Solutions of the time-independent Schrödinger equation

in regions where particles are free to move and not subject to forces, or V(x)=0 and E>0, are of the form

which makes the complete solution including the time dependence

A single wave solution is associated with a continuous flux of particles in the positive or negative x direction, with momentum p and energy E. Solutions that are linear combinations of multiple waves form wave packets that are associated with single particles.

Or in the continuous limit

Look at Gaussian wave packets where

The parameter *k*_{0} is the center wave number
of the distribution.
The parameter *a* controls the width of the Gaussian distribution
in *k* space around *k*_{0},
the larger the value of *a*, the thinner the distribution.
The discrete number of waves that will be summed by the application is also
a parameter to set. You may see how many discrete waves need to be included
in the packet sum to achieve a desired localization in space when all the
wave phases line up.

You may start the time simulation by pressing the "Run" button, and see the wave packet propagate in the positive x direction. Running the simulation for a long time also allows you to observe the phenomenon of wave packet spreading or dispersion.

There are three plot modes to choose from, the default where you can
see both the real and imaginary components of the complex valued wave
packet. Also available is a view of the magnitude squared of the wave
function, which is of course a measure of the probability distribution in space
of the particle. The first plot mode shows the distribution of the discrete wave
numbers in *k* space, directly controlled by the parameters
*k*_{0} and *a*.

While the wave packet is propagating during a simulation, statistical computations are also run on the mean and variance of the wave number and position distributions. These computations are displayed across the top of the plot in real time, along with the product of the variances. The Heisenberg Uncertainty Principle states that the product of the variances must be at least 1/4. Watching this computation as the time is advanced allows you to verify this fundamental law of Quantum Mechanics, and quantitatively observe the wave packet spreading.

Run from a shell or command terminal in the download directory with:

java -jar WavePacket.jar

This application lets you explore the construction of quantum wave packets with various adjustments of the pivot forcing parameters. You may adjust the various parameters controlling the simulations by either moving the sliders or by typing in a numerical value and hitting the Enter key.

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