- #1
id00022
- 4
- 0
Hello,
I am writing a Fortran 95 program to model the scattering of a wavepacket by a potential step of height V0 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)=k2. I want the wavepacket to be in its undispersed state at t=0 at a start position x0. Therefore each component wave is composed of an incident wave, reflected wave, and transmitted wave.
At x≤0 :
ψ(x,t)=A(ei(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]ei(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>V0 otherwise there is discontinuity. Can anyone help me as to why? I don't think this is a Fortran programming problem, but more of a physics one...
I am writing a Fortran 95 program to model the scattering of a wavepacket by a potential step of height V0 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)=k2. I want the wavepacket to be in its undispersed state at t=0 at a start position x0. Therefore each component wave is composed of an incident wave, reflected wave, and transmitted wave.
At x≤0 :
ψ(x,t)=A(ei(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]ei(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>V0 otherwise there is discontinuity. Can anyone help me as to why? I don't think this is a Fortran programming problem, but more of a physics one...