HW6 - Quantum Mechanics I Physics 3143 Assignment 6 Fall...

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Quantum Mechanics I, Physics 3143: Assignment 6, Fall 2010 1 . Consider scattering from a step potential V ( x ) = 0 , x < 0; V 0 , x 0 . Suppose that incoming particles from x = -∞ , with energy E , are incident on the interface. Calculate the incident, reflected and transmitted particle currents in the case E > V 0 , and hence determine the reflection coefficient R from the barrier in terms of the wave numbers of the incident and transmitted particles, k 1 and k 2 , respectively. 2 . Make a sketch/plot of the potential V ( x ) = V 0 / (1 + e - x/l ), and state under what condi- tions the potential approaches the potential of problem 1. For the smoothly varying poten- tial Schrodinger’s time-independent equation may be solved exactly to yield the reflection coefficient R = sinh [ π ( k 1 - k 2 ) l ] sinh [ π ( k 1 + k 2 ) l ] 2 . (a) Discuss the limiting cases E = V 0 and E >> V 0 , respectively. (b) In the case E >> V 0 , R does not agree with the reflection coefficient found in the same limit in problem 1. What is the reason for this ?
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