hw203a - Physics 512 Homework Set #3 Solutions Winter 2003...

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Physics 512 Winter 2003 Homework Set #3 – Solutions 1. Let S ( k 1 ) q 1 and T ( k 2 ) q 2 be two irreducible spherical tensor operators of ranks k 1 and k 2 , respectively. We may form the tensor product W ( k ) q deFned by W ( k ) q = X q 1 + q 2 = q h k 1 k 2 ; q 1 q 2 | k 1 k 2 ; kq i S ( k 1 ) q 1 T ( k 2 ) q 2 Using the deFnition of spherical tensors in terms of their commutation properties [ J z ,T ( k ) q ]= q ¯ hT ( k ) q [ J ± ( k ) q p k ( k +1) q ( q ± 1) ¯ ( k ) q ± 1 prove that W ( k ) q is a spherical tensor operator of rank k . To prove that W ( k ) q is a spherical tensor, we show that it satisfes the commutation properties above. We start with the J z commutator (which is the simpler one). [ J z ,W ( k ) q X q 1 + q 2 = q h k 1 k 2 ; q 1 q 2 | k 1 k 2 ; i [ J z ,S ( k 1 ) q 1 T ( k 2 ) q 2 ] = X q 1 + q 2 = q h k 1 k 2 ; q 1 q 2 | k 1 k 2 ; i ± [ J z ( k 1 ) q 1 ] T ( k 2 ) q 2 + S ( k 1 ) q 1 [ J z ( k 2 ) q 2 ] ² = X q 1 + q 2 = q h k 1 k 2 ; q 1 q 2 | k 1 k 2 ; i ± q 1 ¯ hS ( k 1 ) q 1 T ( k 2 ) q 2 + q 2 ¯ hS ( k 1 ) q 1 T ( k 2 ) q 2 ² = q ¯ h X q 1 + q 2 = q h k 1 k 2 ; q 1 q 2 | k 1 k 2 ; i S ( k 1 ) q 1 T ( k 2 ) q 2 = q ¯ hW ( k ) q In the penultimate line, we have used the Fact that both S ( k 1 ) q 1 and T ( k 2 ) q 2 are spherical tensors in order to compute the commutators with J z . And in the fnal line, we note that since q 1 + q 2 = q we can pull out an overall Factor oF q ¯ h . ±or the other components, we proceed in a similar manner [ J ± ( k ) q X q 1 + q 2 = q h k 1 k 2 ; q 1 q 2 | k 1 k 2 ; i ± [ J ± ( k 1 ) q 1 ] T ( k 2 ) q 2 + S ( k 1 ) q 1 [ J ± ( k 2 ) q 2 ] ² = X q 1 + q 2 = q h k 1 k 2 ; q 1 q 2 | k 1 k 2 ; i ± p k 1 ( k 1 q 1 ( q 1 ± 1) ¯ hS ( k 1 ) q 1 ± 1 T ( k 2 ) q 2 + p k 2 ( k 2 q 2 ( q 2 ± 1) ¯ hS ( k 1 ) q 1 T ( k 2 ) q 2 ± 1 ² Since we wish to relate the right hand side as much as possible to W ( k ) q , we shiFt q 1 q 1 1 in the frst term, and q 2 q 2 1 in the second term. The two terms then 1
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combine, giving the result [ J ± ,W ( k ) q ]= X q 1 + q 2 = q ± 1 ± p k 1 ( k 1 +1) q 1 ( q 1 1) ¯ h h k 1 k 2 ; q 1 1 q 2 | k 1 k 2 ; kq i + p k 2 ( k 2 q 2 ( q 2 1) ¯ h h k 1 k 2 ; q 1 q 2 1 | k 1 k 2 ; i ² S ( k 1 ) q 1 T ( k 2 ) q 2
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This note was uploaded on 02/15/2011 for the course PHYS 512 taught by Professor Unknow during the Winter '03 term at Cornell College.

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hw203a - Physics 512 Homework Set #3 Solutions Winter 2003...

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