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
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]
I think he’s looking for a solution of the time-independent Schrodinger equation where the energy is positive (an unbound state).
Thank you Michael, Actually I’m looking for what Frank said.
So, something like this demonstration but with slightly rounded edges on the potential using any solution method?