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

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Unformatted text preview: PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 10 Topics covered: Greens 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 | i = | i + GV | i , (1) to solve the one-dimensional problem of tunneling through delta potentials. Take ( z ) = e ikz , and let V ( z ) = g ( z ) + g ( z- L ) . (2) (a) Express Eq. (1) as an integral equation for ( z ), and then use the delta-functions to perform the integral. It might be helpful to introduce the dimensionless parameter = 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 ( z ) = te ikz . (3) (c) In the strong-scatterer limit 1, at what k-values is the transmission maximized?...
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852_2010hw10 - PHYS852 Quantum Mechanics II, Spring 2010...

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