Transmission Through a Delta Function Potential

A particle of mass m moves in one dimension where the only potential V(x) = Cδ(x) is at the origin with C > 0. A free particle of wave vector k approaches the origin from the left. Derive an expression for the amplitude T of the transmitted wave as a function of k, C, m, and h.

SOLUTION

On the left the particle has an incident intensity, which we set equal to unity, and a reflected amplitude R. On the right the transmitted amplitude is denoted by T.

(1)

At the point x = 0, we match the value of ѱ(x) on both sides. We match the derivative according to an expression with C = C'. This yields two equations for T and R which can be solved for T:

(2)

(3)

(4)