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Numerical solution of one dimensional Schrodinger equation

Posted 10 years ago
I want to solve one dimensional Schrodinger equation for a scattering problem. The potential function is 1/ ( 1+exp(-x) ). So at -? it goes to 0 and at ? it's 1. The energy level is more than 1. I used Numerov's method and integrated it from +? (far enough) backwards with an initial value =1 . But I believe it's wrong b/c squared wave function is oscillating on whole interval and it's supposed to be constant after the jump in potential. I know that I'm doing somewhere wrong in my solution So I would appreciate you if you help me by this or introduce me some sources.
Thanks,
Moji
POSTED BY: Mojtaba M
4 Replies
Posted 10 years ago
I'm not a physicist, but Numerov's method is used to finding stationary solutions (time-independent Schrodinger equation). When you say scattering and give a potential with that shape it makes me think you want to see how a wave packet is dynamically affected (time-dependent Schrodinger equation). The documentation for NDSolve gives some examples. Here you can see how your potential pushes a wave packet to the left.
NDSolve[{I D[u[t, x], t] + D[u[t, x], x, x] ==
   1/(1 + Exp[-x]) u[t, x],
  u[0, x] == PDF[NormalDistribution[0, 3], x],
  u[t, -100] == u[t, 100]}, u, {t, 0, 10}, {x, -100, 100}]
DensityPlot[Abs[%[[1, 1, 2]][t, x]]^2, {x, -20, 20}, {t, 0, 10},
PlotPoints -> 100]
POSTED BY: Michael Hale
I think he's looking for a solution of the time-independent Schrodinger equation where the energy is positive (an unbound state).
POSTED BY: Frank Kampas
Posted 10 years ago
Thank you Michael, Actually I'm looking for what Frank said.  
POSTED BY: Mojtaba M
Posted 10 years ago
So, something like this demonstration but with slightly rounded edges on the potential using any solution method?
POSTED BY: Michael Hale
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