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Physics 195bProblem set number 16Due 2 PM, Thursday, February 20, 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 “Scattering” course note.PROBLEMS:75. Potential well/bump and spherical solutions: Do Exercise 2 of the Scat-tering course note.76. The “fundamental” and “effective” cross sections: Do Exercise 4 of theScattering course note.77. Parity conservation and scattering amplitudes: Do Exercise 5 of theScattering course note.78. Resonant scattering of light on an atom: Do Exercise 6 of the Scatteringcourse note.4379. When we talked about the hyperfine splitting in atoms, we mentionedthat the magnetic dipole moment of the proton is:µµµp= gpe2mpsp, (139)with a measured magnitude corresponding to a value for the gyromag-netic ratio of gp=2× (2.792847337 ± 0.000000029). Recall also thatI mentioned that the prediction of the Dirac equation for a point spin-1/2 particle is g = 2. We may understand the fact that the protongyromagnetic ratio is not two as being due to its compositeness: Inthe simple quark model, the proton is made of three quarks, two “ups”(u), and a “down” (d). The quarks are supposed to be point spin-1/2particles, hence, their gyromagnetic ratios should be gu= gd=2(upto higher order corrections, as in the case of the electron). Let us seewhether we can make sense out of the proton magnetic moment.The proton magnetic moment should be the sum of the magnetic mo-ments of its constituents, and any moments due to their orbital motionin the proton. The proton is the ground state baryon, so we assumethat the three quarks are bound together (by the strong interaction)in a state with no orbital angular momentum. By Fermi statistics, thetwo identical up quarks must have an overall odd wave function underinterchange of all quantum numbers. We must apply this with a bit ofcare, since we are including “color” as one of these quantum numbershere.Let us look a little at the property of “color”. It is the strong interactionanalog of electric charge in the electromagnetic interaction. However,instead of one fundamental dimension in charge, there are three colordirections, labelled as “red” (r), “blue” (b), and “green” (g). Unitarytransformations in this color space, up to overall phases, are describedby elements of the group SU(3), the group of unitary unimodular 3 ×3 matrices. Just like combining spins, we may combine three colorsaccording to a Clebsch-Gordan series, with the result:3 × 3 × 3=10+8+8+1. (140)We haven’t studied this group, so this decomposition into irreduciblerepresentations of the product representation is probably new to you.44However, the essential aspect here is that there is a singlet in the decom-position. That is, it is possible to combine three colors in such a way asto get a color-singlet state, i.e., a state with no net color charge. Theseare the states of physical interest for our observed baryons, according toa postulate of the quark model. After some thought (perhaps involvingraising and lowering operators along different directions in this colorspace), you could probably convince yourself that the singlet state inthe decomposition above must be antisymmetric under the interchangeof any two colors. Assuming this is the case, write down the colorportion of the proton wave function.Now that you know the color wave function of the quarks in the proton,write down the spin wave function.Since the proton is uud and its isospin partner the neutron is ddu,andmp≈ mn, let us make the simplfying assumption that mu= md.Giventhe measured value of gp, what does your model give for mu? Recallthat the up quark has electric charge 2/3, and the down quark haselectric charge −1/3, in units of the positron charge.Finally, use your results to predict the gyromagnetic moment of theneutron, and compare with


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CALTECH PH 195 - Problem set number 16

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