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The modes are the eigenvalues and eigenvectors (psi(x))of the second order Schrodinger equation (SE) , where h with the bar ℏ is the Planck constant, m is the mass and E is the energy of the particular mode whose wave function is the eigenvector. For this program I have chosen to set the Planck constant (h divided by 2*pi), ℏ, equal to 1 Joule second and the mass, m, equal to 1 kilogram. Of course the potential in quantum field theory is 3 dimensional but three dimensions are hard to depict so I have chosen 1 dimension, x, here. The black trace is the potential that the particle is in. The animation shows the propagation of a quantum wave packet like

`psi_0(x,0)=exp(ikx-(x/sigma)^2)`

in a swaged potential well. The swaged edges of the potential well have neither value or slope discontinuities and these are both more physical and cause less propagation glitches than the the finite depth square well which is usually the one taught in physics courses. The wave packet that I use is a simple complex sinusoid embedded in various envelopes (see "Starting Wave Packet Envelope Shapes"). Its propagating function changes are most accurately computed by solving the time dependent Schrodinger equation (TDSE)`(d^2psi)/(dx^2)+V(x)psi=iℏ(dpsi)/dt`

using finite element analysis (FEA) but this animation just uses finite difference (FD) methods to get adequate accuracy. I give the learner the choice of 3 wave packet envelope shapes:1: Gaussian (standard bell curve used in statistics)

2: Triangular

3: Square

The evolution of the gaussian is the slowest but it eventually gets spread out over the full potential well width; however this spreading is much slower than that in the finite square well. The Triangle, which has a slope discontinuity, remains intact for a cycle or two but it definitely loses its resemblance to its original shape faster than the gaussian. The Square, which has a value discontinuity, loses its identity after only 1 cycle. Note that both of these latter two packets tend toward a gaussian shape. Also note that the wave packet's return to the center of the potential well is delayed a few percent with respet to that of the Classic Particle. The evolution of these shapes is in stark contrast to that seen in the case of a parabolic potential where, within numerical error, the packet probagation is entirely periodic.

Since these packets are all composed of higher mode numbers, one should expect, by the correspondence principle, that their behavior should emulate that of a particle bouncing between two walls but it does not.