On a Quantum Toy

Virtually all physicists are familiar with a toy consisting of an array of pegs oriented orthogonal to the gravitational field. When a large number of balls are dropped, one at a time, onto the top middle peg, the balls bounce from peg to peg downward until they collectively end up in a Gaussian curve at the toy’s bottom.

This raises the interesting question of why the balls don’t all end up at the same spot in the array’s bottom. Axiomatic to classical theory is the requirement that, if the initial conditions are identical at the start of each "drop," then all the balls should end up at the same position at the conclusion of a "run" of any number of drops.

In discussing initial conditions, let us suppose that each peg is perfectly rigid and hence always free of any vibration. And let us suppose the same ball is repeatedly dropped from a non-spinning state of rest at the same position above the middle top peg. We of course assume that gravity is constant. Classically each drop should trace an identical path on its trip to the toy’s bottom. Yet in practice (to the author’s knowledge) no one has ever achieved this singular result. When N (the number of drops in a given run) is large enough, one always obtains the Gaussian result.

The determinist may argue that the failure to obtain a singular result lies in the fact that, no matter how careful we are, we can never attain the required initial conditions. There will always be small uncertainties in the release point of the ball, the vibration of pegs, etc. But Poincare (if still alive) might argue that the impossibility of ever attaining identical initial conditions is a law of nature!

Indeed such a law was stated by Heisenberg when he proposed his Uncertainty Principle. According to that principle it is fundamentally impossible to simultaneously set the exact momentum and position of an observable physical object. In the case of subatomic particles this uncertainty may be relatively large (e.g. the diameter of an atom). But it is fashionable to suggest that, in cases of macroscopic objects (e.g. our ball and pegs) the uncertainties may be too small to measure.

This may be true. But the realities of our toy suggest that the effects may be greatly magnified. Let us consider the ball dropped onto the topmost peg. We might assume that this peg is perfectly rigid and at rest at all times. And we might assume that the ball is perfectly round, does not vibrate, etc. Classically, if the ball is precisely above the peg and its momentum is truly zero at drop time, then it should rebound exactly upward after colliding with the peg. But of course even if this may appear to be the case, following the initial collision it quickly becomes clear ( during subsequent collisions) that the ball has a momentum in one direction or another, since it eventually clears the topmost peg and arcs down toward the second tier of pegs.

In practice we cannot guarantee that the pegs are not vibrating when struck, again owing to the Uncertainty Principle if nothing else. The bottom line is that, all other things being perfect, the Uncertainty Principle guarantees the Gaussian outcome of a run when the number of drops in each run is adequately large. In a "perfect" world our toy can always be considered, at least in part, to be a manifestation of the Uncertainty Principle at work on a macroscopic scale!