The quantum mechanical tipping pencil—a caution for physics teachers

Abstract

Since at least as long ago as 1960, quantum mechanics has been applied with varying degrees of rigour to the problem of a pencil balancing on its point. Although the analysis of the problem may have some merit in terms of solving for the evolution of and the final state of a similarly shaped (rod-like) but microscopic object, it is misleading when students are given the idea that quantum mechanics is useful in describing the behaviour of macroscopic objects.

The quantum pencil revisited

“…We analyze a model problem [2] and solve it fully in a small angle approximation to the time dependent Schrödinger equation (henceforth denoted TDSE). We conclude with Easton [3] and with Lynch [4] that quantum effects due to balance stability are not observable in this situation, and also in fact, we show that the application of the HUP can be easily misinterpreted. The double slit experiment [1] seems to remain as the nearest contact with an observable, 'hands-on', quantum phenomenon.…”

“…This is realistic, but not the case implied by Dicke and Wittke [1]. Studies here include Shegelski et al [8] and the application to the pencil case by Easton [3]. The quantum solution can also be obtained by path integral procedures [9], which give corresponding results.…”

“…In possession of this simple fact, one begins to suspect that quantum effects might be observable, but only if it were possible to prepare the initial alignment. At least two researchers have cautioned of this situation [3,4]. The real chemical and physical effects mentioned earlier that are neglected in the ideal model are certain to become important as dimensions approach and fall below atomic size, which can either increase or decrease the stability.…”

“…The initial position and momentum preparations, Δr = r(0) = w o and Δp = ṙ(0)M = ẇo M, are set in equation (5). The solution for radial motion, equation (3), is now used to find the fall time for these initial conditions. The hyperbolic functions can be replaced by the single dominant exponential because of the large argument.…”

“…It is certain that Dicke and Wittke did not intend their quantum mechanics exercise [1] to explore these perturbations. We must mention that the analysis of Easton [3] is based on an exploration of some of these real effects and obtains a result with a balance time depending on point geometry and quantum tunneling. Lynch's analysis [4] follows the spirit of the original problem of Dicke and Wittke and parallels ours.…”