# Wave-packet backwards emission

I am trying to reproduce the results of a TDSE two wave-packet algorithm described in "Computational Physics: Problem Solving with Python" by Landau, Paez and Bordeianu, in section 22.4 (Wave Packet–Wave Packet Scattering). The chapter references this report and this page of results.

In pretty much all my own results and in some of those provided by the author (like this), you can see a small fragment of a wave-packet being "emitted" backwards in the very beginning of the computation. (The mass of the left packet is a fraction of the right packet's.)

The interaction potential used in these pictures is of gaussian form (attractive, but short range; shouldn't affect the packets at the initial distance). The method assumes that the wave-packets initially are far enough such that the total wavefunction is a product of two gaussians.

My question is, how does one explain this emission? This doesn't seem to make much sense physically.

• These are numerical results? I've seen similar effects due to numerical instabilities in equations allowing for multiple solutions. When propagating one solution an admixture of others may occur due to rounding errors. Unfortunately i may not provide a reference. Apr 25 '19 at 6:45
• Yes, these are numerical results. That is an interesting point. I don't know whether it is that. Different values for masses and momenta produce various degrees of this effect and it only occurs right at the beginning. Perhaps some kind of unnatural initial condition forces the numerical solution to adapt, but I really have no idea. Apr 25 '19 at 7:02

At early times in Fig. 3, as well as in other animations, we can see very small wavepackets moving in opposite directions to the larger wavepackets for each particle. These are numerical artifacts. While wavepackets with reversed values of $$k$$ are valid solutions of the Schrödinger equation, they should be elminated by the initial conditions. For example, if $$\exp(ikx)$$ is a valid solution, then so is $$\exp(−ikx)$$, yet it is hard to get rid of it completely.