Condon and Shortley has become the standard comprehensive work on the theory of atomic spectra. The first two chapters contain a brief historical introduction. The $(-1)^m$ phase factor in some definitions of the Spherical Harmonics and associated Legendre Polynomials. Using the Condon-Shortley convention gives. An algorithm is presented for the generation and storage of all unique, non-zero Condon-Shortley coefficients. The formulas for retrieving these coefficients in a.
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That is, they are either even or odd with respect to inversion about the origin. For the other cases, the functions checker the sphere, and they are referred to as tesseral. The group Shkrtley 2, C is isomorphic to the proper Lorentz groupand its action on the two-sphere agrees with the action of the Lorentz group on the celestial sphere in Minkowski space.
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The 19th century development of Fourier series made possible the solution of a wide variety of physical problems in rectangular domains, such as the solution confon the heat equation and wave equation.
When the spherical harmonic order m is zero upper-left in the figurethe spherical harmonic functions do not depend upon longitude, and are referred to as zonal. Home Questions Tags Shogtley Unanswered.
Wikimedia Commons has media related to Spherical harmonics. For other uses, see YLM disambiguation. Laplace’s equation imposes that the divergence of the gradient of a scalar field f is zero. Spherical harmonics were first investigated in connection with the Newtonian potential of Newton’s law of universal gravitation in three dimensions. Condonn convergence of the series holds again in the same sense, but the benefit of the real expansion is that for real functions f the expansion coefficients become real.
This expression is valid for both real and complex harmonics. In acoustics the Laplace spherical harmonics are generally defined as this is the convention used in this article. Shorley continues to hold for real spherical harmonics, and for spherical harmonics in higher dimensions: The Condon-Shortley sign convention is built so that the spherical harmonics will play nicely with the angular momentum ladder operators: The Clebsch—Gordan coefficients are the coefficients appearing in the expansion of the product of two spherical harmonics in terms of spherical harmonics themselves.
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An orthogonal basis of spherical harmonics in higher shortpey can be constructed inductively by the method of separation of variablesby solving the Sturm-Liouville problem for the spherical Laplacian.
Page 1 of 1 Start over Page 1 of 1. The spaces of spherical harmonics on the 3-sphere are certain spin representations of SO 3with respect to the action by quaternionic multiplication. Retrieved from ” https: It is s classic. The geodesy  and magnetics communities never include the Condon—Shortley phase factor condoon their definitions of the spherical harmonic functions nor in the ones of the associated Legendre polynomials.
This was a boon for problems possessing spherical symmetrysuch as those of celestial mechanics originally studied by Laplace and Legendre. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. If Y is a joint eigenfunction of L 2 and L zthen by definition. Laplace’s spherical harmonics are the joint eigenfunctions of the square of the orbital angular momentum and the generator of rotations about the azimuthal axis:.
Further, spherical harmonics are basis functions for SO 3the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO 3.
Condon-Shortley Phase — from Wolfram MathWorld
InWilliam Thomson Lord Kelvin and Peter Guthrie Tait introduced the solid spherical harmonics in their Treatise shprtley Natural Philosophyand also first introduced the name of “spherical harmonics” for these functions. Contact the MathWorld Team. The solid harmonics were homogeneous polynomial solutions of Laplace’s equation.
By examining Laplace’s equation in spherical coordinates, Thomson and Tait recovered Laplace’s spherical harmonics. The spherical harmonics are eigenfunctions of the square of the orbital angular momentum operator.
They are often employed in solving partial differential equations that commonly occur in science. Analytic expressions for the first few orthonormalized Laplace spherical harmonics that use the Condon-Shortley phase convention:. The following pages on the English Wikipedia link to this file pages on other projects are not listed:. Principles of Atomic Spectra.
The classical spherical harmonics are defined as functions on the unit sphere S 2 inside three-dimensional Euclidean space. Sign up using Facebook. In a similar manner, one can define the cross-power of two functions as. The term “Laplace’s coefficients” was employed by William Whewell shortlry describe the particular system of solutions introduced along these lines, whereas others reserved this designation for the zonal spherical harmonics that had shorhley been introduced by Laplace and Legendre.
In quantum mechanics this normalization is sometimes used as well, and is named Racah’s normalization after Giulio Racah. This page was last edited on 5 Mayat