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# hw5 - oF the total spin v S 3 Consider addition oF two...

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HOMEWORK # 5 Physics 6572 Friday, 10/10/08; due 10/17/08 1. a) Let R 1 and R 2 be two infinitesimal rotations parametrized with δω 1 and δω 2 , and K a three-vector. If Δ K is the change induced in K by R = R - 1 2 R - 1 1 R 2 R 1 , show that to leading order Δ K = - ( δω 1 × δω 2 ) × K . (1) b) The corresponding unitary operators satisfy the relation U ( R - 1 2 ) U ( R - 2 1 ) U ( R 2 ) U ( R 1 ) = e ( R 1 ,R 2 ) U ( R - 1 2 R - 1 1 R 2 R 1 ), (2) where a possible phase factor is included, and for infinitesimal rotations U ( R i ) = e - i δω i · J 1 - i δω i · J . (3) Show that the above relation among the U ’s leads to the commutation relations for the angular momentum operators, and λ must be zero on the basis of symmetry con- siderations. (Hint: λ has to be symmetric in δω 1 δω 2 , but the commutator is antisymmetric.) 2. A system consisting of two spins is described by the Hamiltonian H = avectorσ 1 · vectorσ 2 + b ( σ 1 z - σ 2 z ), (4) where a and b > 0 are constants. a) Is the total spin vector S = 1 2 ( vectorσ 1 + vectorσ 2 ) conserved? Which components of vector S , if any, are conserved? b) Find the eigenvalues of H and the corresponding eigenstates in terms of the eigenstates
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Unformatted text preview: oF the total spin v S . 3. Consider addition oF two angular momenta v J = v J 1 + v J 2 with j 1 = j 2 = 1. ±ind the eigenstates oF the total angular momentum | jm a in terms oF the product states | j 1 m 1 j 2 m 2 a in two ways: a) Make use oF the tables oF the Clebsch-Gordan coe²cients in Appendix A oF GottFried & Yan. b) The state with m 1 = m 2 = 1 must be a state with j = m = 2 (why?). Apply J-repeatedly to this state to obtain all other states oF j = 2. ±orm an appropriate linear combination oF the two states with m 1 + m 2 = 1 to obtain the state with j = 1 and m = 1. ±ind the other j = 1 states by applying J-repeatedly. ±inally, fnd the j = m = 0 state by Forming an appropriate linear combination oF the three states with m 1 + m 2 = 0. c) Compare the results in (a) and (b)....
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