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Unformatted text preview: Quantum Mechanics Problem Sheet 4 (Solutions are to be handed in for marking before the lecture on Monday, 14 Feb 2005) 1. A stream of particles is scattered by an obstacle located at x = 0. The wave function describing the particles is Ψ( x, t ) = A e- i Et/ ¯ h e- i kx for x < , e- i Et/ ¯ h ( C e i kx + D e- i kx ) for x > , where E = ¯ h 2 k 2 / (2 m ) and k > 0. A , C , and D are constant complex coefficients. (a) Calculate the probability density ρ ( x, t ) for x < 0. (b) Calculate the probability current density j ( x, t ) for x < 0. (c) Calculate the probability density ρ ( x, t ) for x > 0. (d) Calculate the probability current density j ( x, t ) for x > 0. (e) The wave function given above is the sum of three distinct parts, labelled by the coefficients A , C , and D . For each of them, determine whether it is right- or left-moving. They represent the incident, reflected, and transmitted parts of the particle current. Which is which? (A little sketch with arrows labelled...
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