To inform our expectation for the shape of energy eigenfunctions, we can partly rely on concepts from classical mechanics. We begin by constructing a classical probability distribution.[i][ii]. This construction is a classical probability map of time spent[iii] in a region of space, Δx, vs. position. The time a classical particle spends in a region of space is inversely related to its speed (the faster an object is going the less time it spends in a given region of space). Because the speed is related to the particle’s kinetic energy, it is also useful to construct a classical energy diagram, since the kinetic energy can be determined from the difference between the total energy and the potential energy, E − V(x). This can be demonstrated as an experiment using carts and motion sensors as described in Refs. 15 and 16, or as an animation using a motion diagram from kinematics as shown in Figure 2.
See the interactive exercise by following this link: Time Spent.
Figure 2: A classical probability distribution for a particle in an asymmetric infinite well. Beneath the probability distribution is a motion diagram showing the motion of the classical particle. Shown on the right is a classical energy diagram for the same scenario showing the change in potential energy (green) and hence the change in kinetic energy (blue) of the particle when x > 0, while the total energy (orange) remains the same.
[i] This initial focus is suggested in L. Bao and E. Redish, “Understanding Probabilistic Interpretations of Physical Systems: A Prerequisite to Learning Quantum Physics,” Am. J. Phys. 70, 210-217 (2002).
[ii] M. C. Wittmann, J. T. Morgan, and R. E. Feeley, “Laboratory-tutorial Activities for Teaching Probability,” Phys. Rev. ST Phys. Educ. Res. 2, 020104 (2006).
[iii] R. W. Robinett, Quantum Mechanics: Classical Results, Modern Systems, and Visualized Examples, Oxford, New York, 1997.