# solution4 - 12 Consider the 3 3 rotation operators Au which...

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12. Consider the 3 × 3 rotation operators A ˆ u ( α ) which rotate vectors in R 3 . For in f nitesimal rotations these take the form A ˆ u ( δα )=1+ M ˆ u . (a) A vector ~v i sro ta tedinanin f nitesimal rotation A ˆ u ( )in tothev ec to r + u × ) , Thus, the transformation law can be written 0 = + M ˆ u = + u × ) . Equating the i th component of this vector equation we deduce that X k M ik v k = X j,k ε ijk u j v k and hence that M ik = X j ε ijk u j Thus, M ii =0 , and M 12 = ε 132 u 3 = u z = M 21 M 13 = ε 123 u 2 =+ u y = M 31 M 23 = ε 213 u 1 = u x = M 32 , which gives, explicitly M = 0 u z u y u z 0 u x u y u x 0 = 3 X i =1 u i M i = u x M x + u y M y + u z M z where M x = 00 0 00 1 01 0 M y = 001 000 100 M z = 0 10 . (b) De f ne the operators/matrices J k = iM k ( k =1 , 2 , 3) , and show that as operators on R 3 they obey angular momentum commutation rules. Introducing the matrices J i = iM i J x = i 0 i 0 J y = i i J z = 0 i 0 i . we f nd that [ J x ,J y ]isequa lto i 0 i 0 i i i i i 0 i 0 = 010 the right hand side of which is M z = i ( iM z )= iJ z . In a similar fashion we

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solution4 - 12 Consider the 3 3 rotation operators Au which...

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