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Physics Tutorial: Quantum Physics and Billiards

Quantum physics takes us away from the materialist determinism of classical physics and returns chance and serendipity to our outlook.

Physics of Billiards

The game of billiards as we know it says a lot about our understanding of the physical world and perhaps

our place in it. Normally we see billiards as a game in which our control of the outcome shows our level of skill. We firmly believe that we can get the billiard balls to do stunts if we know how strike the ball with the cue stick in just the right way. We would expect a physicist to be able to predict with precision the path of a ball or of several balls. We would expect the agreement between the predicted paths and the actual outcome to depend only on our ability to make a reasonable measurement of the mass, position and velocity of each of the balls on the table.

Classical Physics of Billiards

Our Newtonian view of an orderly mechanical universe leads us to apply cause and effect with confidence, knowing that the outcomes over time are predetermined by our actions now. The harder we hit a ball, the greater the energy we impart to it, and the faster it moves. Billiard balls without horizontal spin will sit still, or roll in a straight line with constant speed, until interfered with. Although mathematically more complicated and harder to measure, balls with a horizontal spin (called “english” or “side”) will curve in a predictable manner. In a head-on collision between two balls of equal mass, momentum is transferred; one ball taking velocity from the other. In a glancing collision between a moving ball with no english and a stationary ball, the balls part at right angles. Even something as complex as the break can be said to agree with thermodynamics: systems tend toward more diffused energy. Our confidence in causality, the laws of motion, conservation of momentum, conservation of energy, and thermodynamics lead us to believe in our ability to compete for domination among ourselves and with nature. We feel snubbed or cheated if things don’t turn out in a predetermined way.

Quantum Physics of Billiards

Billiards at the atomic or subatomic level, however, is a different game. Things change when observed, and there are several possible outcomes for any event. Certainty is replaced by probability, reliability is replaced by chaos, and quantization rules. In this world of the very small, if an electron was the size of a grain of sand, a proton would compare to a billiard ball, and the atom (our new billiard table) would larger than an arena. This is the realm of quantum physics.

Quanta

Quantum mechanical billiard balls would be finicky. They could only accept certain amounts of energy

because they could only exist with certain amounts of energy. The difference between classical and quantum billiard balls is like the difference between climbing a ramp compared to climbing a flight of stairs. On a ramp, with a little more energy a climber can go a little bit higher. On stairs, a little more may not be enough to get to the next level, and the climber falls flat on his face. In Einstein’s terms, our billiard balls would only absorb discreet bundles (quanta) of energy. If not struck hard enough, they would not move at all. If struck too much, they might release the excess energy in the form of radiation. To make matters worse, even if they were struck with just the right amount of energy, sometimes they would move and sometimes they would not. Place a large number of quantum billiard balls on a table, supply appropriate energy, and only a certain number of the balls will move. Quantum physics would predict how many would move, but not which ones. Our ability to compete in such an environment now depends on our gambling skills.

Heisenberg Uncertainty Principle

In this new game one constant we can depend on with mathematical precision is uncertainty:

Planck's constant = (uncertainty in energy) x (uncertainty in position)

Quantum physics billiard balls are elusive. One can not define both the position of the ball and the momentum of the ball at the same time. The more certain you are of the position of the balls, the less certain you are of their masses and velocities. If you are very certain of the mass and velocity of a ball, you can not at the same time know where it is. Taking a shot is more like a shot in the dark. The billiard table surface is constantly changing, and any one of the balls may be focused at one point then at the next moment be smeared out over a large area. This inability to observe reliably is not due to inadequate technology; it is a fundamental property the universe of the very tiny. Luckily we can see both the position and the velocity of a speeding car at the same time. Uncertainty is not an issue for cars, grains of sand, or strands of DNA because their masses are large, and Planck’s constant is extremely small. Even the mass of a protein molecule in a bacterium is astronomical compared to an electron. Quantum physics billiard balls are subatomic particles and photons and outside our normal day-to-day experience.

Wave-Particle Duality

Our ability to conceive and communicate concepts on the fringe of our experience is limited by the symbols (language) we use. “Particle” and “wave” are constructs we take from day-to-day experience. We run into problems trying to apply them at the subatomic level because “particle” and “wave” are not mutually exclusive in the realm of quantum physics. Our conceptual problem is not as much one of complex phenomena, as it is of inappropriate coding. In other words . . .

Fundamental particles (quantum billiard balls) of light and matter are described as particles (bundles, packets) with wave-like character. The amplitude of the wave measures the probability that the particle exists at a particular point. The wavelength of the wave is a measure of the particle’s momentum or energy. The shorter the wavelength, the higher the energy. A slow moving electron would have a long wavelength compared to a fast moving helium nucleus. Our quantum physics billiard balls are perhaps best described as wave-particles.

Since waves smear out over a distance, we should not be surprised at our inability to pinpoint a particle’s location. Since we can not specify the mass, velocity or position of the wave-particle, we can not expect to specify exactly where our quantum billiard ball will end up when we give it a whack. The best we can do is calculate the probabilities of it ending up at different places, such as in or around the side pocket. This has nothing to do with the shooter's skill or lack of skill. Uncertainty is simply a fundamental property of this realm.

Suppose we were to shoot millions of photons (like bullets from a machine gun) through two slits toward a target. What follows is a rather amazing result. The pattern on the target matches the probability distribution predicted by quantum mechanics. Areas of high probability are brightly lit, areas of low probability are dark.

Quantum Tunneling

Now let’s attempt a bank shot being careful not to give the billiard ball enough energy to hop over the cushion. In our everyday experience, the path of the reflected billiard ball is precisely predicted from the angle at which the ball hits the cushion, and the elasticity of the padding. However, our quantum physics billiard ball has a probability of leaking through (not over) the reflective barrier. A wave-particle can appear on the other side of an energy barrier without going through the barrier and without damaging the barrier. Instead of bouncing off the cushion, our quantum billiard ball could appear on the other side of the cushion. We do not observe this in an ordinary pool hall because the mass of everyday billiard balls is so much larger than Planck’s constant. The miniscule mass of our wave-particles creates large uncertainties in position which is another way of saying a high probability of being smeared out. This is an everyday occurrence for electrons and is the basis of operation of solid state technologies such as computer chips.

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