Spherical harmonics

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Categories: Partial differential equations | Special hypergeometric functions | Atomic physics | Geophysics | Rotational symmetry

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

Laplace's equation in spherical coordinates is:

<math>{1 \over r^2}{\partial \over \partial r}(r^2 {\partial f \over \partial r})

 + {1 \over r^2\sin\theta}{\partial \over \partial \theta}(\sin\theta {\partial f \over \partial \theta}) 
 + {1 \over r^2\sin^2\theta}{\partial^2 f \over \partial \varphi^2} = 0</math>

(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 <math>\theta\!\,</math> and <math>\varphi\!\,</math> in this article are used in the physicist's way, as opposed to the mathematician's definition of spherical coordinates. That is, <math>\theta\!\,</math> is the colatitude or polar angle, ranging from <math>0\leq\theta\leq\pi</math> and <math>\varphi\!\,</math> the azimuth or longitude, ranging from <math>0\leq\varphi<2\pi</math>.

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

<math> r^{-1-\ell} \cos (m \varphi) P_\ell^m (\cos{\theta} ) </math>

and

<math> r^{-1-\ell} \sin (m \varphi) P_\ell^m (\cos{\theta} ) </math>

where <math>P_\ell^m</math> are the associated Legendre polynomials, and with integer parameters <math>\ell \ge 0</math> and m from 0 to <math>\ell</math>.

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

<math> U_{\ell,m}(r,\theta , \varphi ) = r^{-1-\ell} Y_\ell^m( \theta , \varphi )</math>

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

<math> 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} ) </math>

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

<math>\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</math>

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

<math>Y_\ell^0\quad\quad for\ 0\le\ell\le\infin</math>

and

<math>{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</math>

and

<math>{1\over i\sqrt2}\left((-1)^mY_\ell^m-Y_\ell^{-m}\right)\quad\quad

\mbox{ for } \ 0\le\ell\le\infin,\ -\ell \le m\le -1</math>

The spherical harmonics in cartesian coordinates may be obtained by substituting

<math>\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}</math>.

First few spherical harmonics

These are the first few spherical harmonics:

<math>Y_{0}^{0}(\theta,\varphi)={1\over 2}\sqrt{1\over \pi}</math>

<math>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}</math>

<math>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}</math>

<math>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}</math>

<math>Y_{2}^{-2}(\theta,\varphi)={1\over 4}\sqrt{15\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta</math>

<math>Y_{2}^{-1}(\theta,\varphi)={1\over 2}\sqrt{15\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot\cos\theta</math>

<math>Y_{2}^{0}(\theta,\varphi)={1\over 4}\sqrt{5\over \pi}\cdot(3\cos^{2}\theta-1)</math>

<math>Y_{2}^{1}(\theta,\varphi)={-1\over 2}\sqrt{15\over 2\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot\cos\theta</math>

<math>Y_{2}^{2}(\theta,\varphi)={1\over 4}\sqrt{15\over 2\pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta</math>

<math>Y_{3}^{0}(\theta,\varphi)={1\over 4}\sqrt{7\over \pi}\cdot(5\cos^{3}\theta-3\cos\theta)</math>

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

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

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 Physicsworldde:Kugelflächenfunktionen