2. (a) (c) Clebsch-Gordan series and Gaunt coefficients. Given Du) = DG) D2), prove the Clebsch-Gordan...

 

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2. (a) (c) Clebsch-Gordan series and Gaunt coefficients. Given Du) = DG) D2), prove the Clebsch-Gordan (CG) series DG) (R) Di2) mi,m D2m (R) = (m, m j, m) Dom (R) (m, m | j, m'). (5) j,m,m' Use the CG series to show that [ dSY* (0,6) Y (0, 6) Y (0, 6) = (211 + 1) (2/2+1) 4T (21 + 1) (0,01 1,0) (m, m | l, m), (7) where is the solid angle. The integral over three spherical harmonics is known as the Gaunt coefficient). Hint: Recall that D 4 ym. (0,0). m,0 (0,o) 21+1 2. (a) (c) Clebsch-Gordan series and Gaunt coefficients. Given Du) = DG) D2), prove the Clebsch-Gordan (CG) series DG) (R) Di2) mi,m D2m (R) = (m, m j, m) Dom (R) (m, m | j, m'). (5) j,m,m' Use the CG series to show that [ dSY* (0,6) Y (0, 6) Y (0, 6) = (211 + 1) (2/2+1) 4T (21 + 1) (0,01 1,0) (m, m | l, m), (7) where is the solid angle. The integral over three spherical harmonics is known as the Gaunt coefficient). Hint: Recall that D 4 ym. (0,0). m,0 (0,o) 21+1

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To prove the ClebschGordan CG series a Given D6 D1 D2 we want to prove the CG series expression D1D2 ... View the full answer