This java applet is a quantum mechanics simulation that shows the behavior of a particle in a one-dimensional crystal (or lattice, or periodic potential). Periodic potentials are used to model electron behavior in a crystal.

At the top of the applet you will see a graph of the potential, along with horizontal bands showing the energy levels. By default, the potential is an array of square wells (the Kronig-Penney model). Below that you will see the probability distribution of the particle's position. Below the particle's position you will see its band structure, displayed as a dispersion curve; this is a graph of energy versus the crystal momentum k, with the energy bands shown as gray in the background. The particle's energy is shown as a red horizontal line.

To view a state, click on one of the energy bands in either the potential graph or the dispersion curve.

You may select a different potential from the Setup menu at the top right. You can also adjust the parameters of the potential (like the width of the wells) using the sliders on the right.

Between each graph is a horizontal line which may be dragged up and down to adjust the size of each graph.

The Setup Popup allows you to select a predefined potential. The choices are:

• Finite Wells: this is an array of square wells (the Kronig-Penney model). The width of the wells are adjustable.
• Well Pairs: this is an array of pairs of wells. You can adjust the separation between each pair of wells, and the width and depth of each well.
• Coupled Well Pairs: these are square wells with an adjustable wall between them.
• Harmonic: this is an array of harmonic oscillators.
• Coulomb-Like: this is vaguely similar to a coulomb potential, except that it has finite depth.
• Free Particle: this is a constant zero potential.
• Sinusoidal: here, the potential function is a sine wave.

The Mouse Popup determines what happens when the mouse is clicked. The choices are:

• Set Eigenstate: if you click on the energy or dispersion graph, then the particle will be put in an energy eigenstate.
• Edit Function: this allows you to edit the potential energy graph by clicking and dragging with the mouse. By doing this, you can define a custom potential.

The Ground State button selects the ground state wave function.

The Stopped checkbox stops the evolution of the wave function.

The Simulation Speed slider changes the speed of the wave function evolution.

The Particle Mass slider changes the mass of the particle.

The # of Wells Shown slider changes the number of wells displayed. There are an infinite number of wells extending off to the left and right, but the applet shows a finite number of them.

The Energy Scale slider lets you display more of the energy spectrum by adjusting the scale used to graph energy.

The View Menu has the following items:

• Energy: show the energy/potential graph (on by default)
• Position: show the position graph (on by default)
• Momentum: show the momentum graph
• Bloch Function: show the Bloch function (u).
• Dispersion: show the dispersion graph (on by default)
• Wave Function: display the wave function in one of four ways: as a probability (magnitude squared), as a probability with the phase shown using colors, as real and imaginary parts graphed separately, or as a magnitude with the phase shown using colors.

The Zones Menu allows you to specify which zone scheme to use when displaying the dispersion curve.