Classic and Quantum 1D Objects

This animation uses the Lennard-Jones potential (LJ) as well as a parabolic potential to show the motion of a 1D pair of particles and a pair of 1D wave packets as a function of time. I've previously shown that the period of a particle wave packet in a parabolic potential is the same as a classic particle in that potential and is entirely periodic. The particles and wave packets of the object are initially separated by the distance, dx0, where the LJ force between any two adjacent neighbors is zero. My definition of an object is that its components must be robust to disturbances which are encountered when the object turns around at the point where its kinetic energy equals the potential energy of the parabola. To do that, the object must have both attractive and repulsive bonding forces between its components. The LJ potential provides that combination of bonding forces with very strong repulsion when the particles are close together. Without this bonding, the particles of the object would exhibit first in-first out behavior which means that the initially left hand particle would become the right hand particle at every turn around. Another way of expressing this is that the left particle would pass through the right particle after a turn around and we know that is not the observed behavior of an object. The variables provided by the sliders are the minimum value of LJ potential, a small initial random displacement from dx0, a small initial velocity of each particle, an initial random velocity of each particle, an initial velocity of the entire object center of mass, a coefficient aP for the parabolic potential, V(x)=aP*x*x/2, the value of sigma which is the initial width of the wave packets, the half width of the gridded region, and the number of parabola periods for the animation to run.

I have deliberately allowed the limits of the sliders to be large enough to cause the wave packet iteration to run beyond the gridded length of the wave packet so that the learner can see what happens when the packet runs into the gridded boundary which is essentially an infinite potential change.

The outputs are the graphics of the wave packet magnitudes, the particle position graphics, vertical wide lines at the centroids of the packet, and the particle spacings and the wave packet spacings as a function of time. Note that the wave packet spacings do not exactly follow the particle spacings and their period is about 10% longer than the particle spacing period. Also note that neither the particle or the wave packets are not perfectly periodic with the parabola period. This is to be expected since there are 2 periods in effect here, both the parabola period and the vibrational period associatede with the LJ bonding. If the bonding (vibration) period were an integer multiple of the parabola period, then I would expect to see better periodicity from the object. For this animation of the object I assume that its two wave packets can be described by individual linear array rather than the 2D matrix that would be required for indistinguishable particles. The latter will be the subject of a coming animation.