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I am writing a Fortran 95 program to model the scattering of a wavepacket by a potential step of height V_{0}at x=0. My wavepacket is formed by the superposition of numerous travellling waves of different k values. The wavepacket has the dispersion relation ω(k)=k^{2}. I want the wavepacket to be in its undispersed state at t=0 at a start position x_{0}. Therefore each component wave is composed of an incident wave, reflected wave, and transmitted wave.

At x≤0 :

ψ(x,t)=A(e^{i(k(x-x0)-ωt)}+[itex]\frac{k-k'}{k+k'}[/itex]e^{-i(k(x+x0)-ωt)})

At x>0 :

ψ(x,t)=A[itex]\frac{2k}{k+k'}[/itex]e^{i(k'(x-[itex]\frac{k}{k'}[/itex]x0)-ωt)}

The factor of [itex]\frac{k}{k'}[/itex] in the bottom equation was found analytically to ensure continuity in the wavefunctions at the boundary. Well, at least that's what I thought: It works as long as E>V_{0}otherwise there is discontinuity. Can anyone help me as to why? I dont think this is a Fortran programming problem, but more of a physics one...

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# Wavepacket incident on a potential step

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