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Physics 5002 (Spring 2017) Discussion Problem (4/20) Consider one-dimensional scattering of a particle of mass m by the potential barrier ( V (x) = V0 , 0 < x < L, 0, elsewhere. (1) The energy of the particle is E = h̄2 k 2 /(2m) > V0 . Take the incident wave to be exp(ikx) in the region of x < 0. 1. Derive the amplitude A of the reflected wave A exp(−ikx) in the region of x < 0. 2. Derive the amplitude D of the transmitted wave D exp[ik(x − L)] in the region of x > L. 3. Show that |A|2 + |D|2 = 1. 4. Show that |A|2 = 0 when the imaginary part of D vanishes. Due to the geometric differences, the Optical Theorem for three-dimensional scattering does not have a simple counterpart in one dimension. The results in parts 3 and 4 are close to such a counterpart.