Demonstration Version
OSP-based Programs for Quantum Mechanics:
Time Evolution and ISW Revivals
Mario Belloni and Wolfgang Christian
(Davidson College)
A screen shot of a fractional revival of a wave packet in the
infinite square well (t = Trev/8) showing 4 mini-packet clones of the
original packet.
We have created a number of Java programs (both applications and applets) for the visualization of
quantum-mechanical time development, specifically quantum-mechanical revivals in the infinite square well. Since
in time-dependent quantum mechanics it is necessary to describe the wave function in terms of complex (real and
imaginary) functions, we use an amplitude and phase representation of the wave function. The amplitude is drawn
centered on the y-axis and is the height of the envelope shown from top to bottom. The phase, or phase angle, of
the wave function is shown as color and we often show a color-conversion image, as shown above, to help students
determine the phase of the wave function.
The program we use, SuperpostionApp,
inputs the expansion coefficients, cn, of a general quantum-mechanical superposition:

The program does so by separately inputting the real, Re[cn], and imaginary, Im[cn]
parts of the expansion coefficients, cn. An XML file, such stores these expansion coefficients
and other appropriate initial conditions for the program.
The wave functions are then calculated either numerically for any user-defined potential energy
function, V(x), or calculated analytically for the special cases of the infinite square well, the periodic infinite
well, and the simple harmonic oscillator. Depending on the analysis one wishes to perform, one of the following programs
based on QMSuperpositionApp (which itself only shows the wave function) can be chosen:
-
QMSuperpositionProbabilityApp: adds a view of the probability density
-
QMSuperpositionExpectationXApp: adds a view of the expectation value of x
-
QMSuperpositionExpectationPApp: adds a view of the expectation value of p
-
QMSuperpositionCarpetApp: adds a view of the position-space quantum carpet
-
QMSuperpositionMomentumCarpetApp: adds a view of the p-space quantum carpet
-
QMSuperpositionFFTApp: adds a view of the momentum-space wave function
The use of analytic solutions for each eigenstate allows the simulation to run for a long period of
time and yet never accumulate numerical error since the wave function is calculated anew at each time step. In addition,
within the OSP control, we can set the energy scale, which determines the time scale for the animation. For example,
setting the energy scale to 2/π forces Trev = 1, which is a convenient time scale for the study of
revivals and fractional revivals which occur at Trev and fractions of Trev,
respectively.
The applications described here can be organized using Launcher. Once Launcher is opened, the user can double
click on a node to run a program with the initial conditions stored in the associated XML file. Once the launcher
is opened, the user can launch a file which opens a control and two views of a given model.
From the control, an animation can be run or changed. In addition, using
the read command, any saved animation can be loaded and viewed. The
configuration that controls an animation is saved as an XML file that can itself
be changed.
The links on the right step you though the use of this set of programs.
In addition, a link is provided to a visual tutorial on wave packet revival
phenomena.
While most of the pre-set files contained herein have been
created to study wave packet revivals in the infinite square well, the
quantum-mechanical dynamics
of any system can be studied too.
© Mario Belloni and Wolfgang Christian (2004).