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Unformatted text preview: 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 x < ¯ V x ≥ , (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 ≤ A 2 e + ik 2 x + B 2 e- ik 2 x x ≥ (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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