# 10.13 The Uncertainty Principle

Imagine there is a ball somewhere in a completely darkened room. How would you find it? You could blunder around in the dark, but it is very likely you would bump into it and send it off in some unknown direction. A smarter way would be to search for it with a flashlight. However, light in the form of photons carries energy. So when the photons bounce off the ball, which allows you see it, the ball gains a tiny amount of energy. In the macroscopic world full of big objects, this tiny amount of energy carried by photons is negligible. As far as we are concerned, the flashlight lets us see the position of the ball as accurately as we want. But what about when we are looking for a tiny particle, instead of a ball? It turns out that we can no longer be sure of the position of the tiny object.

Searching for tiny objects with a light beam, the energy and momentum of the photons become significant. Light reflecting off the object gives it energy and so moves it slightly. Now if we are looking for a tiny subatomic particle, we might try to illuminate it with a single photon. But the photon hitting the particle moves it, and we no longer know where the particle is! So our knowledge of its position comes at the expense of our knowledge of its motion. We could be clever, and shine a photon of lower and lower energy at the particle. But at some point the photon, which must have a long wavelength if it has a low energy, becomes too big to interact with the particle. Thus, there is a fundamental uncertainty in the subatomic world: the act of measurement always alters the objects being measured. In the everyday world, this effect is too small to be noticed, but it dominates our understanding of the microscopic world.

Werner Heisenberg. Click here for original source URL.

Werner Heisenberg expressed the fundamental indeterminacy of measurement in a simple and elegant form. The equation describes a trade-off between measurements of time and motion. The product of the uncertainty in position times the uncertainty in velocity (or momentum) is related to the Planck constant. In equation form, it is stated

(Δx) x (Δv) x (Mass) ≥ h / 4π

Max Planck. Click here for original source URL.

where the Greek letter Δ is the commonly used symbol for the error or uncertainty in a quantity. In this equation, Δx is the uncertainty in the position in one direction, Δv is the uncertainty in the velocity in that same direction, and h is the universal constant named after the early 20th century physicist Max Planck. The equation is an inequality, saying that the product of the uncertainty in the position and the uncertainty in the momentum is about equal to or greater than a very tiny number.

What does the mathematical form of the Heisenberg uncertainty principle mean? It means that we can know the position of a tiny particle, but only at the cost of not knowing its motion at all. Or we can know its motion, but at the cost of not knowing where it is! The amount of the uncertainty is given by Planck's constant, a tiny number that has little or no effect in the everyday world.

Suppose that a pitcher throws a 90 mile per hour fastball (about 40 meters per second). The radar gun measures its speed as it passes through the 10 centimeter beam of radar. What is the quantum uncertainty in the speed of the baseball? Rearranging the equation above, the uncertainty in speed Δv = h / (4π x Mass x Δx). In matching units, the position uncertainty is 10 centimeters or 0.1 meter, the mass is about 0.3 kilograms, and h = 6.63 x 10^{-34} Joule seconds. So, Δv = 6.63 x 10^{-34} / (12.6 x 0.3 x 0.1) = 1.7 x 10^{-33} meters per second. In other words, we could theoretically measure the speed of the fastball to thirty-two decimal places! The quantum uncertainty plays no role in this measurement.

How does a tiny object behave? How accurately could we measure the speed of an electron in a hydrogen atom? An electron has a mass of 9 x 10^{-31} kilograms, and a hydrogen atom has a rough size of 10^{-10} meters. Working as before, Δv = 6.63 x 10^{-34} / (4π x 9 x 10^{-31} x 10^{-10}) = 5.8 x 10 ^{5} meters per second. Our uncertainty is larger than the velocity of light! This means that we have no knowledge of the electron's velocity once we have determined that it is in a hydrogen atom.

Heisenberg believed his result expressed a fundamental aspect of nature; it has nothing to do with poor observations or imperfect measuring equipment. There is a limit to the precision with which we can know the physical world. The uncertainty principle is not just an exotic theoretical idea. It is demonstrated in practice every day in the physics laboratory. Measurements of subatomic particles are always limited by the Heisenberg uncertainty equation. In terms of measuring position, the uncertainty principle implies an unavoidable "fuzziness" to the subatomic world.

Heisenberg’s uncertainty principle has larger philosophical implications. The subject of astronomy contains many examples of how science works to gain knowledge. Scientists have explored space, delved into the world of the atom, and opened up the electromagnetic spectrum. We know far more about the natural world than we did one hundred years ago. Surely this long march of progress will just continue? Heisenberg’s uncertainty principle places a limit on knowledge. Some scientists do not like the fact that there is a limit to our knowledge, a veil beyond which we cannot see. Other people are reassured, because it means that determinism has no place in physical theories. In other words, since the behavior of atoms is not completely predictable, we can never predict the future behavior of any physical system with certainty. Either way, there is no escaping the strange world of the quantum