Skimming a Barrier: Wave Function and Matching Conditions
In this problem, we lay the groundwork for computing the transmission
coefficient of a rectangular barrier in the very special case that the energy
E of the incident particles is exactly equal to the barrier height Vo Let *
denote the wavenumber of the incident particles and a denote the barrier
a) Write down the time-independent Schrodinger equation in regions (1)
I < 0. (II) 05 I < a and (III) I a, assuming a wave incident on the
barrier from the left. Label the coefficients as follows: A.B in region (I) with
A the incident amplitude; F. G in region (II); and C in region (III). Indicate
the solution in each region, and show explicitly that is satisfies the corre-
sponding Schrodinger equation. In this way, derive the relation between *
and E in regions (1) and (III).
b) write down the complete set of matching conditions at each of the two
interfaces (x = o and x=a).
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