Spherical harmonics

From Exampleproblems

(Redirected from Spherical harmonic)
Jump to: navigation, search

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.

Contents

Introduction

Image:Rotating spherical harmonics.gif
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 Yl,m and Yl,-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:

{1 \over r^2}{\partial \over \partial r}\left(r^2 {\partial f \over \partial r}\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).

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^{-1-\ell} \cos (m \varphi) P_\ell^m (\cos{\theta} )

and

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

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_{\ell,m}(r,\theta , \varphi ) = r^{-1-\ell} 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{{(2\ell+1)\over 4\pi}{(\ell-m)!\over (\ell+m)!}} \cdot e^{i m \varphi } \cdot P_\ell^m ( \cos{\theta} )

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

\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
Y1 Image:Legendre Y1 xy.png Image:Legendre Y1 polaire.png
Y2 Image:Legendre Y2 xy.png Image:Legendre Y2 polaire.png
Y3 Image:Legendre Y3 xy.png Image:Legendre Y3 polaire.png

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

Y_\ell^0\quad\quad for\ 0\le\ell\le\infin

and

{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

and

{1\over i\sqrt2}\left((-1)^mY_\ell^m-Y_\ell^{-m}\right)\quad\quad \mbox{ for } \ 0\le\ell\le\infin,\ 1\le m\le \ell

The spherical harmonics in cartesian coordinates may be obtained by substituting

\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}.

First few spherical harmonics

Image:Harmoniques spheriques positif negatif.png
Representation as ρ = ρ0 + ρ1·Ylm(θ,φ)
then the representative surface looks like a "battered" sphere;
Ylm is equal to 0 along circles (the representative surface intersects the ρ = ρ0 sphere at these circles). Ylm is alternately positive and negative between two circles.
Image:Traces harmonique spherique.png
the Y32 with four sections

These are the first few spherical harmonics:

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


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}
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}
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}


Y_{2}^{-2}(\theta,\varphi)={1\over 4}\sqrt{15\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta
Y_{2}^{-1}(\theta,\varphi)={1\over 2}\sqrt{15\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot\cos\theta
Y_{2}^{0}(\theta,\varphi)={1\over 4}\sqrt{5\over \pi}\cdot(3\cos^{2}\theta-1)
Y_{2}^{1}(\theta,\varphi)={-1\over 2}\sqrt{15\over 2\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot\cos\theta
Y_{2}^{2}(\theta,\varphi)={1\over 4}\sqrt{15\over 2\pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta


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

References

Retrieved from "http://www.exampleproblems.com/wiki/index.php/Spherical_harmonics"
Argan Oil
Natural Skin Care
Organic Skin Care
visitor stats