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Unformatted text preview: Physics 212: Problem Set 12 1. Show that the s-wave phase shift for scattering off a potential of the form V ( r ) = C 2 μ δ ( r- r )¯ h 2 , where C is a constant and δ ( r- r ) is a Dirac delta function, gives the phase shift relationship tan ( kr + δ ) = tan ( kr ) 1 + ( C/k ) tan ( kr ) . From this show that in the low energy limit σ = 4 πr 2 Cr 1 + Cr 2 . 2. Consider a one dimensional potential barrier V ( x ) which tends to zero for both x < a and x > b . The energy E of the particle which will be interacting with this potential intersects the potential first at x = a . After that, the potential remains higher than E until it intersects with E again at x = b . The potential is otherwise irregularly shaped but satisfies the WKB requirements. Here you will calculate how the coefficients describing solutions for x < a are related to those describing solutions for x > b . You are given the following “connection” formulas (which we will derive in class): The solution for x < a...
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