↑

x_{2}

x

Probability densities

x_{1},
x_{2}
→

Run
Speed:

Interparticle purity = 1.00
Hide

Brightness:

Interaction energy | = | 0.060 | |

Interaction range | = | 4.0 |

This program simulates collisions of two quantum-mechanical particles moving in one dimension. Initially the particles are localized and moving toward each other, but the interaction can “split” each of them into reflected and transmitted pieces, “entangled” so they are either both reflected or both transmitted.

The system’s wavefunction, shown at left, is a function of the coordinates of both particles,
x_{1} and x_{2}. To help you visualize what’s happening in physical space,
the particles’ probability densities are plotted at right. But a lot of
information is lost in calculating the probability densities, so to really see what’s happening
you need to look at the actual wavefunction.

The brightness of the wavefunction image indicates the wavefunction’s magnitude, while the color hue indicates the phase, going from red (pure real and positive) to light green (pure imaginary and positive) to cyan (pure real and negative) to purple (pure imaginary and negative) and finally back to red. Use the slider to adjust the overall brightness.

The two particles interact via a “square” potential energy that is constant out to a certain range and zero beyond that. You can adjust both the strength and the range. The configurations with nonzero potential energy are shown as a diagonal gray band superimposed on the wavefunction image.

Each particle’s initial state is a Gaussian wavepacket whose average energy you can adjust. Compare the energies of the wavepackets to the interaction strength, to distinguish between ordinary transmission and quantum tunneling. The particles have the same mass, but the simulation assumes that they are distinguishable in some other way so there is no symmetry constraint on the wavefunction.

The simulation uses natural units in which the particles’ masses, Planck’s constant ℏ, and the (nominal) screen pixel width are all equal to 1. It works by integrating a discretized version of the time-dependent SchrÃ¶dinger equation, with a grid spacing of one unit and a grid size of 320 × 320. The wavefunction is always zero along all edges of the square grid, so the particles are effectively trapped inside an infinite square well of width 320.

The “interparticle purity” is measure of the independence of the two particles’ states. A value of 1.0 indicates that the states are completely independent of each other, so the system wavefunction is a simple product of the wavefunctions of the two separate particles. A purity value less than 1.0 indicates that the particles’ states are correlated or “entangled” with each other, so it is not meaningful to ask what their separate wavefunctions are, and a measurement made on one particle will alter the probability distribution for a subsequent measurement on the other particle. The smaller the purity value, the bigger this change is likely to be.

This simulation is computationally intensive. Although you *can* run it on a mobile
device, I recommend using a reasonably powerful personal computer. Computing the interparticle purity
takes a moment, so you may want to hide the value (suppressing the computation) to get smoother
animation.

Besides studying collisions and entanglement, you might enjoy just letting the simulation run for a while and watching the kaleidoscopic interference patterns. (I sure do!)

This is free, open-source software. Use your browser to view the source code and license.

Daniel V. Schroeder, Physics Dept., Weber State University

See also this simulation of a single particle scattering from a fixed potential energy barrier.