Spherical harmonics

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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.

Contents

1 Introduction
2 First few spherical harmonics
3 Addition theorem
4 Generalizations
5 See also
6 References

Introduction

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 Y_l,m and Y_l,-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:

[\left(r^2 \right) + \left(\sin\theta \right) + = 0]

(see nabla in cylindrical and spherical coordinates).

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

[\left(sin\theta \right) + + l(l+1)\Theta(\theta)\Phi(\phi) = 0]

Separation of variables leads to solutions expressed in terms of trigonometric functions and Legendre polynomials. Note that the spherical coordinates [\theta\!\,] and [\varphi\!\,] in this article are used in the physicist's way, as opposed to the mathematician's definition of spherical coordinates. That is, [\theta\!\,] is the colatitude or polar angle, ranging from [0\leq\theta\leq\pi] and [\varphi\!\,] the azimuth or longitude, ranging from [0\leq\varphi<2\pi].

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

[ r^ \cos (m \varphi) P_\ell^m (\cos ) ]

and

[ r^ \sin (m \varphi) P_\ell^m (\cos ) ]

where [P_\ell^m] are the associated Legendre polynomials, and with integer parameters [\ell \ge 0] and m from 0 to [\ell].

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

[ U_(r,\theta , \varphi ) = r^ Y_\ell^m( \theta , \varphi )]

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

[ Y_\ell^m( \theta , \varphi ) = \sqrt \cdot e^ \cdot P_\ell^m ( \cos ) ]

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

[\int_^\pi\int_^Y_\ell^mY_^\,d\Omega=\delta_\delta_\quad\quad d\Omega=\sin\theta\,d\varphi\,d\theta]

Y₁
Y₂
Y₃

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

[Y_\ell^0\quad\quad \mbox\ 0\le\ell\le\infin]

and

[\left((-1)^mY_\ell^m+Y_\ell^\right)\quad\quad \mbox \ 0\le\ell\le\infin,\ 1\le m\le \ell]

and

[\left((-1)^mY_\ell^m-Y_\ell^\right)\quad\quad \mbox \ 0\le\ell\le\infin,\ 1\le m\le \ell]

The spherical harmonics in cartesian coordinates may be obtained by substituting

[\cos\theta=,\qquad e^\cdot\sin^n\theta=,\qquad r=\sqrt].

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 [\theta\!\,] and [\varphi\!\,] can be expanded as a linear combination of spherical harmonics.

[f(\theta,\varphi)=\sum_^ \sum_^ c_Y_(\theta,\varphi)]

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

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

[c_=\int_f(\theta,\varphi)Y_^*(\theta,\varphi)d\Omega]

First few spherical harmonics

Representation as ρ = ρ₀ + ρ₁·Y_l^m(θ,φ)
then the representative surface looks like a "battered" sphere;
Y_l^m is equal to 0 along circles (the representative surface intersects the ρ = ρ₀ sphere at these circles). Y_l^m is alternately positive and negative between two circles.

the Y₃² with four sections

These are the first few spherical harmonics:

[Y_^(\theta,\varphi)=\sqrt]

[Y_^(x)=\sqrt\cdot e^\cdot\sin\theta\quad=\sqrt\cdot]

[Y_^(x)=\sqrt\cdot\cos\theta\quad=\sqrt\cdot]

[Y_^(x)=\sqrt\cdot e^\cdot\sin\theta\quad=\sqrt\cdot]

[Y_^(\theta,\varphi)=\sqrt\cdot e^\cdot\sin^\theta]

[Y_^(\theta,\varphi)=\sqrt\cdot e^\cdot\sin\theta\cdot\cos\theta]

[Y_^(\theta,\varphi)=\sqrt\cdot(3\cos^\theta-1)]

[Y_^(\theta,\varphi)=\sqrt\cdot e^\cdot\sin\theta\cdot\cos\theta]

[Y_^(\theta,\varphi)=\sqrt\cdot e^\cdot\sin^\theta]

[Y_^(\theta,\varphi)=\sqrt\cdot(5\cos^\theta-3\cos\theta)]

More spherical harmonics up to Y₁₀

Addition theorem

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,\theta,\phi)] and [(r ',\theta ',\phi ')],respectively, have an angle [\gamma] between them. The addition theorem expresses a Legendre polynomial of order [l] in the angle [\gamma] in terms of the products of the spherical harmonics of the angles [\theta,\phi] and [\theta',\phi']:

[P_l( \cos \gamma ) = \frac\sum_^l Y_^*(\theta',\phi')Y_(\theta,\phi) ]

where [\cos\gamma=\cos\theta\cos\theta'+\sin\theta\sin\theta'\cos(\phi-\phi')\,]

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.

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.
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]