The Reality Program

Chapter Three

The Uncertainty Principle
Quantum Properties, and More on the Measurement Effect

Tink was not all bad: or, rather, she was all bad just now, but, on the other hand, sometimes she was all good. Fairies have to be one thing or the other, because being so small they unfortunately have room for one feeling only at a time. They are, however, allowed to change -- only it must be a complete change.
J.M. Barrie, Peter Pan

Properties. In the last two chapters, we have spoken of "properties" of waves and particles in our experience, and "properties" of quantum units. Let us try to define the word "property" for our purposes. In a very general sense, a "property" is something about an object or a phenomenon that you can measure or describe. Weight, height, color, speed, symmetry, density, location, temperature, wetness, and so forth, are all "properties" in this sense. If I hold a billiard ball, say the 8-ball, in my hand, I can start to tell you something about its properties: it is spherical; it weighs about 6 ounces; it is mostly black; it has the number 8 painted on it. These are properties that we think of as being part of a description of this particular billiard ball; that is, if I set the billiard ball down, I believe I can still say that it weighs about 6 ounces, and is mostly black. What is more, if I had put it down so that I couldn't actually see the number 8, I still would be prepared to go way out on a limb and say it had the number 8 painted on it. Why should the number change? The painted number 8 is one of the properties that I think of as defining this particular billiard ball.

    I can describe other properties of the billiard ball, variable properties that might change over time. For example, I can pick it up and tell you that it is located in my hand. That will certainly change if I put it down. I could tell you that the painted number 8 is positioned at the top of the billiard ball as I hold it. That, too, will change if I turn it upside down. If I put it down on the table and give it a spin, I can tell you it is spinning clockwise when viewed from above; if I stop it and spin it the other way, I could tell you it is spinning counter-clockwise.

    Sometimes we measure properties directly, which is what I have been doing so far - looking at it and seeing the visible properties, or holding it and feeling the weight. On the other hand, sometimes we measure properties indirectly. Suppose I spun the 8-ball so fast it was a blur and I couldn't tell if it was spinning clockwise or counter-clockwise. There are lots of ways I could still measure which direction it was spinning. For example, I could watch it head straight for the bumper, and then see whether it bounced off to the left or to the right; if to the left, then it was spinning clockwise; if to the right, then it was spinning counter-clockwise. In this example, I am measuring the spin direction indirectly, by observing how the billiard ball behaves and using my rudimentary knowledge of physics to deduce that the behavior (a deflection in the way it rebounds) is caused by spinning in a certain direction.

    By making many observations (measurements) of the billiard ball, I can give you a fairly complete description of the 8-ball and what it is doing at any given time. Here is an exercise in describing many properties of the billiard ball:

Its existence property is: existing (as opposed to all the billiard balls the manufacturer plans to make next year)
Its shape property is: spherical
Its weight property is: 6 ounces
Its color property is: black
Its number property is: 8
Its number orientation property is: at the top
Its spin property is: spinning
Its spin direction property is: clockwise
Its location property is: on the table
Its temperature property is: about room temperature
Its motion-with-respect-to-the-table property is: standing still (but spinning)

And so on. After a while, I can give you a very complete description of this billiard ball. And even if it starts to move across the table (perhaps because of the spin), so that its motion-with-respect-to-the-table property changes and its number orientation property changes; and even if it falls off the table so that its location property changes, and it slows down and stops spinning - I could still pick a single instant and tell you all about the billiard ball as it existed at that moment.

Predictability and certainty in our day-to-day, ground level lives. I mentioned before that when I put the 8-ball down, I will still believe it has an 8 painted on it, even if I can't see the number because of the way I put it down. The same is true of the shape property and the weight property and the color property. Once I've looked at it and felt it, I can put it down and close my eyes and, depending on my memory, give you what I think is an accurate description of the billiard ball.

    Perhaps the most basic of these properties of which I am certain is the existence property. If I put the ball down on the table and then turn my back, I believe it will continue to exist. I turn around again, and sure enough it is still there, so this belief is reinforced. Having held it in my hand, I will stubbornly continue to believe it exists even if it is not there when I turn around again. In fact, if it is not there when I turn around again, I will speculate that it must have fallen off the table, so I will look around on the floor; or I will call out to my children and ask whether one of them has taken it. If all else fails - that is, if I check and the room was locked, and there is nothing on the floor, and there aren't any piles of papers that it could be under, and so forth - I will probably begin to wonder whether I saw it at all, or whether I really put it on the table just now or perhaps I am remembering something I did yesterday. In short, I will probably question my own sanity rather than consider that the billiard ball may have simply "disappeared" into nonexistence. I am that sure of the continued existence of things. Once we have passed the age of eight or nine, we don't believe for an instant that a magician can make something disappear; we may be impressed with the cleverness of the trick, but it's still just a trick, and we know that because we know that things do not actually "disappear."

    We live our lives according to a kind of "certainty principle" in this regard. For the most part, we expect that once we have measured a property, it will stay measured unless and until something changes it. However, we have already seen that quantum mechanics does not always conform to our expectations.

The uncertainty principle in quantum mechanics. As with all aspects of QM, the uncertainty principle is not a statement of philosophy, but rather a mathematical model which is exacting and precise. That is, we can be certain of many quantum measurements in many situations, and we can be completely certain that our results will conform to quantum mechanical principles. In QM, the "uncertainty principle" has a specific meaning, and it describes the relationship between two properties which are "complementary," that is, which are linked in a quantum mechanical sense (they "complement" each other, i.e., they are counterparts, each of which makes the other "complete"). Put simply, the relationship is this:

For any two complementary properties, any increase in the certainty of knowledge of one property will necessarily lead to a decrease in the certainty of knowledge of the other property.

