: Matlab software archive]
* [http://www-user.tu-chemnitz.de/~potts/nfft/ NFFT: C subroutine library (fast spherical Fourier transform for arbitrary nodes)]
* [http://www.spice-rtn.org/library/software/shansyn Shansyn: spherical harmonics package for GMT/netcdf grd files]
* [http://www.embl-heidelberg.de/~khairy/links.html SHAPE: Spherical HArmonic Parameterization Explorer]

External links

*Interactive calculator of spherical harmonics on [http://wm.eecs.umich.edu:8180/webMathematica/tcarmon/sh2.jsp Tal Carmon's Research Homepage]
*Spherical harmonics applied to Acoustic Field analysis on [http://www.trinnov.com/research.php#concept Trinnov Audio's research page]
* [http://demonstrations.wolfram.com/SphericalHarmonics/ Spherical Harmonics] by Stephen Wolfram and [http://demonstrations.wolfram.com/NodalDomainsOfSphericalHarmonics/ Nodal Domains of Spherical Harmonics] by Michael Trott, The Wolfram Demonstrations Project
* [http://mysite.du.edu/~jcalvert/math/harmonic/harmonic.htm An accessible introduction to spherical harmonics (by J. B. Calvert)]
* [http://en.citizendium.org/wiki/Spherical_harmonics Spherical harmonics entry at Citizendium]


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