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+ δα M ˆ u . (a) Usingthefactthatinanin f nitesimal rotation A ˆ u ( δα ) the vector ~v is rotated into the vector ~v + δα (ˆ u × ~v ) , construct the matrix representing M ˆ u in any standard coordinate system, and show that it can be written in the form M ˆ u = u x M x + u y M y + u z M z in terms of matrices M x ,M y ,M z . [Hint: this part is done in the notes.] (b) De f ne the operators/matrices J x = iM x ,J y = iM y , and J z = iM z , (these are operators on the space of ordinary vectors in R 3 ). Show that these three matrices obey angular momentum commutation rules.[It su ﬃ ces to compute [ J x ,J y ] , and to obtain the others by cyclic permutations. (c) Determine the eigenvalue spectrum of any one of the matrices J x ,J y ,J z . 13. Let
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