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clebrpp - 32 Clebsch-Gordan coecients 1 32 CLEBSCH-GORDAN...

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32. Clebsch-Gordan coefficients 1 32. 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 θ e Y 0 2 = r 5 4 π 3 2 cos 2 θ - 1 2 Y 1 2 = - r 15 8 π sin θ cos θ e Y 2 2 = 1 4 r 15 2 π sin 2 θ e 2 Y - m = ( - 1) m Y m * h j 1 j 2 m 1 m 2 | j 1 j 2 JM i = ( - 1) J - j 1 - j 2 h j 2 j 1 m 2 m 1 | j 2 j 1 JM i d m, 0 = r 4 π 2 + 1 Y m e - imφ d j m 0 ,m = ( - 1) m - m 0 d j m,m 0 = d j - m, - m 0 d 1 0 , 0 = cos θ d 1 / 2 1 / 2 , 1 / 2 = cos 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 = 1 + cos θ 2 cos θ 2 d 3 / 2 3 / 2 , 1 / 2 = - 3 1 + cos 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 = 3 cos θ - 1 2 cos θ 2 d 3 / 2 1 / 2 , - 1 / 2 = - 3 cos θ + 1 2 sin θ 2 d 2 2 , 2 = 1 + cos θ 2 2 d 2 2 , 1 = - 1 + cos 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 = 1 - cos θ 2 2 d 2 1 , 1 = 1 + cos 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 2 cos 2 θ - 1 2 + 1 5/2 5/2 3/2 3/2 +
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