Simplified Model of Quantum Scattering

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This Demonstration simulates the quantum dynamics of two interacting nonrelativistic particles. The goal is to demonstrate inelastic scattering in which a particle hits a target (a bound particle), excites the internal dynamics of the target and flies away with its state changed. The challenge is to arrange things in such a way that the essential features of inelastic scattering become visible in accessible observation times. We must make a considerable reduction in state-space dimension while conserving basic scattering geometry.


The solution presented here is the "crossway system," which I devised in 2006. Here the particles "live" in one-dimensional discrete finite spaces. Each particle has its own space, its "biotope" as it may be called. The two biotopes are arranged in Euclidean space orthogonal to one another, resembling two crossing roads. The particles interact through a potential depending on the distance between the particles. The potential has finite range, so that interaction becomes effective only if both particles are close to the intersection point. Apart from the interaction between particles, particle 1 moves freely and particle 2 moves under the influence of a parabolic potential that has its minimum just at the intersection point. Particle 2 is thus a finite-dimensional version of a harmonic oscillator. In order to keep the particles in the system, we give the two biotopes a toroidal topology by declaring the next neighbor of the last point to be the first point.

There are controls for sizes of the biotopes, numbers of their discretization points, masses of the particles and strengths of oscillator potential and interaction potential. Further controls set the initial condition: an oscillator eigenstate for particle 2 and a bell-shaped position distribution together with a "boost" velocity for particle 1. The velocity with which particle 1 is sent toward the crossing is set primarily by setting the oscillator level, which is just reachable by absorbing all kinetic energy of particle 1. Detailed explanation of all control parameters is given in Tooltips that appear as you mouse over them.

The "run" box triggers time evolution of the two-particle wavefunction, which is shown as a color distribution on a rectangular display area. In this display, colors correspond to complex numbers, pixel position along the horizontal direction to discrete position of particle 1, and pixel position along the vertical direction to discrete position of particle 2. Thus each color pixel encodes a quantum amplitude for a situation in which the discrete position of each particle has a definite value. The display area is set such that the path crossing is the center of the area. Taking this as the origin of a conventional coordinate system, particle 1 runs along the axis and particle 2 along the axis.


Contributed by: Ulrich Mutze (December 2020)
Open content licensed under CC BY-NC-SA



The following Snapshots all deal with the same system (the standard system in the bookmarks) for two values for the interacting coupling constant .

Snapshot 1: here we have : strong repulsive interaction

Snapshot 2: : strong attractive interaction

Snapshot 3: Again , but in a representation mode that is explained in the annotation to the selected show box. Here the black-and-white mode is used, since it improves the visibility of the dark stripe patterns, which are reminiscent of photographs of stellar spectra.

The motivation for considering the crossway system is more fully described in [1, Section 5]. It should be noted that the default integrator of this Demonstration is different from the one employed in [1]. It is the method DALF of [2]. The numbers in this program may refer to any system of units for length, time and mass, setting .

Some remarks concerning usage of the controls: The controls fall into two groups, separated by a horizontal black line. If the Manipulate is evaluated the first time it computes the default initial state and prepares it for executing the time evolution algorithm, without starting it. To start it, one has to check the 'run'-box. To start from a different initial state one may perform any modification to the controls. If the settings are as wanted, one has to check the 'initialize'-box. Again, checking the 'run'-box asks for starting the time evolution. Before it actually starts (there is a counter to show this) it may take a while if the integers and were chosen quite large. While running the program allows to manipulate the controls below the black horizontal line. For inspection of details one may toggle the 'run'-box arbitrarily often. By changing 'countMax' one may continue the run if it arrived to its preset end.


[1] U. Mutze. "The Direct Midpoint Method as a Quantum Mechanical Integrator." (Oct 4, 2019)

[2] U. Mutze, "An Asynchronous Leapfrog Method II."

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