Wavepacket incident on a potential step

1. May 19, 2013

id00022

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) +$\frac{k-k'}{k+k'}$e-i(k(x+x0)-ωt))

At x>0 :
ψ(x,t)=A$\frac{2k}{k+k'}$ei(k'(x-$\frac{k}{k'}$x0)-ωt)

The factor of $\frac{k}{k'}$ 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 dont think this is a Fortran programming problem, but more of a physics one...

2. May 19, 2013

sweet springs

Exactly. Aren't you satisfied yet?

3. May 19, 2013

id00022

Well not really: when I work through the maths, this factor should work fine in all situations, but in practice it only works when E>V0. If you have any idea why this might be then I would be very grateful. I'm fairly certain its not a problem with my code as I have applied it in many different ways all with the same result.

4. May 19, 2013

sweet springs

Hi

for E<V0, ψ for x>0 is no longer a plane wave but an exponentially dumping function as e^-Kx where k'=iK pure imaginary number.

Last edited: May 19, 2013
5. May 19, 2013

id00022

Hi,
Thanks, but this is built into my program already: k' becomes complex reducing ψ=eik'x to e-αx the dumping function you describe.

6. May 19, 2013

sweet springs

Hi.

In both of the cases, e^ik'x = e^-Kx = 1 for x=0. So continuity conditions should be the same in the both cases.
What is the discontinuity you are worrying about?

7. May 19, 2013

id00022

Because if I do not offset by x0, then the wavepacket impacts the step at exactly t=0. Therefore to watch it scatter you must set the time range to be from -t to t. In the period -t to 0 the wavepacket starts off slightly dispersed, becoming less dispersed until it hits the barrier. This is unphysical so I have offset the position of the wavepacket at t=0 to x=x0. Although eikx and e-αx both equal 1 at x=0, eik(x-x0) and e-α(x-x0) are not equal there. So the factor $\frac{k}{k'}$ has to be introduced as above. This works fine on paper but not in practice.

8. May 19, 2013

sweet springs

Hi.

The dumping depends on the distance from the wall, x-0, not the distance from x0, x-x0.
The wave function for x>0 is e-αx+ikx0.