Motion of Particle Wave Packet in a 1D Square Potential Well of Infinite Depth

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`

(ℏ) is the reduced Planck constant, `m` is the mass and `E` is the energy of the particular mode whose wave function is the eigenvector. For the square well of infinite depth, the modes are

`Psi_n(x)=sin(npix/w)`

where n is an integer from 1 to infinity and is the mode number and w is the width of the well and x runs between 0 and w. This leads to energies E_n=1/2*(n*π/w)² 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. In a following animation I will use the actual values of ℏ and for the mass I will choose that of an electron so that the learner can see the evolution for actual elementary particles. The animation shows the propagation of a quantum wave packet

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

in a square potential well. The wave packet that I used 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)

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

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_0^ooc_nPsin(x)exp(-i(E_n/ℏ)t)`

where En are the eigenvalues of the potential, `Psin(x)` are the stationary state solutions, and `c_n` are the coefficients needed to fit the initial wave packet. For reasonably small initial wave vectors, an upper limit of n=20 or so is sufficient. I give the learner the choice of 3 initial 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 classic period 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 classic period. 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 walls but it does not. For the parabolic potential, the motion of the wave packet centroid exactly matches the motion of a particle in the same potential and is therefor periodic at the same period. For the square well potential, the motion of the packet centroid starts out approximately matching the particle motion but with a lag. However, since the time dependence of the packet is `exp(-i(En/ℏ)t)` it turns out that the packet motion is indeed periodic with a period `t_P=2piℏ/E_1` where `E_1=1/2*(pi/w)^2`. Thus, although the wave packet propagation in a square well is not periodic with the same period as an equivalent classic particle, it is periodic with the period `(2piℏ)/E_1` where `E_1` is the energy of the lowest stationery state. This period, which I call a quantum period, is usually much longer than that of a classic particle. At times distant from the start and end of a quantum period the centroid of the packet stays reasonably well centered in the square well so its location is unknown.