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The SphericalHarmonics - Welcome to SCIPP

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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 , (.)

Physics 116C Fall 2012 The SphericalHarmonics 1. Solution toLaplace’s equation in spherical coordinates In spherical coordinates, the Laplacian is given by

  The sphericalharmonics

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