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852_2010hw10

# 852_2010hw10 - PHYS852 Quantum Mechanics II Spring 2010...

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PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 10 Topics covered: Green’s function, Lippman-Schwinger Eq., T-matrix, Born Series. 1. T-matrix approach to one-dimensional scattering: In this problem, you will use the Lippman- Schwinger equation | ψ i = | ψ 0 i + GV | ψ i , (1) to solve the one-dimensional problem of tunneling through delta potentials. Take ψ 0 ( z ) = e ikz , and let V ( z ) = ( z ) + ( z - L ) . (2) (a) Express Eq. (1) as an integral equation for ψ ( z ), and then use the delta-functions to perform the integral. It might be helpful to introduce the dimensionless parameter α = Mg ~ 2 k . To solve for the two unknown constants, generate two equations by evaluating your solution at z = 0, and z = L . (b) Compute the transmission probability T = | t | 2 , with t defined via lim z →∞ ψ ( z ) = te ikz . (3) (c) In the strong-scatterer limit α 1, at what k -values is the transmission maximized? (d) Consider an infinite square-well of length L . What are the k -values for each bound-state? How do these compare with the transmission resonances in the strong-scatterer limit?

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