Two Particle Object Propagation in a Parabolic Potential

Here I define an object as having more than 1 particle which means that it must have more than 1 quantum wave packet.

This animation shows the one dimensional (1D) movement of coupled gaussian envelope wave packets of two particles as governed by the Schrodinger equation. The form of the Schrodinger solution for 2 particles is

`psi(x_1,x_2,t)=f_1(x_1,t)f_2(x_2,t)`

where `x_1` and `x_2` are independent variables and t is time. The Schrodinger equation must be iterated to determine the time dependence of `f_1` and `f_2`. Since `x_1` and `x_2` are independent except for wave packet interaction, the discretized expression of `psi` is a square matrix.

Another 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 Lennard Jones 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.

For the initial wave function we just choose gaussian wave packets for f1 and f2. Since the solution is a product we are forced to make the calculation as a 2D matrix. The wave packet is discretized as the rows and columns of a square matrix. If there were 3 particles, then a cubic matrix would be required even for 1D calculations

Any single wave packet propagation in a parabolic potential has the very special quality that it is perfectly periodic with the same period as a classical particle. In fact the centroid of the absolute value squared of the wave packet follows the path of a classic particle with the same initial conditions.

When two wave packets are coupled by a central force like that of Lennard-Jones, their motion is no longer that of simple un-coupled particles in a parabolic potential because the coupling results in an additional periodicity due to the vibrational motion of the packets. Many parameters are variable in this animation.

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.

I have chosen to present 3 different solutions of the time dependent Schrodinger equation (TDSE).

1. The motion of a single wave packet in the parabolic potential just to demonstrate its periodicity at the particle period. 2. The motion of two separate interacting particles/packets that are not entangled and therefore do not require a square matrix for their discretization. 3. The motion of two separate interacting particles/packets that are entangled and therefore requires a square matrix for their discretization.

For this 2 particle animation there is a distance dependent potential that tends to hold the particles together and keeps the particles from crossing each other. This potential is of the Lennard-Jones (LJ) type where the repulsion when near greatly exceeds the attraction when far. Because of this potential, the particles will be started with speeds in the same direction and with spacings close to the potential minimum, In this way they are expected to start to vibrate when they come to a turnaround point associated with the parabolic potential. 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 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 associated with the LJ bonding.

For 2D plot of the third solution it is easy to see that the motion in (x1,x2) space reverses in the lower right quadrant where the in this case x1~x2 and the particles come closest together. This is necessary to avoid particle 2 crossing particle 1. Note that, when plotting real, imaginary, or phase, the wavefronts are always perpendicular to the path as they must be. The total desktop computer time for completion of a 1D period is about 30 seconds. 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. As a note in passing, it seems that some authors seem to think that the wave function has to go to zero at the boundaries of the discretization. I would like to point out that this and their computations are numerical approximations to the wave function and therefore it is only necessary to choose discretized widths within which the wavefunction is negligible at the turnaround of the wave packet.