rpp2010-rev-clebsch-gordan-coefs

# rpp2010-rev-clebsch-gordan-coefs - 36. Clebsch-Gordan...

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36. Clebsch-Gordan coeﬃcients 1 36. CLEBSCH-GORDAN COEFFICIENTS, SPHERICAL HARMONICS, AND d FUNCTIONS Note: A square-root sign is to be understood over every coeﬃcient, e.g. ,for 8 / 15 read p 8 / 15. Y 0 1 = r 3 4 π cos θ Y 1 1 = r 3 8 π sin θe Y 0 2 = r 5 4 π ³ 3 2 cos 2 θ 1 2 ´ Y 1 2 = r 15 8 π sin θ cos Y 2 2 = 1 4 r 15 2 π sin 2 2 Y m ± =( 1) m Y m ± ± j 1 j 2 m 1 m 2 | j 1 j 2 JM ² 1) J j 1 j 2 ± j 2 j 1 m 2 m 1 | j 2 j 1 ² d ± m, 0 = r 4 π 2 ± +1 Y m ± e imφ d j m ± ,m 1) m m ± d j m,m ± = d j m, m ± d 1 0 , 0 =cos θd 1 / 2 1 / 2 , 1 / 2 θ 2 d 1 / 2 1 / 2 , 1 / 2 = sin θ 2 d 1 1 , 1 = 1+cos θ 2 d 1 1 , 0 = sin θ 2 d 1 1 , 1 = 1 cos θ 2 d 3 / 2 3 / 2 , 3 / 2 = θ 2 cos θ 2 d 3 / 2 3 / 2 , 1 / 2 = 3 θ 2 sin θ 2 d 3 / 2 3 / 2 , 1 / 2 = 3 1 cos θ 2 cos θ 2 d 3 / 2 3 / 2 , 3 / 2 = 1 cos θ 2 sin θ 2 d 3 / 2 1 / 2 , 1 / 2 = 3cos θ 1 2 cos θ 2 d 3 / 2 1 / 2 , 1 / 2 = θ 2 sin θ 2 d 2 2 , 2 = ³ θ 2 ´ 2 d 2 2 , 1 = θ 2 sin θ d 2 2 , 0 = 6 4 sin 2 θ d 2 2 , 1 = 1 cos θ 2 sin θ d 2 2 , 2 = ³ 1 cos θ 2 ´ 2 d 2 1 , 1 = θ 2 (2 cos θ 1) d 2 1 , 0 = r 3 2 sin θ cos θ d 2 1 , 1 = 1 cos θ 2 (2 cos θ +1) d 2 0 , 0 = ³ 3 2 cos 2 θ 1 2 ´ + 1 5/2 + 3/2 + 1/5 4/5 1/2 3/5 2/5 1 2 5/2 3/2 21 + + 1 2/5 3/5 2 + 2 + + + 1/3
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## This note was uploaded on 06/07/2011 for the course PHYS 4132 taught by Professor Kutter during the Spring '11 term at University of Florida.

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