731F11_EXAM1 - 1 Rather than starting with a semiclassical...

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1. Rather than starting with a semiclassical approximation, a quantum mechanical approach to screening in the electron gas would first solve Schroedinger’s equation and then determine the induced charge density from the resulting wavefunction state occupation. a. Write down the wavefunctions resulting from a perturbing potential V( r ) using first- order time-independent perturbation theory. Use your knowledge of the unperturbed wavefunctions to simplify. Your answer will be in terms of an undetermined sum over states. b. Using this first-order wavefunction and the Fermi-Dirac occupation function f(k), determine the real-space charge density δρ ( r ) to first order in V( r ) by summing again over all states. Do not attempt to evaluate either sum (yet). Subtract the unperturbed background. c. Now find the Fourier transform g2012g2025 g3562 g4666g2199g4667g3404g1516g2012g2025g4666g2200g4667g1857 g2879g3036g2199uni22C5g2200 g1856 g2871 g1870uni002E Note that g1516g1857 g3036g3044 g4594 g3045 g1857 g2879g3036g3044g3045 g1856g1870g3404g2012g4666g1869g3398g1869 g4593 g4667uni002E , which enables the evaluation of the sum over one state index.
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