The SphericalHarmonics
Physics 116C Fall 2012. The spherical harmonics 1. Solution to Laplace's equation in spherical coordinates In spherical coordinates, the Laplacian is given by 2. . ~ 2 1 2 1 1. = 2 r + 2 2 sin + 2 2 . (1). r r r r sin r sin 2. We shall solve Laplace's equation, ~ 2 T (r, , ) = 0 , (2). using the method of separation of variables, by writing T (r, , ) = R(r) ( ) ( ) . Inserting this decomposition into the Laplace equation and multiplying through by r 2 /R . yields 1 1 d2 . . 1 d 2 dR 1 1 d d . r + sin + = 0. R dr dr sin d d sin2 d 2. Hence, 1 d2 sin2 d . 2 dR sin d d . 2. = r sin = m2 , (3). d R dr dr d d . where m2 is the separation constant, which is chosen negative so that the solutions for ( ) are periodic in , (. eim . ( ) = for m = 0, 1, 2, 3.)
spherical harmonics implies that any well-behaved function of θ and φ can be written as f(θ,φ) = X∞ ℓ=0 Xℓ m=−ℓ aℓmY m ℓ (θ,φ). (12) for some choice of coefficients aℓm. For convenience, we list the spherical harmonics for ℓ = 0,1,2 and non-negative values of m. ℓ …
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