Quantum Field Theory - Homework Set 7 Solutions | PHY 389K, Assignments of Quantum Mechanics

Download Quantum Field Theory - Homework Set 7 Solutions | PHY 389K and more Assignments Quantum Mechanics in PDF only on Docsity! PHY 389K QM1, Homework Set 7 Solutions Matthias Ihl 10/30/2006 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 (a) For Ej = Ej0 = ~2 2I j0(j0 + 1), we have W = (TrΛj0)−1Λj0 = 1 2j0 + 1 Λj0, (1) so that the Born probability is given by PW (|j0, m0〉〈j0, m0|) = Tr(|j0, m0〉〈j0, m0| 1 2j0 + 1 Λj0) = 1 2j0 + 1 . (2) (b) If j 6= j0, then obviously, W j = 1 2j + 1 Λj 6=j0 ⇒ PW j(|j0, m0〉〈j0, m0|) = 0, (3) since states with different j are orthogonal. 2 Problem 2 Using the results given in the supplementary notes 10/17/2006 and the text- book (chapter V on the Wigner-Eckart theorem), this is a straightforward exercise. 1 (a) For Vκ = Jκ, the spherical components of the angular momentum operator, we obtain 〈r′, l′0; l′, m′|J0|r, l0; l, m〉 = 〈r′, l′0; l′, m′|J3|r, l0; l, m〉 = ~mδ(r − r′)δl′0,l0δl,l′δm′,m, 〈r′, l′0; l′, m′|J±1|r, l0; l, m〉 = 〈r′, l′0; l′, m′| ∓ (J1 ± iJ2)|r, l0; l, m〉 = ∓ ~√ 2 √ l(l + 1) − m(m ± 1)δ(r − r′)δl′ 0 ,l0δl,l′δm′,m±1. (b) For Vκ = Qκ, the spherical components of the position operator, we get 〈r′, l′0; l′, m′|Q0|r, l0; l, m〉 = (√ l2 − m2clδl′,l−1 − malδl′, l − √ (l + 1)2 − m2cl+1δl′,l+1 ) × δ(r − r′)δl′ 0 ,l0δm′,m, 〈r′, l′0; l′, m′|Q±1|r, l0; l, m〉 = 〈r′, l′0; l′, m′| ∓ (Q1 ± iQ2)|r, l0; l, m〉 = ∓ 1√ 2 ( ± √ (l ∓ 1)(l ∓ m − 1)clδl′,l−1δm′,m±1 − √ (l ∓ m)(l ± m + 1)alδl′,lδm′,m±1 ± √ (l ± m + 1)(l ± m + 2)cl+1δl′,l+1δm′,m±1 ) δ(r − r′)δl′ 0 ,l0, where cl := r i l √ l2 − l20 4l2 − 1 , al := l0r l(l + 1) , l0 := 0,± 1 2 ,±1,±3 2 , r ∈ [0,∞[. 3 Problem 3 The original Hamiltonian of the “perturbed” rotator is given by H = ~J2 2I + αJ3 + βJ1. (4)