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American Scientist

The Way the Ball Bounces

Brian Hayes

Published Jul 1, 1996• 1 citation
OpenAlex

Abstract

The billiard ball rolls smartly along the baize, aimed for another ball at the far end of the table. The player watching its trajectory anticipates the solid tlock! of contact, to be followed by the familiar textbook demonstration of the conservation of energy and momentum. But something has gone awry on this billiards table. The moving ball passes right through the stationary one, without altering its own motion or disturbing the repose of its target. And then the rolling ball glides wraithlike through the rail at the edge of the table and continues imperturbably sailing across the room, with no inclination to fall to the floor. What’s going on here? It’s not quantum tunneling;

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The billiard ball rolls smartly along the baize, aimed for another ball at the far end of the table. The player watching its trajectory anticipates the solid tlock! of contact, to be followed by the familiar textbook demonstration of the conservation of energy and momentum. But something has gone awry on this billiards table. The moving ball passes right through the stationary one, without altering its own motion or disturbing the repose of its target. And then the rolling ball glides wraithlike through the rail at the edge of the table and continues imperturbably sailing across the room, with no inclination to fall to the floor. What’s going on here? It’s not quantum tunneling;
One day Mr Tompkins was going home, feeling very tired after the long day's work in the bank, which was doing a land office business. He was passing a pub and decided to drop in for a glass of ale. One glass followed the other, and soon Mr Tompkins began to feel rather dizzy. In the back of the pub was a billiard room filled with men in shirt sleeves playing billiards on the central table. He vaguely remembered being here before, when one of his fellow clerks took him along to teach him billiards. He approached the table and started to watch the game. Something very queer about it! A player put a ball on the table and hit it with the cue. Watching the rolling ball, Mr Tompkins noticed to his great surprise that the ball began to ‘spread out’. This was the only expression he could find for the strange behaviour of the ball which, moving across the green field, seemed to become more and more washed out, losing its sharp contours. It looked as if not one ball was rolling across the table but a great number of balls, all partially penetrating into each other. Mr Tompkins had often observed analogous phenomena before, but today he had not taken a single drop of whisky and he could not understand why it was happening now. ‘Well,’ he thought, ‘let us see how this gruel of a ball is going to hit another one.’
One day Mr Tompkins was going home, feeling very tired after the long day's work in the bank, which was doing a land office business. He was passing a pub and decided to drop in for a glass of ale. One glass followed the other, and soon Mr Tompkins began to feel rather dizzy. In the back of the pub was a billiard room filled with men in shirt sleeves playing billiards on the central table. He vaguely remembered being here before, when one of his fellow clerks took him along to teach him billiards. He approached the table and started to watch the game. Something very queer about it! A player put a ball on the table and hit it with the cue. Watching the rolling ball, Mr Tompkins noticed to his great surprise that the ball began to ‘spread out’. This was the only expression he could find for the strange behaviour of the ball which, moving across the green field, seemed to become more and more washed out, losing its sharp contours. It looked as if not one ball was rolling across the table but a great number of balls, all partially penetrating into each other. Mr Tompkins had often observed analogous phenomena before, but today he had not taken a single drop of whisky and he could not understand why it was happening now. ‘Well,’ he thought, ‘let us see how this gruel of a ball is going to hit another one.’
We started to investigate quantum mechanics by considering only the quantization of magnetic arrows. In our explorations we found out that the magnetic arrow had some funny properties (for example, it was possible that m x did not have a definite value), but at first it seemed that other properties, such as the position of an atom, behaved in the familiar classical way. Eventually (section 9.3) we found that it was also possible to have an atom without a definite value for its position. In this chapter, we investigate what happens when we apply quantum mechanics to a particle's position. Ball bouncing from a floor This chapter will show our framework for quantum mechanics in action, by applying it to the problem of a ball bouncing from a floor. Let us use a very fast ball, such as an electron, so that we can ignore the force of gravity. (We restrict ourselves to an electron that is moving fast on a human scale but slow compared to the speed of light, so that relativistic considerations don't come into play. Also, the magnetic arrow associated with the electron has no effect on the phenomena described in this chapter, so I won't mention it again.) Imagine a source of balls that could send a ball flying in any direction, for example a hot tungsten filament that boils off electrons. Suppose a ball begins at point P , bounces off the floor, and ends up at point Q . (Points P and Q are equally distant from the floor.