The applet above shows the potential energy for an electron confined to a finite square well of width 0.200 nm and height 100 eV (these values are recorded on the Formulas tab, along with the electron mass m = 511 keV/c²). The listing to the right of the graph includes placeholders for three stationary states for the electron in the well. In this exercise we will find the two lowest-lying bound states and determine the total number of bound states this well can support. The exercise illustrates energy quantization, i.e., only certain energies are permitted for the electron, corresponding to states which satisfy the acceptability criteria for quantum wavefunctions.

• Show one of the waveforms by right-clicking its placeholder in the list, clicking the visibility icon beside the "Real" label in the Colors | Visibilities field, then choosing the "OK" button. The displayed wavefunction has a noticeable discontinuity. Since quantum wavefunctions must be everywhere continuous, the energy of this state cannot be one of the allowed values for the electron.
• Adjust the energy of this state upward to eliminate the discontinuity. Right-click on the list entry, then select "Change Energy". Use the slider to change the highlighted digits in the text field; the first digit highlighted can be moved left (right) using the up (down) arrows to the right of this field. [Extra digits can be added before or after the decimal by typing directly in the text field.] The button at the lower right resets the matching point, and should be used when nearing a correct energy --see Technical Notes below.] When no discontinuity is evident, the energy is "allowed" and the wavefunction is one of the bound states for the electron in this potential well. Count the number of nodes for the wavefunction to see which bound state you have found. Finish by selecting "Close", then "OK" to close all dialogs with the current settings.
• Repeat the above procedure for the second placeholder in the list. Search for and display the acceptable states for the lowest and next-lowest energy; these are the ground state (no nodes) and first-excited state (one node), respectively. Note the different symmetries for these two states: the ground state is symmetric about the midpoint of the potential well, the first-excited state is antisymmetric about this point.
• Repeat the procedure once more for the third placeholder, this time searching for an acceptable wavefunction with the highest possible energy, but still "localized" in the well. Count the number of nodes to discover which excited state you have found. Adding one to this number (for the ground state) gives the total number of electron bound states this particular well can support.

 Final form of applet...

1. The Numerov method is used to contruct stationary waves for a given (trial) energy. Using boundary conditions derived from the correct asymptotic form first at the left endpoint, and then again at the right endpoint, the Schrödinger equation is integrated inward to a preset match point. The right-side wave is then scaled to make its slope agree with that of the left-side wave at the match point. If the trial energy is allowed, the wavefunction values also will agree at the match point; otherwise, a discontinuity results. The [fractional] discontinuity is recorded in the tolerance text box labeled δψ. The technique is most accurate when the match point coincides with an extremum of the wavefunction. Each press of the reset button  re-positions the match point to an extremum of the current waveform.