E_{1} = |
E_{2} = |
E_{3} = |
E_{4} = |

p = 1.00 |
p = 1.00 |
p = 1.00 |
p = 1.00 |

Interaction strength = 0
Interaction range = 0.30
Attractive
Repulsive
Smooth
Abrupt |
Green/magenta Red/cyan |

These images show some of the quantum behavior of two interacting particles confined inside a one-dimensional “box”. You can adjust the strength and range of the interaction between the particles, make it attractive or repulsive, and make it vary with their separation distance either smoothly or abruptly. Try adjusting all the controls and watch the images respond.

Each of the square images plots the position of one particle horizontally and the other vertically. Because the particles become entangled, we need a two-dimensional configuration space plot to show how their behaviors are correlated with each other. Note that the main diagonal, running from lower left to upper right, represents configurations in which the two particles are in the same place. The two corners farthest from the main diagonal represent the two configurations in which one particle is at one end of the box and the other particle is at the opposite end. The calculations assume that the particles are different from each other, so it is meaningful to say which one is where. However, for simplicity this simulation gives both particles the same mass.

The plot to the left of the controls shows the interaction energy, which is larger in magnitude when the particles are close together and smaller (or zero) when they are far apart. Each of the four plots across the top shows a wavefunction of definite energy, with the lowest-energy state at the left and the higher-energy states to the right. In general the wavefunctions tend to be larger in magnitude where the interaction energy is lower, and smaller in magnitude where the interaction energy is higher. However, each successive wavefunction must be different from (orthogonal to) all the ones before it, and this requirement forces the higher-energy wavefunctions to vary over shorter distances, with nodes (lines of zero values) across which the sign changes.

All the plots show positive values in one color (green) and negative values in another (magenta). Brighter areas represent larger magnitudes, while black represents zero. So in the wavefunction plots, the brightest areas represent the most probable configurations of the two particles.

Underneath each wavefunction plot is displayed the wavefunction’s energy *E*,
measured in natural units in which Planck’s constant ℏ, the particles’ masses,
and the width of the box are all equal to 1. In these units a single particle has a minimum
energy of π^{2}/2 or a little under 5. The interaction strength is measured
in these same units, and the interaction range is in units of the box width.

Also displayed for each wavefunction is its “interparticle purity” *p*, which
is a measure of its degree of entanglement. Smaller *p* values indicate more entanglement,
while an unentangled state has *p* = 1. Physically, a smaller *p* indicates that a
measurement made on one particle will tend to have a greater effect on the probability
distribution for a subsequent measurement on the other particle. When *p* = 1 the two
measurements are uncorrelated, so the first has no effect on the second. In that case
the wavefunction can be factored into a product of separate wavefunctions for each particle.

Mathematically, the problem of two equal-mass but distinguishable particles in a one-dimensional
box is equivalent to that of a *single* particle in a *two*-dimensional square box.
Under the latter interpretation, the “interaction” energy becomes an externally
applied potential energy that depends only on the distance from the main diagonal. Sometimes
this reinterpretation makes it easier to guess what the wavefunctions should look like.
To see how a single particle in two dimensions responds to a broader range of potential
energy functions, use the related Quantum Bound States in Two
Dimensions web app.

The wavefunction calculations use a variational-relaxation algorithm on a 50 × 50 grid. As the images change you are seeing the algorithm at work—not an actual dynamical process playing out in simulated time. When two wavefunctions have nearly the same energy the algorithm can sometimes fail, getting them out of order or mixing them to produce asymetrical combinations. Sometimes you can fix these problems by clicking/tapping on a wavefunction plot to nudge it slightly, or by clicking the Reset button.

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