Physics 195bProblem set number 12 – Solution to Problem 58Due 2 PM, Thursday, January 23, 2003READING: Finish reading the “Angular Momentum” course note.PROBLEMS:56. Application of SU(2) to nuclear physics: Isospin. Do Exercise 12 ofthe Angular Momentum course note. This is not a problem on angularmomentum, but it demonstates that the group theory we developed forangular momentum may be applied in a formally equivalent context.The problem statement claims that there is an attached picture. Thisis clearly false. You may find an appropriate level scheme via a googlesearch (you want a level diagram for the nuclear isobars of 6 nucleons),e.g.,at:http://www.tunl.duke.edu/nucldata/figures/06figs/06is.pdfFor additional reference, you might find it of interest to look up:F. Ajzenberg-Selove, “Energy Levels of Light Nuclei, A = 5-10,” Nucl.Phys. A490 1-225 (1988)(see also http://www.tunl.duke.edu/nucldata/fas/88AJ01.shtml).57. Symmetry and broken symmetry: Application of group theory to levelsplitting in a lattice with reduced symmetry. Do exercise 14 of theAngular Momentum course note. This is an important problem – itillustrates the power of group theoretic methods in addressing certainquestions. I hope you will find it fun to do.58. In class, we have discussed the transformation between two differenttypes of “helicity bases”. In particular, we have considered a system oftwo particles, with spins j1and j2,intheirCMframe.One basis is the “spherical helicity basis”, with vectors of the form:|j, m, λ1,λ2, (116)where j is the total angular momentum, m is the total angular mo-mentum projection along the 3-axis, and λ1,λ2are the helicities of thetwo particles. We assumed a normalization of these basis vectors suchthat: j,m,λ1,λ2|j, m, λ1,λ2 = δjjδmmδλ1λ1δλ2λ2. (117)34The other basis is the “plane-wave helicity basis”, with vectors of theform:|θ, φ, λ1,λ2, (118)where θ and φ are the spherical polar angles of the direction of particleone. We did not specify a normalization for these basis vectors, but anobvious (and conventional) choice is: θ,φ,λ1,λ2|θ, φ, λ1,λ2 = δ(2)(Ω− Ω)δλ1λ1δλ2λ2, (119)where d(2)Ω refers to the element of solid angle for particle one.In class, we have obtained the result for the transformation betweenthese bases in the form:|θ, φ, λ1,λ2 =j,mbj|j, m, λ1,λ2Djmδ(φ, θ, −φ), (120)where δ ≡ λ1− λ2. Determine the numbers bj.Solution: To select a particular bj, i.e., a particular j,letusinvertthe basis transformation:4πdΩDj∗mδ(φ, θ, −φ)|θ, φ, λ1,λ2 = (121)j,mbj|j,m,λ1,λ24πdΩDj∗mδ(φ, θ, −φ)Djmδ(φ, θ, −φ)=j,mbj|j,m,λ1,λ24πdΩdjmδ(θ)djmδ(θ)exp[−i(mφ − δφ)+i(mφ − δφ)]=j,mbj|j,m,λ1,λ21−1d cos θdjmδ(θ)djmδ(θ)2π0dφei(m−m)φ(122)=2πjbj|j,m,λ1,λ21−1d cos θdjmδ(θ)djmδ(θ) (123)=2πjbj|j,m,λ1,λ22δjj2j +1(124)=4π2j +1bj|j, m, λ1,λ2. (125)Note that we should probably justify the interchange of the order ofsummation and integration in the very first step above. Thus,|j, m, λ1,λ2 =2j +14πbj4πdΩDj∗mδ(φ, θ, −φ)|θ, φ, λ1,λ2. (126)35Now,1= j, m, λ1,λ2|j, m, λ1,λ2 (127)= 2j +14π|bj|24πdΩDj∗mδ(φ, θ, −φ)4πdΩDjmδ(φ,θ, −φ) θ,φ,λ1,λ2|θ, φ, λ1,λ2= 2j +14π|bj|24πdΩDj∗mδ(φ, θ, −φ)4πdΩDjmδ(φ,θ, −φ)δ(cos θ− cos θ)δ(φ− φ)= 2j +14π|bj|24πdΩDj∗mδ(φ, θ, −φ)Djmδ(φ, θ, −φ) (128)=2π 2j +14π|bj|21−1d cos θdjmδ(θ)2(129)=4π2j +1 2j +14π|bj|2. (130)Therefore, |bj|2=(2j +1)/4π, or picking a phase convention,bj=4π2j +1. (131)where we assume that it is all right to interchange the summation andintegration. Since each term is non-negative (and each finite), there isno potential for cancellations. Hence, if we find convergence for oneordering of the operations, we will for the other as well.Note that we have used the result of Eqn. 348 of the notes to obtain:1−1d cos θdjmδ(θ)2=22j +1. (132)59. Clebsch-Gordan coefficients, an alternate practical approach: Exercise16 of the Angular Momentum course note.60. Application to angular distribution: Exercise 18 of the Angular Mo-mentum course note. While you may apply the formula we derived inclass, I urge you to do this problem by thinking about it “from thebeginning” – what should be the angular dependence of the spatialwave function? That is, I hope you will try using some “physical intu-ition” first, and use the formula as a check if you wish. Note that you36are intended to assume that the frame is the rest frame of the