## Curricular Design and Features

These exercises are intended to be used as a tutorial[i] that could be used in class as an interactive demonstration[ii] or as part of a hands-on laboratory exploring the role of the potential energy function in energy-eigenfunction shape.

The tutorial begins with a brief discussion of the theory and background at an introductory level along the lines suggested by French and Taylor.  These exercises have the following features.

Multiple Representations of the Energy: The current state’s energy is depicted numerically in a table, on an energy-level diagram with a red horizontal line (the rest of the energy levels are shown in green), and as an orange horizontal line on an energy diagram along with the potential energy function.  The energy-level diagram, besides allowing students to change state (see Designer Infinite Wells), provides a visual framework to understand the structure of energies for bound states.

Energy Diagram and Energy Eigenfunctions Plotted Separately:  Most textbooks, to save space, plot energy eigenstates (or even multiple states) on the energy diagram with the potential energy function.[iii]  This depiction can confuse students since while the horizontal axes are the same, the vertical axes are not.  This approach leads to a student-perceived vertical offset on the energy eigenstate that corresponds to the state’s energy, which incorrectly gets attributed to the state.  In other words, in this type of depiction, the energy eigenfunctions are not shown crossing the horizontal axis which may be partly responsible for students’ misconceptions regarding the energy loss in quantum-mechanical tunneling.[iv]  We plot the energy diagram (with potential energy function and the current state’s energy) on a separate graph from the energy-eigenfunction plot.

Ability to Change the Potential Energy Function with Sliders: Since only a few quantum-mechanics problems can be solved exactly, we use a standard numerical technique (the shooting method) to determine energy eigenfunctions based on a given potential energy function.  We then use sliders to change the potential energy function so that the resulting effect of this change can be immediately seen in the shape of the energy eigenfunctions.  By seeing a wide variety of situations students can come to understand energy eigenfunction shape.  This control over the parameters in the potential energy function also allows us to focus student exploration on the parameters that are the most valuable for a given exercise.

Ability to Change State by Selecting an Energy Level: This, like the ability to change the potential energy with sliders, allows students to quickly see the energy eigenfunction for a variety of states.

[i] L. McDermott and P. Schaffer and the Physics Education Group, Tutorials in Introductory Physics (Prentice Hall, Upper Saddle River, NJ, 2001).

[ii] D. Sokoloff and R. Thornton, “Using Interactive Lecture Demonstrations to Create an Active Learning Environment,” The Physics Teacher, 35, 340 (1997).

[iii] Note that we say potential energy function so as to avoid confusion with the electric potential as suggested by Kenneth Krane.

[iv] M.C. Wittmann, J.T. Morgan, and L. Bao, “Addressing Student Models of Energy Loss in Quantum Tunneling,” Eur. J. Phys. 26, 939–950 (2005).