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, Δ*x*, 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.