QMTools solves the eigenvalue problem posed by the [time-independent] Schrödinger equation. The eigenfunctions are the stationary state waves and the eigenvalues are the associated energies. For confined systems, the demands that the wavefunction be continuous and bounded everywhere limit the eigenvalues to a discrete set: finding those values is the subject of this section.

Finding Eigenvalues

1. Begin by creating a stationary state wave in the applet or the Applet Editor, as described in QM Waves; graph this wave (right-click in its formula field and select Plot Function from the popup menu).
2. The energy of this stationary state is represented by a variable assignment on the Math Palette. Right-click anywhere in the value field of this entry and select Edit Parameter... from the popup menu; this activates the Energy Editor.
3. The name of the energy variable, say E, appears to the immediate left of the energy input field near the top of the editor. Type a new value into this field and press enter to re-calculate the waveform using the new 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 an eigenvalue, the wave values also will agree at the match point; otherwise, a discontinuity results. The [fractional] discontinuity is recorded in the editor tolerance field, labeled δψ, and is visually evident on the graph.
4. A noticeable discontinuity in the waveform means a physically acceptable wavefunction is not possible for the chosen energy. To find a suitable energy, we must 'tweak' the energy and repeat the previous steps. This trial-and-error process is informed by the following considerations:
   The sense of discontinuity in the waveform at the matching point (i.e., which side is higher) indicates the direction in which the energy must be adjusted. Start by making relatively large corrections and watch for 'overshoot' (what was the high side now is low). Proceed this way systematically, making ever smaller corrections until no discontinuity is evident. The resulting energy is an approximate eigenvalue, and the associated waveform is the eigenfunction.
   Adjustments to the energy are conveniently carried out using the editor's slider control, which affects only the highlighted digits in the energy input field. The first digit highlighted can be moved left or right by clicking the arrows to the right of this field. If needed, extra digits can be added before or after the decimal by typing directly in the energy field. The sum of leading and trailing digits is limited to 9.
5. The search process just described may be automated by entering a non-zero value in the auto-search range field (labeled E±) at the bottom of the editor, and pressing the enter key. In this configuration, an automatic search is performed within the specified range of the value entered in the energy input field. An automated search also requires an acceptable level of error, which is specified in the editor's tolerance field (labeled δψ) (defaults to 10⁻⁶). If the search succeeds, the correct eigenvalue overwrites the value in the energy field, and the tolerance field is updated with the [fractional] wave mismatch at the new energy. An unsuccessful search results in the requested tolerance being displayed in red to indicate that the desired wave discontinuity was not realized.
   When an automated search finishes (successfully or not), the editor reverts to manual mode; as a consequence, after editing any search parameters, you must return to the auto-search range field and press the enter key to initiate another search using the new values.

Back to Top

Search Tips and Techniques

1. Bound states are characterized by the number of nodes they exhibit (the nth excited state has exactly n nodes). Even before the correct eigenvalue is reached, you should count the nodes in the waveform to be sure you are closing in on the desired state.
2. The precision with which eigenvalues are found is related to the location of the match point, the plot interval, and the number of data points at which the function is calculated. The match point is chosen automatically at the start of the editing session; subsequently, it can be reset by clicking the reset button , which re-positions the match point to an extremum of the current waveform. The extremum location assures the greatest accuracy, and thus is preferred during the final phase of the search.
3. The plot interval and number of data points can be changed from the GraphProperties Editor, activated by right-clicking anywhere in the graph background and selecting Graph Properties... from the popup menu.
   Some fluctuation in the eigenvalue can be expected as the plot interval is adjusted. For best results, be sure the wave penetrates several decay lengths into the classically forbidden region on either side of the match point.

Used carefully, QMTools can determine eigenvalues to an accuracy better than one-half of one percent.

Back to Top