# Basic Concepts: Measurement

It’s been a while since I did a “Basic Concepts” post. They tend to take a good bit of effort, and things have been hectic enough at work that I haven’t had the energy. To make up for the blizzard of basketball-related stuff over the last week (with more to come), here’s a look at the concept of measurement.

This one is almost more philosophy than physics, and there is a distinct possibility that I’ll accidentally say something really stupid. The question of measurement in physics, particularly modern physics, is a very subtle and fascinating one, though, and it’s worth a little discussion.

The one-sentence descriptions would be something like:

Measurement is the process by which we determine the properties of the world around us.

That sounds kind of obvious, but it’s more literally true than you might think on first reading.

The obvious reading of that is pretty much consistent with the idea of measurement in classical physics (“classical” here means roughly “before 1900”). In the classical view, the world around us has definite properties, and we can make quantitative observations of those properties which we call measurements. Before we make a measurement of some physical quantity– say, the mass of the book sitting on the desk in front of me– the exact value of that mass is not known to us, but it has a definite value. Classical measurements, then, serve only to reveal the pre-existing and absolute properties of the world around us.

Note that this does not mean that classical measurements are perfectly certain. There is still uncertainty in classical measurements, because there are always factors beyond the control of the experimenter that limit the precision with which a quantity can be determined. When I try to measure the mass of the book sitting in front of me, I’m limited by the resolution of the scale that I use to make the measurement. If I get a better scale, then I may need to worry about air currents in the room upsetting the reading, or the local mass distribution changing the magnitude of the gravitational force on the book, and so on.

In classical physics, the properties of objects are pre-existing and absolute properties, and it’s possible in principle to measure them to infinite precision, but in practice, real measurements are always limited. There’s always an uncertainty associated with a real measurement, reflecting the uncontrolled variables that may have affected the measurement, but that uncertainty is the result of experimental limitations, not anything fundamental. In principle, any observer provided with the appropriate apparatus would get exactly the same value, as the measurement is just revealing properties that are fixed in advance.

This concept of measurement starts to fall apart with the introduction of Special Relativity. Relativity tells us that the results of some very fundamental measurements will be different according to different observers. In particular, moving observers will disagree about the size of objects, and the timing of events.

This is a very disturbing idea, philosophically. It’s not a simple matter of error in measurement, either– Relativity tells us that an observer moving relative to an object will measure its length to be shorter than the length seen by an observer at rest relative to that object, and there is absolutely no measurement that the moving observer can do that will give any other result. It’s not a limitation of the apparatus used for measuring the length, it’s a limitation due to the structure of the universe we live in.

In a sense, Einstein’s real achievement with Relativity. He didn’t come up with the mathematics of Relativity out of thin air– the basic ideas of length contraction and time dilation were around, and the math had been worked out by Lorentz and FitzGerald and Poincare and others. They hadn’t caught on because nobody really bought into the idea that the properties of objects were not fixed and absolute. Einstein’s main contribution was to show that this is how the universe has to be– that you can’t simply assume that all observers measure things the same way, but you need to think about the physical processes involved in making measurements, and the limitations placed on those processes by the structure of the universe. If you’ve read older textbook descriptions of Relativity, this is what’s going on with all the tedious business about infinite lattices of clocks and meter sticks, and discussions of how to synchronize spatially separated clocks– the point is to demonstrate that when you think about the details, there’s no physical way to make measurements in such a way that observers moving relative to one another will agree about the size of objects and the timing of events.

Relativity was a serious blow to the classical conception of the meaning of measurement, but quantum mechanics absolutely buried it. Quantum mechanics really delivers a sort of one-two punch to the classical worldview: first, there’s the Uncertainty Principle, which says that the structure of the universe places strict limits on what kinds of things we’re allowed to know, and after that, there’s the whole problem of superposition and the active role of measurement.

The Uncertainty Principle is the more straightforward of the two. You can see the most famous form of the uncertainty principle at the upper left of the banner for this weblog: it says that the product of the uncertainty in the position of a particle and the uncertainty in the momentum of that particle must be greater than or equal to Planck’s constant over two. The lower limit is a very small number– less than 10-34 in SI units– but it’s emphatically not zero.

The usual introduction to this idea comes through thought experiments that demonstrate that attempts to measure the position will change the momentum, and vice versa, but it’s actually a deeper statement than that. Like the differences between moving observers in Relativity, this is not a simple matter of doing insufficiently clever measurements: it’s an inescapable result of the deep structure of the universe.

The easiest way to understand the position-momentum uncertainty relationship is as a result of the wave nature of matter. In quantum theory, particles have wave-like properties, with the wavelength being inversely proportional to the momentum– that is, the faster the particle is moving, the shorter the wavelength, and the slower it’s moving, the longer the wavelength. Of course, particles are also localizable to a small region of space, which can only be done by combining a bunch of different wavelengths together, as in this picture:

The upper graph shows the beginning of a wave packet– something with wave characteristics that are confined to a small region of space– while the lower graph shows the different sine waves that are added together to get the upper graph.

This picture shows how both position and momentum have to be uncertain. The particle can only be found in the region where the total wave function has a substantial amplitude, and there’s some width to that region, giving you an uncertainty in position. The momentum is related to the wavelength, and lots of different wavelengths contributed to the total wave function, giving you an uncertainty in the wavelength, and thus the momentum. If you look at this in more detail, you find that the two are inversely related– that is, if you want a narrower wave packet, you need to add more waves, increasing the uncertainty in momentum. And this is a fundamental uncertainty, not dependent on how you go about measuring the position or the momentum.

The Uncertainty Principle tells us that there are limits on the sorts of things that we can hope to measure, but quantum theory is weirder than that. It turns out that the act of measuring the properties of an object in some sense create those properties. In quantum theory, it’s possible to put a particle in a state in which its properties are indeterminate– that is, the particle is in two or more states at the same time. It’s only at the instant of measurement that the particle chooses a single outcome, and is found in a single state.

In a very real sense, then, quantum mechanics tells us that measurement determines the state of the world around us, not just in the sense of revealing previously established properties, but by causing the particles around us to take on definite values, where they had previously not had any. If relativity was bad, this is much, much worse– not only do different observers disagree about what the properties of an object are, the object doesn’t even have definite properties until they’re measured. And yet, this has been experimentally confirmed.

So, while “measurement” may seem like a term whose meaning is too obvious to rate as a Basc Concept, I hope this give you some idea of the shadings of meaning it carries in a physics content. Measurement in physics is limited, not just by practical considerations, but by fundamental limitations due to the deep structure of the universe. The results of any given measurement can be applied only for a very limited number of observers, and they also constrain the possibilities for any future measurements. And, in a certain sense, the process of measuring the world creates the reality we see.

Keep that in mind, the next time you break out the tape measure…