HW7 - Consider transmission through two delta-function...

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PHYS851 Quantum Mechanics I, Fall 2008 HOMEWORK ASSIGNMENT 7: 1D Scattering: 1. [10pts] Step-down: An incident wave with wave-vector k 1 approaching from the left encoun- ters a step-down potential, V ( x ) = - V 0 u ( x ) where u ( x ) is the unit step function. Calculate the quantum reFection probability, R . Under what conditions does it agree to a good approximation with the classical reFection probability? 2. [15pts] Square barrier plus step potential Consider a plane wave incident from the left onto the following potential V ( x ) = 0 , x < 0 V 0 , 0 < x < a V 1 , x > a , where V 0 > V 1 > 0. Using the transfer-matrix approach, ±nd the reFection and transmission probabilities R and T as a function of the incident wave-vector, k 1 > 0. Show explicitly that T = | t | 2 k 3 k 1 , where k 3 is the wave-vector in the region x > a . 3. [15pts] Inverted Delta function potential: Consider a particle subject to the potential V ( x ) = - ( x ) with g > 0. ²or a plane wave incident from the left with wave-vector k 1 > 0 what are the transmission and reFection probabilities T , and R ? 4. [30pts] Delta function Fabry Perot Resonator:
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Unformatted text preview: Consider transmission through two delta-function barriers, V ( x ) = g ( x ) + g ( x-L ), where g &gt; 0. Use transfer matrix approach to nd the transmission probability T for a particle incident from the left, as a function of the incident momentum, k , and the barrier separation L . Make a plot of T versus kL for dierent values of the parameter ka = p 2 k mg . Compare the location of the resonances to the energy eigenvalues of an innite square well of width L . In what limits to they match? 5. [20pts] Inverted Double Delta Potential: Consider a pair of inverted delta potentials, V ( x ) =-g ( x )-g ( x-L ), where g &gt; 0. ind the two bound-state energies and corresponding wavefunctions. What happens to each of the levels in the limit L 0? Also in this limit, make a connection with the bound state of a single delta potential, V ( x ) =-2 g ( x )?...
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