HOMEWORK ASSIGNMENT 9 PHYS852 Quantum Mechanics II, Spring 2008 1. Use the Lippman-Schwinger equation: | ψ i = | ψ0 i + GV | ψ i (1) to solve the one-dimensional problem of resonant tunneling through two delta-potentials. Take ψ0 ( x ) = e ikx and V ( x ) = g [ δ ( x ) + δ ( x-L )] . (2) a.) Express (1) as an integral equation for ψ ( x ) and use it to ﬁnd the general solution ψ ( x ) for arbitrary k and g . b.) use your result from a.) to compute the transmission probability T = | t | 2 as a function of k . c.) Consider an inﬁnite square-well V ( x ) = 0 from x = 0 to x = L , and V ( x ) = ∞ otherwise. Show that the bound states are of the form ψ n ( x ) ∝ sin( k n x ), where k n = nπ/L . d.) Under what conditions does the transmission probability have a maximum at or near the k n corresponding to the square-well bound states? 2. S-wave resonance scattering: use the method of boundary conditions to solve the 3-d resonant scat-tering problem of a spherical delta-shell potential, given by
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This note was uploaded on 11/26/2010 for the course PHYSICS PHYS 852 taught by Professor Michaelmoore during the Spring '10 term at Michigan State University.