lec14-2 - Spherical harmonics • In solving the Laplace...

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Unformatted text preview: Spherical harmonics • In solving the Laplace equation in spherical coordinates, we encountered solutions of the form u = r l P m l (cos θ )[ a lm cos m φ + b lm sin m φ ] u = r- l- 1 P m l (cos θ )[ a lm cos m φ + b lm sin m φ ] • Recall that we had Z 1- 1 P m l ( x ) P m l ( x ) dx = δ l , l 2 2 l + 1 ( l + m )! ( l- m )! • We also see that if we use e im φ instead of the sin m φ and cos m φ above, Z 2 π e i ( m- m ) φ d φ = 2 πδ m , m Spherical harmonics,continued • We often instead use spherical harmonics Y m l ( θ,φ ), defined by Y m l ( θ,φ ) = s (2 l + 1) 4 π ( l- m )! ( l + m )! P m l (cos θ ) e im φ • We can show these are orthonormal when integrated over all solid angles Z 2 π Z π Y m * l ( θ,φ ) Y m l ( θ,φ )sin θ d θ d φ = δ l , l δ m , m Use of spherical harmonics • Having derived the spherical harmonics to have these useful properties, we see then that we can take as the solutions for Laplace’s equation in spherical coordinates, u = r l Y m l ( θ,φ ) u = r- l- 1 Y m l ( θ,φ ) • For example, if we are interested in solutions away from the origin to infinity, we have u ( r ,θ,φ ) = ∞ X l =0 m = l X m =- l a lm r- l- 1 Y m l ( θ,φ ) Use of spherical harmonics, continued...
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This note was uploaded on 07/30/2011 for the course PHZ 3113 taught by Professor Staff during the Spring '03 term at University of Central Florida.

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lec14-2 - Spherical harmonics • In solving the Laplace...

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