Quantum Behavior of a Particle Wave Packet in a 1D Mexican Hat Potential (Time Dependent)

The goal of this program is to explore the boundary between classical physics and quantum physics. This boundary was first hypothesized by Niels Bohr in 1920 and is called the Bohr Correspondence Principle (BCP) . The BCP states that for quantum mode numbers that are much greater than 1, the quantum and classical physics of a particle will be essentially the same. To do this I will show the classical motion of the wave packet at a selected energy as well as the usually periodic variation of the particle's quantum wave packet. The quantum wave packet that I will use has much higher spatial frequencies than the lowest quantum modes of the potential that I will use. Since the eigenvectors of any potential are a complete orthonormal set, the gaussian wave packet can easily be expressed as a linear sume of these eigenvectors. The time dependent Schrodinger equation was published in 1926 which is 93 years from the present year, 2019. Its time dependent solutions that I have been exposed to are for very simple potentials such as square wells with infinite potential energy at each bound or, perhaps, with a square potential barrier in the middle. The reasons for these choices of potential are to demonstrate "tunneling" of the probability of position which is the same as evanescent waves at total internal reflecting boundaries in optics. This program will show the behavior of a wave packet in a much more interesting potential. The quantum wave packet that will be used in this program had the following equation at time=0 and was centered at x=0: where , p is momentum and sigma is chosen to be considerably greater than 1/k so that we have several waves under the gaussian envelope. The wave packet's initial velocity is going to be p/m, where m is the mass of the particle, but this velocity will vary as kinetic energy is gained and lost during the packet's transit through the potential. The quantum wave packet's time evolution will be shown as three YouTube videos (see links below) because it takes a long time to compute its time dependence. The quantum time evolution of the wave packet was computed using finite element analysys (FLexPDE) because the accuracy required is far beyond what could be achieved with a finite difference program. The parameters used for the quantum wave packet were the normal value of the Planck constant and particle mass was that of an electron. The magnitude of the potential and the initial kinetic energy were scaled so that the particle's period was 10^-17 seconds and its turnaround was at about 10^-10 meters which is 1 Angstrom. The period and range of the particle is in approximate agreement with that in an atom and therefore what is expected for an electron. I also show a much faster running classical physics model of the propagation of the wave packet which very closely emulates the quantum wave packet's time dependence. The classical physics model is an array of particles in bins where the number per bin is a gaussian whose width is chosen by the viewer. These particles obey the classical physics laws of motion as they move through the potential, speeding up when the potential is smaller and slowing down when they reach a maximum of the potential. The speed of these particles also goes to zero at the turn around where their total energy reaches the local value of the potential. See the YouTube video at the link below for a talk through explanation of the wave packet model. The potential V(x) here is a standard harmonic oscillator parabola with a guassian "bump" in the middle so it looks like the cross section of a Mexican hat. Of course the potential in most quantum problems is 3 dimensional but three dimensions are hard to depict so I have chosen 1 dimension, x, here. Besides, the behavior of a particle in a quantum "wire" would resemble what is shown here. As labeled, the black trace is the potential that the particle is in, the red trace is the force it feels at any given location, the orange trace is the signed velocity of the particle, and the green trace is the time spent per unit distance moved by the particle, 1/|Velocity|. The green trace value should be proportional to the amplitude of the quantum mechanical wave packet and that is indeed the behavior seen for the classical model shown here as well as the quantum wave packet depicted in the videos. In quantum mechanics, the particle will sometimes "tunnel" through the center hill of the potential, so a simple relation to the classical value of 1/|vx| cannot be expected but we have none of that for the potentials used here. The learner has the option of downloading any of these graphics by pressing the "Download Canvas" link at the top left of the screen.