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3890ps5_10 - Physical Chemistry I Chem 3890 Fall 2010...

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Chem 3890 Physical Chemistry I Fall 2010 Chemistry 3890 Problem Set 5 Fall 2010 Due: in class, Friday, October 15 Last revised: October 8, 2010 Problem 1 Consider the quantum mechanics of a particle, mass m , subject to the step/shelf potential: V [ x ] = braceleftBigg 0 x< 0 ¯ V x 0 , (1) where the (real) parameter ¯ V can be positive, negative or zero. Write the particle wavefunction at energy E> 0 in the form ψ ( x ) = braceleftBigg A 1 e + ik 1 x + B 1 e - ik 1 x x 0 A 2 e + ik 2 x + B 2 e - ik 2 x x 0 (2) where k 1 = 1 planckover2pi1 2 mE (3a) k 2 = 1 planckover2pi1 radicalBig 2 m ( E - ¯ V ) . (3b) 1. Is the quantity k 2 necessarily real? If not, for which values of ¯ V will it not be real? 2. By imposing suitable matching conditions on the wavefunction and its derivative at x = 0, derive a linear relation between the coefficients ( A 1 ,B 1 ) and ( A 2 ,B 2 ). Note that this relation is valid for all values of ¯ V . 3. By imposing the BC appropriate for a scattering experiment in which a beam of particles with E> 0, k 1 > 0 incident ‘from the left’ on the step/shelf, obtain an expression for the reflection coefficient R . Your expression will in principle be a function of E , ¯ V m and planckover2pi1 .
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