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

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

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

where h with the bar (ℏ) is the Planck constant, `h`, divided by `2pi`, m is the mass and E is the energy of the particular mode whose wave function is the eigenvector `Psi`. For the square well of infinite depth, the modes are

`Psi_n=sin(npix/w)`

where `n` is an integer from 1 to infinity and is the mode number, `w` is the width of the well and `x` runs between 0 and `w`. This leads to energies

`E_n=(nℏ/mpi/w)^2`

For this program I have chosen to set the Planck constant (divided by 2*pi), ℏ, equal to its actual value, 1.056e-34 Joule seconds and the mass, m, equal to that of an electron 0.911e-30. The animation shows the propagation of a quantum wave packet (gassian shown here):

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

in a infinitely deep square potential well. The wave packet that I use is a simple complex sinusoid embedded in various envelopes (see "Starting Wave Packet Envelope Shapes"). Here the time dependence of a given starting packet will be obtained by use of a series of the stationary states with their respective time factor

`Psi(x,t)=sum_0^ooc_npsi_n(x)exp(-iE_n/ℏt)`

where `E_n` are the eigenvalues of the potential, `psi_n(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 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 classic 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)`.

To show that the packet returns to its original shape and position after this period I have given the option of either Quantum Period or Classic Period. The Quantum Period is much longer than the Classic Period so the learner will have to ' be patient to see that the Quantum Wave Function is exactly periodic. 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 `2pi&hbar/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 spread out in the square well so its location is unknown. The packet spends most of its time on the flat portion of the potential. On this portion of the domain, the Schrodinger equation reads

`(dPsi)/dt=-(iℏ)/(2*mass)*(d^2Psi)/(dx^2)`

With our choice of `psi=exp(-ikx)`, the second derivative is just `(d^2psi)/(dx^2)=-k^2psi`. Therefore decreasing the value of `ℏ/(mass)` causes the change of `psi` (or wave packet speed) to be decreased by the same factor. It turns out that this decreases both the rate of motion of the centroid of the packet as well as the rate of spreading of the width of the packet by the same factor. i.e. `v_(packet)=v_(packet1)*ℏ/(mass)` and

`w_(packet)=sqrt(w_(packet0)+i*ℏ/(mass)*time)`

where `v_(packet1)` is the rate of speed of the packet when hbar/mass=1 and w_packet is the initial width of the packet which becomes a constant width when ℏ goes to zero. Recall that the velocity of a particle packet is related to its wave vector, k, as `v=(ℏ/(mass))k`. For a square well of width w, the lowest value of `k` is `pi/w`. Therefore the speed of all of the states of a square well scales as `ℏ/mass`. The width of the packet grows as

`sigma(t)=sqrt(sigma_0^2+i*ℏ/(mass)t)`

where `i=sqrt(-1)`. For the lowest mass "regular" particle, the electron, this ratio `ℏ/(mass)`is about 0.000116 `m^2/(sec)`. Reasonable choices of square well width would be that for an electron with one electron volt (eV) of energy. This results in widths that are commensurate with the orbitals of the Hydrogen atom. The average wave vector, `k`, for the wave packet should result in many waves within the square well width. A choice of average energy of the wave packet should then be a few hundred eV and that is what I have provided here.