Skimming a Barrier: Wave Function and Matching Conditions

Question:
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 V0.

Let k denote the wavenumber of the incident particles and a denote the barrier
width.

a) Write down the time-independent Schrodinger equation in regions (I) x < 0, (II) and (III) x > 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 corresponding Schrodinger equation. In this way, derive the relation between k and E in regions (I) and (III).

b) write down the complete set of matching conditions at each of the two interfaces (x = 0 and x= a).

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Quantum: Skimming a Barrier

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