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:

i-1ede026ffc63bad1b361f34c8e9ecf68-wave-packet.gif

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…

Comments

  1. #1 Jonathan Vos Post
    March 5, 2007

    “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….”

    Yes, experimentally confirmed as you have nicely summarized. BUT the interpretation is not unanimous. The Physics community is split between different ways of explaining this, with John von Neumann’s “collapse of the wave function” being a well-known interpretation, the observer being tied to this, disagreement on whether or not the observer needs to be conscious, and whether or not the “many worlds” interpretation makes sense.

    In a “Basics” essay, this might confuse the intended audience. On the other hand, not admitting the differing opinion gives an opportunity for loony critics of the Creationist and Conservapedia variety to say that Physicists are covering up their failures.

  2. #2 Adam
    March 5, 2007

    Jonathan: my experience is that most of the QM community doesn’t worry too much about which interpretation they adopt (given that, at least at present, there is no experiment to distinguish them).

  3. #3 Mike
    March 5, 2007

    Perhaps your explanation of the different measurements made by different observers in relativistic situations is a bit misleading in one small way. While they will make different measurements of space and time, they also have the ability to transform those into the other’s frame, thereby predicting what any other observer would measure. So their “disagreement” is only skin deep.

    Also, did you see the current Physics Today (March)? It has some interesting letters about teaching QM, especially the one from Griffiths. He seems to favor a not so magic sounding way of talking about measurements on quantum systems.

  4. #4 Chad Orzel
    March 5, 2007

    I tried to finesse the question of quantum interpretations, because it doesn’t really matter for our purposes. Whatever interpretation you favor, the state is indeterminate before the measurement, and well-defined after the measurement. Whether you say this is because of a non-unitary “collapse of the wavefunction” or because the universe splits so we only see one part of the unitarily evolving superposition makes no difference in terms of measurable outcomes.

    Mike: Perhaps your explanation of the different measurements made by different observers in relativistic situations is a bit misleading in one small way. While they will make different measurements of space and time, they also have the ability to transform those into the other’s frame, thereby predicting what any other observer would measure. So their “disagreement” is only skin deep.

    Good point.

    Also, did you see the current Physics Today (March)? It has some interesting letters about teaching QM, especially the one from Griffiths. He seems to favor a not so magic sounding way of talking about measurements on quantum systems.

    I have it somewhere, but haven’t read it yet. I’ll take a look for it.

  5. #5 dileffante
    March 5, 2007

    I’m uncertain about principles… and what better place to ask!

    People use “principle” to refer to different kinds of propositions, sometimes theorems, sometimes rules-of-thumb, sometimes well trusted conjectures, etc. And I don’t know exactly what’s the status of the Uncertainty Principle. Is it derived, like a theorem, from the postulates of QM? My doubt is: could we imagine that some smart guy could come up with an alternative to QM, compatible with all current experimental knowledge, and in which the U.P. does not hold? Or is such a thing not only unlikely, but actually ruled out by current empirical restrictions on possible theories?

  6. #6 mike
    March 5, 2007

    Dileffante, good question. I’ve seen in argued by Victor Stenger that the uncertainty principle is a direct result of our requirement that the universe be observer independent. If that’s true, then it’s hard to come up with another version of QM (or any useful science) without that requirement.

    I don’t think I could reproduce his arguement without thinking really hard. I’m pretty sure I saw this in his book, The Comprehensible Cosmos.

  7. #7 Pseudonym
    March 5, 2007

    Chad, in part II, will you be talking about exactly what it means to perform a “quantum measurement”?

  8. #8 Carl Brannen
    March 5, 2007

    The most elegant formulation of quantum mechanics is the one written up by Julian Schwinger in the late 1950s; now called Schwinger’s Measurement Algebra”.

    Regular QM has states live in a Hilbert space, and operators live in a space of things that operate on the Hilbert space. This is two objects and is a bit clunky. The measurement algebra puts the states in operator form. The Hilbert space melts away and is not used, except as a mathematical convenience for calculations.

    You’re already partly familiar with this because when you take a spinor and turn it into a density operator you turn a state into an operator. What Schwinger did was generalize this sort of thing. You end up without a need for bras and kets. And you lose the complex phase gauge symmetry.

    The herd lost interest in the idea when it became necessary to assume that the quantum vacuum had vevs that were non trivial.

