Physics 212 – Problem Set 11 – Spring 2010 1. Scattering in 1-D. (a) Consider a single potential step. The potential is zero from x < 0 and a constant V0 for x > 0. A particle of mass m is incoming from x < 0 with energy E = ¯ hω . i. Write the Schrodinger equation for x < 0. Write the Schrodinger equation for x > 0. Write expressions for the general solution in the region where x < 0 and in the region where x > 0. ii. Remembering that the particle is incoming from x < 0, ﬁnd the probability current for the x <0 region. Find the probability current for the x > 0 region. From this ﬁnd the probability of transmission (= j trans j inc ) and the probability of reﬂection (= j ref j inc ) in terms of the relevant wave numbers and unknown coeﬃcients. (b) The general solution for an arbitrary V ( x ) may be written Ψ( x,t ) = | Ψ( x,t ) | e iS ( x,t ) / ¯ h . Find the probability current associated with this solution. Notice how the current depends upon the phase S ( x,t ) (c) Suppose we could write the general solution for a potential
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