homework4 - 12. Consider the 3 3 rotation operators Au ()...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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 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 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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/19/2010 for the course PHYSICS ph 463 taught by Professor Paule. during the Fall '10 term at Missouri S&T.

Ask a homework question - tutors are online