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CALTECH PH 195 - Problem set number 20

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Physics 195bProblem set number 20Due 2 PM, Thursday, March 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. . .PROBLEMS:92. We obtained the following Hamiltonian, up to a constant, for our su-perconductor:HS=d3(x)12m∇ψ†s(x) ·∇ψs(x) − µψ†s(x)ψs(x) (169)+12d3(x)d3(y)∆ss(x, y)ψs(y)ψs(x)+∆∗ss(x, y)ψ†s(x)ψ†s(y),where(a) ψ†s(x) is a field operator creating an electron with spin projections at point x,56(b) repeated indices are summed over,(c) µ is the “chemical potential” (energy of the Fermi surface here),(d) the “gap function” is:∆ss(x, y)=ψ†s(x)ψ†s(y) U (x − y), (170)(e) and U(x−y) is the (weakly attractive) electron-electron potential.As a first step towards the Bogoliubov transformation, we defined thetwo-component vector field according to:Υα=ψs(x)ψ†s(x), (171)where α subsumes the s and x indices in one symbol.Show that our Hamiltonian may be written in the form:HS=12Υ†αHαβΥβ+ H0, (172)whereH0=12Tr−12m∇2− µ, (173)andHαβ=12m∇x·∇y− µδαβ∆∗αβ−∆αβ−12m∇x·∇y− µδαβ. (174)93. Let us investigate the very low energy scattering limit somewhat fur-ther. In this limit, we expect S-wave scattering to dominate, so let uslook at the S-wave term (considering spinless case for now):f0(k)=12ike2iδ0− 1. (175)In the low energy limit, we can expand k cot δ0in a series in powers ofk2:k cot δ0= −1a+12r0k2+ O(k4), (176)57where a is called the (zero-energy) “scattering length”,1and r0is calledthe “effective range”. In principle, we need to establish this formula,and relate a and r0to the properties of the scattering center. However,let us assume that the energy is sufficiently low that we may neglectall but the first term.(a) Show that, in the low energy limit, we may write δ0= −ak,andfind a simple “physical” picture for a. That is, I want you to relatea to some value of radius r. It is possible to do this by consideringthe wavefunctions only in the region where the potential vanishes,although to actually compute a requires looking at the potential,of course. What is the total cross section, in terms of the scatteringlength?(b) We continue by considering specifically very low energy neutron-proton scattering. These are not spinless particles, and indeed it isobserved that the potential is spin-dependent (can you think whyyou already know this?). Thus, for neutron-proton scattering atvery low energies, we have two potentials to think about (more,if spin flips happen, but we assume that we are at sufficiently lowenergy so that everything is S-wave, with total spin conserved):An effective potential for scattering in the spin-singlet state, andanother for the spin-triplet state. Corresponding to these twopotentials, we introduce two scattering lengths, atfor the tripletinteraction, and asfor the singlet.With the sign convention of part (a), and using what you knowabout the neutron-proton interaction, give a simple discussion ofwhat you expect for the signs of atand as.(c) Continuing with low energy neutron-proton scattering, show thatthe total cross section for low energy scattering of neutrons on atarget of randomly polarized protons is:σ0=4π34a2t+14a2s. (177)94. In practice, the result of the preceding problem may be applied toneutron scattering on a hydrogen target (as our source of protons),when the neutron wavelength is short compared with the hydrogen1Not everyone uses the same sign convention for a!58molecular size, but still long enough to be in the “low-energy” regime,where we may neglect both higher partial waves and the effective rangeterm.(a) Consider a possible target with hydrogen gas at room tempera-ture. Assuming thermal equilibrium, such a gas is a mixture ofparahydrogen (nuclear spins antiparallel) and orthohydrogen (nu-clear spins parallel).2What is the fraction of parahydrogen?(b) Suppose now that we have constructed a target (at 20 K, say)consisting entirely of parahydrogen. We consider the scatteringof “cold” neutrons from this target. By “cold” we mean neutronswith an energy corresponding also to T ≈ 20 K. In this case(as you should convince yourself with a quick computation), theneutrons scatter elastically from the parahydrogen molecule as awhole. This may be thought of as coherent scattering from thetwo protons. Show that the total cross section for the scatteringof cold neutrons on parahydrogen is:σP=4π64934at+14as2. (178)You should assume: (i) The neutron wavelength is much largerthan the molecular size; (ii) The scattered wave is the sum of thewaves scattered from each proton – i.e., there is no “double scat-tering” where the wave scattered from one proton subsequentlyscatters on the second proton; (iii) The scattering is strictly elas-tic.Experimentally (see e.g., R. B. Sutton et al.,Phys.Rev. 72 (1947)1147),σ0=20.4 × 10−24cm2, (179)σP=3.9 × 10−24cm2. (180)Thus, determine atand as. Are your results sensible?2How do you remember which is which? Life is complicated by paradoxical parallelisms,to say nothing of orthogonal orthodoxisms. But in the end, orthodoxy is


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