Physics 195bProblem set number 12Due 2 PM, Thursday, January 23, 2003Notes about course:• Homework should be turned in to the TA’s mail slot on the first floorof East Bridge.• Collaboration policy: OK to work together in small groups, and to helpwith each other’s understanding. Best to first give problems a good tryby yourself. Don’t just copy someone else’s work – whatever you turnin should be what you think you understand.• There is a web page for this course, which should be referred to for themost up-to-date information. The URL:http://www.hep.caltech.edu/˜fcp/ph195/• TA: Anura Abeyesinghe, [email protected]• If you think a problem is completely trivial (and hence a waste of yourtime), you don’t have to do it. Just write “trivial” where your solutionwould go, and you will get credit for it. Of course, this means you arevolunteering to help the rest of the class understand it, if they don’tfind it so simple. . .READING: 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:38F. 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 “spheical helicity basis”, with vectors of the form:|j, m, λ1,λ2 , (131)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. (132)The other basis is the “plane-wave helicity basis”, with vectors of theform:|θ, φ, λ1,λ2 , (133)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, (134)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), (135)where δ ≡ λ1− λ2. Determine the numbers bj.3959. 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 youare intended to assume that the frame is the rest frame of the