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The modes are the eigenvalues and eigenvectors (psi(x))of the second order Schrodinger equation (SE)

`-(ℏ^2)/(2m)(del^2Psi)/(delx^2)+V(x)Psi=EPsi`

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 (divided by 2*pi), ℏ, equal to 1 Joule second and the mass, m, equal to 1 kilogram. Although the SE can also express the time dependence of the wave function, only the steady state eigenmodes will be computed here. 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 eigenvalues of the particular setup are depicted with small black circles at each eigenvalue. By placing the mouse on one of these circles and clicking, the eigenvector (spatial mode) of that eigenvalue will be drawn as a green trace. From the Schrodinger Equation solutions we obtain three important types of information about how a particle moves in the potential V(x).1. The energies, Ei, which are the eigenvalues of the stationary state modes.

2. The square of the amplitudes of the eigenvectors, vi, which is the probability density for the position, x, of the particle.

3. The spatial frequencies, k(x) Vs position x of the eigenvectors to which the particle momentum is proportional.

k(x) is constant and real inside the potential well but imaginary outside its bounds. Since k is constant, that would indicate that the speed of a particle that it represents would also be constant.

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. The time dependence of a given starting packet can also be obtained by use of a series of the stationary states with their respective time factor`psi(x,t)=sum_(i=0)^(i=oo)c_iPsin(x)exp[-i(E_i/ℏ)t]`

where E_i are the eigenvalues of the potential, `Psin(x)` are the stationary state solutions, and `c_i` are the coefficients needed to fit the initial wave packet envelope. For reasonably small initial wave vectors, an upper limit of i=20 or so is sufficient. 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. The Triangle, which has a slope discontinuity, remains intact for a cycle or two but it definitely loses its resemblance to its original shape. The Square, which has a value discontinuity loses its identity after only 1 cycle. The evolution of these shapes is in stark contrast to that seen in the case of a parabolic potential.

Since these 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 wall but it does not. Comparing this behavior with that in a parabolic potential, I would have to declare that this discrepancy is due the fact that a square well potential can not be contrived even in classic physics.

Further, I have grave doubts about the significance of the Kronig-Penney model which uses the finite well as a starting potential.