Solutions to Homework Set 9 in Physics 389K: Angular Momentum and Spin-1 Systems, Assignments of Quantum Mechanics

Download Solutions to Homework Set 9 in Physics 389K: Angular Momentum and Spin-1 Systems and more Assignments Quantum Mechanics in PDF only on Docsity! PHY 389K QM1, Homework Set 9 Solutions Matthias Ihl 11/23/2006 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) The space of physical states of the deuteron is Rj1=1/2 ⊗Rj2=1/2 = Rj=1 ⊕Rj=0. (1) (b) For this, one needs to calculate the Clebsch-Gordan coefficients (table 2.1, p. 173). This leads to: |j = 1, m = 1〉 = |1/2, 1/2; 1/2, 1/2〉, |j = 1, m = 0〉 = √ 1 2 (|1/2, 1/2; 1/2,−1/2〉+ |1/2,−1/2; 1/2, 1/2〉), |j = 0, m = 0〉 = √ 1 2 (|1/2, 1/2; 1/2,−1/2〉 − |1/2,−1/2; 1/2, 1/2〉), |j = 1, m = −1〉 = |1/2,−1/2; 1/2,−1/2〉. (c) Let us investigate the action of ( ~J2 − ~J2p − ~J2n ) on the deuteron states: ( ~J2 − ~J2p − ~J2n ) |j = 1, m = ±1, 0〉 = ~ 2 2 |j = 1, m = ±1, 0〉, ( ~J2 − ~J2p − ~J2n ) |j = 0, m = 0〉 = −3 2 ~ 2|j = 0, m = 0〉. 1 2 Problem 2 (a) Let us look at ~J2 first. ~J2 = ( ~J (a) + ~J (b))2 = ~J (a)2 + ~J (b)2 + 2J (a) i J (b) i = ~J (a)2 + ~J (b)2 + 2J (a) 3 J (b) 3 + J (a) − J (b) + + J (a) + J (b) − . There are two eigenvectors: ~J2|1/2, 1/2; 1/2, 1/2〉 = 2~2|1/2, 1/2; 1/2, 1/2〉, (2) and ~J2|1/2,−1/2; 1/2,−1/2〉 = 2~2|1/2,−1/2; 1/2,−1/2〉, (3) from which we conclude that j(j + 1) = 2 ⇒ j = 1. The ”mixed” states are not eigenvectors of ~J2. (b) Therefore we want to construct linear combinations α|1/2, 1/2; 1/2,−1/2〉+ β|1/2,−1/2; 1/2, 1/2〉, (4) such that they are (normalized) eigenvectors of ~J2. One finds α = ±β = 1√ 2 : • α = β : 1√ 2 (|1/2, 1/2; 1/2,−1/2〉+ |1/2,−1/2; 1/2, 1/2〉), (5) • α = −β : 1√ 2 (|1/2, 1/2; 1/2,−1/2〉 − |1/2,−1/2; 1/2, 1/2〉). (6) The eigenvalues are ~J2 ( 1√ 2 (|+;−〉 + |−; +〉) ) = 2~2 ( 1√ 2 (|+;−〉 + |−; +〉) ) ⇒ j = 1, (7) ~J2 ( 1√ 2 (|+;−〉 − |−; +〉) ) = 0 ⇒ j = 0, (8) where we have introduced some obvious notation. All of the above four eigenvectors are mutually orthogonal, as can be explic- itly checked by calculating the respective inner products.