This page will help you understand how to use Dr. Bongo's Stationary State Lab. Additional help can be found on the Entering Functions help page.
This program lets you enter an arbitrary function to describe a 1-D potential V(x). It can then calculate the energies and wavefunctions of the lowest lying stationary states. In various exercises you may use Dr. Bongo's Stationary State Lab to help you develop intuition for what the energy eigenstate wavefunctions look like in an arbitrary potentials, to help you see how the energies of stationary states scale with n, to explore covalent bonding, or to investigate band theory.
Note that all numbers in this program are assumed to be in SI units; units should not be entered into any of the boxes, since it will just confuse the simulation.
The layout of the applet is divided into three columns.
The left column contains input fields where you can enter the potential V(x), the mass of the particle, and the principle quantum number of the displayed wavefunction. This column also has a buttons to reset the applet and to calculate the energies and waveforms using the entered parameters. Each field in the left column is explained below.
"Reset All" Button: This resets every parameter to its default value.
Principle Quantum Number (n) Box: The simulation only displays one stationary-state wavefunction at a time. Use this box to select which stationary-state to display.
Potential V(x) Box: In this box you enter a function which represents the potential energy which confines the particle. The simulation uses this function to calculate the potential from x=-1 to x=1. To make the simulation work, the program assumes that the potential goes to infinity outside of this range. For information on how to enter functions, see the Entering Functions page.
mass/hbar^2 Box: In this box you enter the mass of the particle. You may need to adjust this in order to get good solutions. Higher mass makes the stationary states "settle" deeper into the potential, lowering their energies.
Recalculate Button: When you make changes to the mass or the potential they are ignored until you press this button. When you press this button energies and wavefunctions are recalculated using the current potential function and mass.
The center column contains two plots. In the upper plot the potential V(x) and the wavefunction of the nth stationary state are plotted. The y axis is marked in energy units. The wavefunction's amplitude is arbitrary --- it is scaled to look nice on the scale used to fit the potential curve. The "y=0" line for the wavefunction is set equal to the energy of the state to make it apparent where the kinetic energy of the wave is negative. The lower plot displays the energy of the lowest lying stationary states vs. the principle quantum number n.
In the rightmost column the energy and the expectation value for x for the nth stationary state is shown. Various warning messages are also shown in this column when necessary.