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# Group Theory PHY 745

WFU

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This 2 page Class Notes was uploaded by Miss Lucienne Hamill on Wednesday October 28, 2015. The Class Notes belongs to PHY 745 at Wake Forest University taught by Staff in Fall. Since its upload, it has received 6 views. For similar materials see /class/230734/phy-745-wake-forest-university in Physics 2 at Wake Forest University.

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Date Created: 10/28/15

PHY745 7 February 3 2009NAWH 1 Notes on transformations of Spherical Harmonic functions These notes use the convention of M E Rose Elementary Theory of Angular Momentum John Wiley amp Sons Inc 1957 which seems to be consistent with your Tinkham text Consider a transformation of the spherical harmonic functions mm 7 Sambamm lt1 Here the transformation RI39 might be a rotation through the 3 Euler angles a about the 2 axis 6 about the new y axis and 7 about the new 2 axis so that Rr M1 ltvgtMy 6Mzar 2 cos y sin y 0 M1 ry 7 sin y cos y 0 3 0 0 1 cos B 0 7 sin 6 MM 7 0 1 0 7 lt4 sin 0 cos B and cos a sin a 0 M2 04 7 sin 04 cos a 0 5 0 0 1 By multiplying these three matrices we nd the 9 components of the rotation matrix to be CT Rm cos acos cosy 7 sinasin y I Rwy sina cos cosy cosa sin y 00 H O VVVVVVVVV Rm 7 sin cosy AAAA Rm 7cosacos sin y 7 sinacosy Ryy 7s1nacos s1n y cosacosy Ryz sin sin y Rm cos a sin Rzy s1n a Sln Ru cos AAAAA H D It can be shown that the spherical harmonic transformation representation takes the form Dinm0 E DinMa a e mmldln mcos 6 m7 15 PHY745 7 February 3 2009NAWH 2 For m 2 m7 I minimum1 1 6 WWW A B quotHquot dinmCOS 7 lt00S S1115 25 gtlt 2F1m7l7m7lm7m1g7tan 5 The hypergeometric function is de ned to be ab 1aa 1bb 1 2 7 2 F 5 El 7 17 2 107 lclz l czl2 cc1 This equation can generate all the rotation matrices needed by use of some of the following identities dinmcos B dam7 cos B 18 PM R 1lDin m77 19 where R E inversion gtltR We can determine the Euler angles a B and y for a given rotation matrix R from the form of the nine components of the rotation matrix given above Therefore7 given the rotation matrix R7 we can determine the Euler angles using cos RH 20 sin 17 R31 21 If sin 31 07 then Rm 7 Z39Rzy 7 22 6 sin and R R in zz Z yz 6 7 sin 39 lf sin 07 then we can choose 7 07 and w 24 6 Ru When there is inversion symmetry7 we can nd 04 B and y for R and then use Eq 19 to determine the complete transformation of R

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