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Course: PHYS 356, Fall 2008
School: UVA
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Word Count: 620

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356 Instructions: Final Phys Exam May 2, 2007 This is an in-class, three hour exam. You may refer to your textbook, class notes, and homework solutions, and you may use a calculator. No other reference materials are permitted. Turn in your solutions stapled to these exam sheets. The exam consists of six problems, each worth 10 points. For full credit, you must explain your reasoning and show all your work. You...

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356 Instructions: Final Phys Exam May 2, 2007 This is an in-class, three hour exam. You may refer to your textbook, class notes, and homework solutions, and you may use a calculator. No other reference materials are permitted. Turn in your solutions stapled to these exam sheets. The exam consists of six problems, each worth 10 points. For full credit, you must explain your reasoning and show all your work. You may cite the result of a calculation in the text, notes, or homework without rederiving it, but state clearly where the result is from. Name: Pledge: Some useful formulas: sin = 2 sin sin2 2 cos 2 = 2 cos = cos2 cos2 2 = 2 sin2 2 1 (1 cos ) 2 1 (1 + cos ) 2 Also, dont forget that the textbook has a few handy integrals on the inside back cover. 1. Suppose a particle is conned in an innite square well, V (x) = 0 if 0 < x < a else and subject to a perturbation H (1) = x. Calculate the resulting rst order energy (1) shifts En . 2. A spin-1/2 electron is held motionless in a magnetic eld B = B(sin x + cos ). z Use the variational principle to estimate the ground state energy, base on the trial wave function | = a | + b | for variable a and b which you may take to be real. Here | and | represent spin up and down along z. How does your answer compare to the exact result? 3. An entagled state of two particles is one that cannot be expressed as a product of single-particle states, (A, B) = a (A)b (B). However, two identical particles must always be in an eigenstate of exchange. So two electrons nominally in orthogonal states a and b are actually in the entangled two-body state 1 (A, B) = [a (A)b (B) b (A)a (B)] . 2 Can this be used for quantum communication purposes? For instance, if Alice has an electron in some denite state a , and Bob has another electron in a denite state b , they could make any measurements that violate Bells inequality? Why or why not? (Im looking for an explanation here, not really a calculation.) 4. Use the Born approximation to calculate the scattering amplitude for the potential V (r) = r2 + a2 Hint: if you extend the radial integral from to using the symmetry of the integrand, it becomes straightforward to evaluate using contour integration. 5. Consider the spontaneous emission rate for the hyperne transition F = 1 to F = 0 in the ground state of hydrogen. (a) Write out the four hyperne ground states of hydrogen in the mS , mI basis. Explain why there is no electric dipole coupling between these states. (b) The magnetic coupling Hamiltonian is H = (S + I ) B where S is the magnetic moment of the electron and I is the magnetic moment of the nucleus. Calculate the matrix elements of this perturbation between the |F = 1, mF = 0 excited state and the |F = 0, mF = 0 ground state, for magnetic elds in each of the x, y, and z directions. You may neglect any small terms in the calculation. (c) If the magnetic eld is incoherent, it can b...

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