Particles incident from the left in Figure P41.59 are

Chapter 41, Problem 59

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Particles incident from the left in Figure P41.59 are confronted with a step in potential energy. The step has a height U at x 5 0. The particles have energy E . U. Classically, all the particles would continue moving forward with reduced speed. According to quantum mechanics, however, a fraction of the particles are reflected at the step. (a) Prove that the reflection coefficient R for this case is R 5 1k 1 2 k 2 2 2 1k 1 1 k 2 2 2 where k 1 5 2p/l1 and k 2 5 2p/l2 are the wave numbers for the incident and transmitted particles, respectively. Proceed as follows. Show that the wave function c1 5 Aeik1x 1 Be2ik1x satisfies the Schrdinger equation in region 1, for x , 0. Here Aeik1x represents the incident beam and Be2ik1x represents the reflected particles. Show that c2 5 Ceik 2x satisfies the Schrdinger equation in region 2, for x . 0. Impose the boundary conditions c1 5 c2 and dc1/dx 5 dc2/dx, at x 5 0, to find the relationship between B and A. Then evaluate R 5 B2/A2. A particle that has kinetic energy E 5 7.00 eV is incident from a region where the potential energy is zero onto one where U 5 5.00 eV. Find (b) its probability of being reflected and (c) its probability of being transmitted.

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