Numerical solution of one dimensional Schrodinger equation

[mojtaba m]

Hi,
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

 

[The_Duck]

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.

Sounds like your solution is a linear combination of a particle incoming from the left and a particle incoming from the right. Interference of $e^{ipx}$ and $e^{-ipx}$ terms would produce this oscillating behavior. Maybe you can find two solutions and take a linear combination to eliminate the left-moving component on the right side of the potential jump?

 

[mojtaba m]

Actually we have two wave function A exp(ipx) and B exp(−ipx) at far left and one C exp(iqx) at far right. I need to write my program in a way which the coefficients A, B and C been determined self-consistently.

 

[The_Duck]

Actually we have two wave function A exp(ipx) and B exp(−ipx) at far left and one C exp(iqx) at far right.

Right, but the most general solution to the Schrodinger equation also has a D exp(-iqx) component at the far right. Are you doing anything to prevent this component from appearing in your numerical solution?