## Classical Probability and Energy

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, EV(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.