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Unformatted text preview: HOMEWORK ASSIGNMENT 9 PHYS852 Quantum Mechanics II, Spring 2009 New topics covered: Scattering amplitude, cross-section . 1. Use the Lippman-Schwinger equation, | ψ ) = | ψ ) + GV | ψ ) , (1) to solve the one-dimensional problem of resonant tunneling through two delta-potentials. Take ψ ( x ) = e ikx and V ( x ) = g [ δ ( x ) + δ ( x − L )] . (2) a.) Express eq. (1) as an integral equation for ψ ( x ). Then solve this equation to find the general solution ψ ( x ) for arbitrary k and g . It might be helpful to express your answer in terms of the dimensionless parameter α = Mg/ ( planckover2pi1 2 k ). b.) Compute the transmission probability T = | t | 2 as a function of k , defined via lim x →∞ ψ ( x ) = te ikx . c.) In the strong-scatterer limit α ≫ 1, at what k values is the transmission maximized? d.) Consider an infinite square-well V ( x ) = 0 from x = 0 to x = L , and V ( x ) = ∞ otherwise. What are the k-values associated with each bound state? 2. Find the scattering amplitude, f ( vector k ′ | k ), for a Gaussian scattering potential V ( vector r ) = V e − ( r/r ) 2 , using the first Born approximation. Within the first Born approximation, what are the differentialusing the first Born approximation....
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