Physics 212 – Problem Set 11 – Spring 20101. Scattering in 1-D.(a) Consider a single potential step. The potential is zero fromx <0 and a constantV0forx >0. A particleof massmis incoming fromx <0 with energyE= ¯hω.i. Write the Schrodinger equation forx <0. Write the Schrodinger equation forx >0. Write expressionsfor the general solution in the region wherex <0 and in the region wherex >0.ii. Remembering that the particle is incoming fromx <0, find the probability current for thex <0region. Find the probability current for thex >0 region. From this find the probability of transmission(=jtransjinc) and the probability of reflection (=jrefjinc) in terms of the relevant wave numbers and unknowncoefficients.(b) The general solution for an arbitraryV(x) may be written Ψ(x, t) =|Ψ(x, t)|eiS(x,t)/¯h. Find the probabilitycurrent associated with this solution. Notice how the current depends upon thephaseS(x, t)(c) Suppose we could write the general solution for a potentialV(x) as Ψ(x, t) =AeiS(x,t)/¯h
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