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homework6

# homework6 - 1 Problem set 6 Sharp potentials Due on...

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1 Problem set 6 - Sharp potentials Due on Wednesday, October 25, 2011. Posted on Oct. 19 1. To the edge of the cliff and back . In classical mechanics a particle hitting the a wall (i.e. a potential jump) gets reﬂected. In quantum mechanics, a particle will also get reﬂected from a potential drop. Consider an electron moving with momentum p = ¯ hk from x = -∞ towards a drop described by the following potential landscape: V ( x ) = { 0 x < 0 - V 0 x > 0 V 0 > 0 (1) (a) Find the time-independent wave function describing the electron (assume the incoming amplitude is 1). What is the amplitude of the reﬂected wave, and of the transmitted wave? Give your answers in terms of p, V 0 and any constants associated with electrons. (b) What are the reﬂection and transmission probabilities ? How does the transmission probability behave as V 0 → ∞ ? (c) What are the reﬂection and transmission probabilities for an electron moving with group velocity v = 10 6 m/s scattering off a potential drop of V 0 = 30 eV = e · 30 V with e = 1 . 6 · 10 - 19 C being the charge of the electron?
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