    Anyway, this is the theory I’ve been busily applying to the standard model. The mathematical unity of operator and state is very satisfying. One has few opportunities to make stuff up; there is very little freedom to do the things that generate so many unspecified arbitrary constants as happens in the usual way.

  9. #9 Torbjörn Larsson
    March 6, 2007

    Another aspect of measurement is that we need measurement standards to measure against, or rather to calibrate our measurements. Several different systems exists, which provides different base and derived units. The SI system tries to minimize the base units and make them independent of man-made standards.

    This also explains why we separate measurements into unit and value. It also explains why the value even in principle is a, rational, approximation to the real value. These later distinctions of course all disappears as soon as we allow for describing uncertainty and likelihood in the measurements.

    And, in a certain sense, the process of measuring the world creates the reality we see.

    I’m not happy with this unnecessary description. Observations constrain possibilities, as you say.

    Maybe I’m overly touchy about a post on basics. But this way of description is analogous to when biologists describes evolution conveniently as if the animal chooses character to fit the situation, thus making it seem like an agent (or worse, teleology) is in play.

    And I don’t know exactly what’s the status of the Uncertainty Principle. Is it derived, like a theorem, from the postulates of QM?

    Yes, a good question.

    The wikipedia article is quite good and lists many of the individual uncertainty relations. It also mentions that Heisenberg derived his relation from gaussian distributions (independent measurements, I think; http://www.aip.org/history/heisenberg/p08a1.htm ), but that one can make a rigorous but statistical derivation, the Robertson-Schrödinger relation, from the basic properties ( http://en.wikipedia.org/wiki/Uncertainty_principle ). One can also make simpler heuristic derivations from de Broglie’s relation, see the Feynman lectures.

    So perhaps there is room for discussing the value of the limit. OTOH people have made observations with squeezed states to make the most of the measurements, so perhaps it is fairly tested.

    Or is such a thing not only unlikely, but actually ruled out by current empirical restrictions on possible theories?

    My impression is that the linearity of QM is the constraint that prevents other theories. IIRC the linearity is an expression of unitarity, which is basically preserving the observed property of probabilities.

  10. #10 raj
    March 6, 2007

    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.

    Kinda/sorta but not exactly. The HUP (Heisenberg Uncertainty Principle) tells us that there are limits to the precision with which we can measure two variables at the same time–position and momentum on the one hand, and energy and time on the other (the position/momentum formulation and the energy/time formulation are related via a simple transform). I have interpreted that as something a result of the fact that the measurer is actually part of the experimental set-up in which the measurement is being made, and his/her being part of the set-up adds some uncertainty to what might have occurred had the measurer not been there. (And, from a mathematical standpoint, a result of the fact that position and momentum, for example, are related via a Fourier transform, which clearly shows the extent of the uncertainty.)

    What I find most interesting about the HUP is something that I did not learn at university many decades ago. The most interesting thing is that the HUP also predicts that virtual particles will be generated–they will come out of apparently no-where (vacuum energy?), and return so that delta E delta T < h-bar (the energy/time formulation of the HUP) is not violated. In other words, the HUP is not just a limitation on our ability to measure with precision, but it is also a prediction of a “particle generator.” I had wondered whether the particles could be detected, but the most recent Scientific American issue indicates that they can be and that they have been.

  11. #11 raj
    March 6, 2007

    That last didn’t complete. It should be

    “they will come out of apparently no-where (vacuum energy?), and return so that delta E delta T is less than h-bar. I was wondering whether the virtual particles could be observed, but there was an article in the last Scientific American that they could be observed and they have been.”

    I suppose that the html didn’t like the fact that I used an angle bracket for the “less than”.

  12. #12 raj
    March 6, 2007

    #5 | dileffante | March 5, 2007 05:13 PM

    And I don’t know exactly what’s the status of the Uncertainty Principle. Is it derived, like a theorem, from the postulates of QM?

    As to the latter, yes. You could take a look at the rather detailed description at Wikipedia http://en.wikipedia.org/wiki/Uncertainty_principle but the explanations in the QM books that I have read go along the lines of the following. Take a wave packet (you know, Schroedinger’s wave function). The wave packet “defines” the position of the particle to some degree of precision. Take a function of the Fourier transform of the wave packet. That function (usually referred to as “operator” in QM lingo) “defines” the momentum of the particle to some degree of precision. Multiply them together, and you get that the maximum precision that you can get for both measurements when considered together–a value that is greater than or equal to h-bar. That’s the HUP.

