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First I should state that the motion of a mass in a parabolic potential well is, by far, the most prevalent macroscopic motion seen in nature (I purposely exclude motions of the electrons of atoms in this statement since these are nanoscopic and the potentials for their lower energy states cannot be modeled by parabolas.) . Almost all potentials with a smooth minimum can be simulated by a parabola at least over a short range around the minimum. Again, over a short range near the potential minimum, all motions of nucleii in solids can be simulated as motions in a parabolic potential well. This is the basic assumption of all stress modeling of solids and therefore the basis of all modern structural engineering. What is striking to me is that, almost a century since the advent (1926) of the Schrodinger equation, I have not seen any serious attempt to model the propagation of a quantum particle's wave packet in a simple parabolic potential. The present animation will do a lot to rectify this situation. The animation shows the propagation of a quantum wave packet

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

in a parabolic potential well. The wave packet is a simple complex sinusoid embedded in a gaussian envelope. 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. For this analysis we will let the Planck constant, ℏ, and the mass, m, of the particle both equal 1. FEA clearly shows that the wavepacket's propagation is strictly periodic with a period defined by the coefficient of the quadratic potential `V(x)=(b/2)x^2` so that the period is 2*pi/omega where omega=sqrt(b). Note that `V(x)` is the potential energy associated with a mass and linear spring system where the spring constant is b. The behavior of the wave packet is another manifestation of the Bohr Correspondence_principle. Just as a classsic particle moves in a parabolic potential well, the centroid of the magnitude squared of the wave packet moves sinusoidally with function `x_c(t)=x_(Max)sin(omega*t)` where `xMax=k_0/omega` and where `k_0` is the initial wave vector, `k`, of the packet. The Schrodinger equation requires that the integral of the wave packet's magnitude be constant but its amplitude and width can vary and these will be shown by the animation. A very important distinction between what may be expected from classic physics and what happens in quantum physics is the effect of the initial width, `sigma_0`, of the gaussian envelope. When the envelope width is very small compared to the maximum range of the particle, its width grows while its amplitude decays always keeping the integral of the magnitude squared constant as required by the Schrodinger equation. On the other hand when the initial width of the gaussian envelope is a fair fraction of the total range of the particle, the width gets much narrower at the limit of the range while the amplitude grows, again keeping the integral of the magnitude squared constant. Basically, for this latter case, the wave packet becomes very similar to the steady state solution for its own total energy. In order to demonstrate that this is the case, I have also provided a check box "Show Hermite Plots" which plots the steady state solution, eigenfunction, of the parabolic well for the particle energy, `En=1/2 k_0^2` which is `(n+1/2)ℏomega`. Note that the spatial frequencies of the wave packet and the eigenfunction are reasonably equal over the entire range, +/-`x_(Max)`. I let `sigma_0=rx_(Max)` where `r` is less than 1 and is selectable by a slider. I also provide sliders to select Initial Wave Velocity and Parabolic Potential Coefficient, Time Increment for solving the TDSE, and the total number of grid points (Gridding Range). I cases where the quality of the plots seems less than desirable, the learner might want to experiment with these latter two sliders. After a full period of propagation, I provide an option to plot the position and momentum variances for the prseent set of parameters. As a check on the solution accuracy, included in these plots is that of the integral of the magnitude of `Psi"*"Psi`.