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Unformatted text preview: PH 216, Problem set 6, Due 5/19 1. A diatomic molecule, such as H2 or O2 , consisting of two identical atoms, is modeled by two identical spherically symmetric potentials centered on points A and B, which are separated by a displacement R that points from A to B. Suppose the scattering amplitude for scattering an electron off of one of these centers is known to be f (). Assuming an incoming place wave of general wavenumber find the scattering crossk, section for scattering off the molecule both in terms of R, and also averaged over all orientations of R. (Simplification: Neglect the effect of multiple scattering.) 2. Use the WKB approximation to calculate the s-wave phase shift 0 for scattering off of an attractive U (r) = 2mV (r)/2 , for an incident plane wave of wavenumber k. Your result should express 0 as a radial integral involving k and U (r). r 3. In class we wrote the classical vector potential A(, t) as an expansion in fourier modes ^ c ( t) and polarizations : k, 2 1 r r ^ k)c k, ^ k)c k, r, t) = A( d3 k ( ( t)eik + ( ( t)e-ik . (2)3/2 =1 We also defined 1 c (k, t) + c ( t) , k, 4 2 p ( t) = -i k, c ( t) - c ( t) . k, k, 4 Then we promoted all these to operators: x ( , t), p ( , t) = i, 3 ( - ), k k k k x ( t) = k, with all other equal-time commutators (between xs and ps) zero. We then also defined 1 t) = t) + ip ( t) . a (k, x (k, k, 2 r r (a) What are the classical fields E(, t) and B(, t) in terms of the c ( t)s? k, r r (b) What are the quantum fields E(, t) and B(, t) in terms of the a ( t)s? k, (c) Compute [Ei (, t), Bj ( , t)]. r r r r (d) Compute the commutators of E(, t) and B(, t) with the Hamiltonian 1 H= d3 r(E 2 + B 2 ). 8 (e) Find the time dependence of the fields using these commutators and show that they are consistent with Maxwell's equations. 5 ...
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- Spring '09