PHY 389K: HW Set 10 Solutions - Total j, Clebsch-Gordan Coefficients, and Reduced Matrix, Assignments of Quantum Mechanics

Download PHY 389K: HW Set 10 Solutions - Total j, Clebsch-Gordan Coefficients, and Reduced Matrix and more Assignments Quantum Mechanics in PDF only on Docsity! PHY 389K QM1, Homework Set 10 Solutions Matthias Ihl 11/25/2006 Note: I will post updated versions of the homework solutions on my home- page: http://zippy.ph.utexas.edu/~msihl/PHY389K/ We will frequently work in God-given units c = ~ = 1. The casual reader may also want to set 1 = 2 = π = −1. 1 Problem 1 There are two possibilities for the total j, namely j = 5 2 and j = 3 2 , with dimensionality 6 and 4 respectively. There are 18 different CG coefficients altogether (cf. table 2.1). 2 Problem 2 Using table 2.2, one finds 55 different Clebsch-Gordan coefficients for 9 states with total j = 4, 7 states with j = 3 and 5 states with j = 2. 3 Problem 3 There are three possibilities for the reduced matrix element 〈j′||V J=1||j〉, (1) since, in general, j′ = j + 1, j, j − 1. On the other hand, one gets a lot more matrix non-vanishing elements for 〈j′, m′|V J=1κ |j, m〉, (2) 1