Quantum Particle Wave Packet in a Parabolic Potential

In this version I include two features that were not included previously.
1. For a gaussian wave packet envelope, I plot the algebraic expression that agrees with the Time Dependent Schrodinger Equation (TDSE) solution of the width variation as a function of time.
2. The modeling of a three initial envelope shape wave packets in terms of the stationary states of the parabola.
The motivation for including these items was a book "Quantum Mechanics" 3rd Edition, 1998 John Wiley and Sons Inc. by Eugen Merzbacher The animation shows the propagation of a quantum wave packet in a parabolic potential well. 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) 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[cn*Hn(x)exp(-(i+1/2)*omega*t), {n,0,infinity}] where omega is defined below, Hn(x) are the Hermite polynomials, and cn are the coefficients needed to fit the initial wave packet. For reasonably small initial wave vectors, an upper limit of n=100 or so is sufficient. For this analysis we will let the Planck constant, hbar, 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*x 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 xc(t)=xMax*sin(omega*t) where xMax=k0/omega is the turn around value of x and where k0 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, sigma0, of the wave packet envelope. To show how the time dependent solutions relate to the stationary state solutions, I have provided a check box "Show Stationary State Plots" which plots the steady state solution, eigenfunction, of the parabolic well for the particle energy, En=1/2 k0^2 which is (n+1/2)hbar*omega. Note that the spatial frequencies of the moving wave packet and the eigenfunction are reasonably equal over the entire range, +/-xMax. I let sigma0=r*xMax where r is less than 1 and is selectable by a slider. I also provide sliders to select Initial Wave Velocity, 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. The equation that provides the width variation of a gaussian packet is given as dx^2=(c*dx0)^2+(s*dp0)^2 where c=cos(omega*t), s=sin(omega*t), dx0^2 is the position variance initial packet and dp0^2=1/(b*dx0^2) is the momentum variance of the initial packet. When the "gaussian" starting shape is chosen, this expression is plotted as a black trace along with the iterative or series time dependent solutions. Using a Chrome browser, the reader is welcome to press F12 on a Windows computer in order to examine the Javascript code that I have written.