    That really is the long and the short of it. The HUP is a necessary–and interesting–result of the mathematics of QM. Is QM the correct theory? I don’t know. But it makes predictions that have held up pretty well over the last century or so.

  13. #13 Jonathan Vos Post
    March 6, 2007

    VERY cool article by the Math superstar Terrence Tao, on Quantum Mechnics interpretation, using a computer game analogy, and the differing viewpoints of us in the xternal world playing the game, and a character in the game baffled by nonclassical rules.

    Quantum mechanics and Tomb Raider
    http://terrytao.wordpress.com/

    Monday, February 26th in non-technical | 30 comments

    This post is derived from an interesting conversation I had several years ago with my friend Jason Newquist on trying to find some intuitive analogies for the non-classical nature of quantum mechanics. It occurred to me that this type of informal, rambling discussion might actually be rather suited to the blog medium, so here goes nothing…

    Quantum mechanics has a number of weird consequences, but here we are focusing on three (inter-related) ones:

    1. Objects can behave both like particles (with definite position and a continuum of states) and waves (with indefinite position and (in confined situations) quantised states);
    2. The equations that govern quantum mechanics are deterministic, but the standard interpretation of the solutions of these equations is probabilistic; and
    3. If instead one applies the laws of quantum mechanics literally at the macroscopic scale, then the universe itself must split into the superposition of many distinct “worlds”.

    In trying to come up with a classical conceptual model in which to capture these non-classical phenomena, we eventually hit upon using the idea of using computer games as an analogy. The exact choice of game is not terribly important, but let us pick Tomb Raider – a popular game from about ten years ago (back when I had the leisure to play these things), in which the heroine, Lara Croft, explores various tombs and dungeons, solving puzzles and dodging traps, in order to achieve some objective. It is quite common for Lara to die in the game, for instance by failing to evade one of the traps. (I should warn that this analogy will be rather violent on certain computer-generated characters.)

    Read the rest of this entry »
    [at the blog pointed to]

  14. #14 Brunhouse
    April 5, 2007

    It is important not to confuse properties and relationships. Measurements do not determine properties. Measurements are assigned to properties. Long before there were metersticks, there were just… sticks. Measurements determine a relationship between properties. It cannot be assumed that because we understand the relationship between force and acceleration, that we have determined anything profound about the property of acceleration or the property of mass. Measurments standardize an experiment so that laws may be developed that give the world some hope of predictability. Measurement is a tool to observation.

    The experiment is more about observing. By observing I determine the properties of everything. I “understand” the stick has a realitive spatial property akin to my arm. The stick also has the property of being relative to me so that I can always find my way back to it. But just by observing, I am not inclined to believe there is an infinite number of points that may determine an acurate measure between me and the stick– you see that is the tool of measurement, not the observation. It is a very old tool and Discarte’s cartesian coordinate system is based around it. So far as I know waves have only been discribed in a cartesean coordinate system or other very similar geometries. That is too bad. Because as is explained above, when people started to observe quanta they observed quanta doesn’t have exact locations. They observed complimentarity. They learned, that just by observing, what you are observing is fundamentally changed. It seems to me our current concept of measurement (as a tool), of cordinate systems, is headed to the recycling bin, but don’t worry, what ever replaces it will be refasioned so science is still emperical, it is just that the theory will be recast to work in harmony with what we actually observe.

  15. #15 Oldfart
    April 16, 2007

    When the universe was the size of a grapefruit…………….just who in the hell was standing OUTSIDE the universe with a yardstick measuring it?

    I contend the Universe was never the size of a grapefruit – it was always the size it is now.

  16. #16 G. Tingey
    May 13, 2007

    And you have still not gone over, even in “classical” terms, the difference between:
    ACCURACY
    and PRECISION.

    And why this is important, in all forms of measurement.
    This also leaves out he problems of multiple measurement, error-bars, and other reasons for (some) uncertainty in any measurement.

    Perhaps some questions will clarify what I’m driving at:
    How accurate is your measuring stick?
    Are you holding it up straight against the meauree?
    Are you even using the correct stick?

  17. #17 vinay chander chandel
    March 28, 2008

    how we can say that with the moving particle, the wave packet is associated but not a single wave?

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