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Unformatted text preview: Quantum Mechanics Problem Sheet 4 (Individual help with these problems is available in the workshop on Friday, 10 Feb 2006. Solutions should be handed in for marking at the start of the lecture on Tuesday, 14 Feb 2006.) 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 ( B e i kx + C e i kx ) for x > , where E = ¯ h 2 k 2 / (2 m ) and k > 0. A , B , and C 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 , B , and C . For each of them, determine whether it is right or leftmoving. They represent the incident, reflected, and transmitted parts of the particle current. Which is which? (A little sketch with arrowsreflected, and transmitted parts of the particle current....
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 Spring '06
 EBERLEIN
 mechanics, Work, Fundamental physics concepts, wave function, probability current density

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