Uncertainty in properties subject to variation. The uncertainty principle was originally thought to be more statement of experimental error than an actual principle of any great importance. When scientists were measuring the location and the speed (or, more precisely, the momentum) of a quantum unit - two properties which turn out to be complementary - they found that they could not pin down both at once. That is, after measuring momentum, they would determine position; but then they found that the momentum had changed. The obvious explanation was that, in determining position, they had bumped the quantum unit and thereby changed its momentum. What they needed (so they thought) were better, less intrusive instruments. On closer inspection, however, this did not turn out to be the case. The measurements did not so much change the momentum, as they made the momentum less certain, less predictable. On remeasurement, the momentum might be the same, faster, or slower. What is more, the range of uncertainty of momentum increased in direct proportion to the accuracy of the measurement of location.

    The uncertainty principle describes the behavior of quantum units as observed under laboratory conditions. As we have seen in the previous chapter, the behavior of quantum units often seems to defy common sense; and because the uncertainty principle describes this seemingly illogical behavior, the principle itself seems to defy common sense. In order to get some idea of what the uncertainty principle "means," let us see how it might apply to a common situation involving measurement of both momentum and position, to wit, the speed of a car and its location along a highway.

The parable of the conscientious traffic cop. Imagine a state trooper on traffic assignment. He has a radar gun which can be adjusted for more or less accuracy, depending on the need. He has set up a speed trap along a section of the interstate which borders a rural and urban area, and which has some construction in progress. Because of the differing zones and the construction, the speed limit along this stretch of road changes from 65 m.p.h., to 55, to 45, and back to 55, all within a space of, say, 200 yards.

Trooper with radar set to low accuracy

  • When the trooper first uses his radar gun, he sets it only to rough accuracy, and trains it on a car that seems to be in the construction zone. The radar shows that the car is going somewhere between 40 and 70 m.p.h. Obviously, this is not good enough to write a ticket, because if the actual speed is in the lower end of this range (40 m.p.h.), then the car is not speeding in the 45 m.p.h. construction zone.

    Trooper with radar set to medium accuracy

  • For the next car, the trooper increases the accuracy of the radar, and he gets a reading of 47 to 63 m.p.h. However, it seems that this car might have been outside of the construction zone at the time of the measurement, so that the applicable speed limit might be 55 m.p.h., so this is still not good enough.

    Trooper with radar set to high accuracy

  • The trooper increases the accuracy again, and gets a reading of 52 to 57 m.p.h. However, now he can't tell if the car was in the construction zone (45 m.p.h.), or the urban speed zone (55 m.p.h.), or even possibly the rural zone (65 m.p.h.). He still can't write the ticket.

  • When the trooper finally zeros in on a car that is going precisely 54.5 m.p.h. according to the radar gun, he can't even swear that it is in his state.

    Expressing the uncertainty principle as a mathematical relationship. The dilemma faced by the state trooper in trying to measure both the speed and location of the car is the same dilemma faced by the early experimentalists in trying to measure both the speed (momentum) and location of quantum units. In 1925, the German scientist Werner Heisenberg conducted a mathematical analysis of the position and momentum of electrons. His results were surprising, in that they showed a mathematical incompatibility between the two properties. Briefly, the math showed that if you try to relate these two properties, you are left with a logical inconsistency: Property "p" (position) times Property "m" (momentum) equals some number; but Property "m" times Property "p" does not equal the same number. In mathematical terms, the properties are noncommutative. The two numbers, which should be the same, come out different because of the way they are calculated in a matrix of the quantum unit's properties. This was an unusual result for mathematics. We are used to multiplication which is commutative, which means that if four times three equals twelve, then three times four also equals twelve, it doesn't matter which way you put it. But in the mathematics of quantum properties, it is as if four times three equals twelve, but three times four equals fifteen, or some other number (which doesn't make any sense).

        With a little more fiddling, Heisenberg was able to state that there was a mathematical relationship between the properties p (position) and m (momentum), such that the more precise your knowledge of the one, the less precise your knowledge of the other. This "uncertainty" followed a formula which, itself, was quite certain.[1] Heisenberg's mathematical formula accounted for the experimental results far, far more accurately than any notion of needing better equipment in the laboratory. It seemed, then, that uncertainty in the knowledge of two complementary properties was more than a laboratory phenomenon - it was a law of nature which could be expressed mathematically.

        The uncertainty principle with regard to position and momentum has definite and mathematically predictable consequences for quantum units such as electrons, which we will discuss later in this book. For now, just try to imagine that the state trooper, by increasing the accuracy of his radar gun, has caused the car on the interstate to stretch out between the different speed zones. That would be quite a "measurement effect." Yet, it is in almost this way that getting a good fix on the momentum of, say, an electron unit actually causes a spreading of the range of possible locations in which we may expect to find the electron unit when we ask for (measure) its location. Vice versa, by confining an electron unit to a smaller and smaller space in which it can be located (i.e., by determining its position more and more precisely), the range of its possible momentum becomes larger and larger. It will still have just one momentum when we get around to measuring it, but that momentum will turn out to be a number within a wider and wider range. Every once in a while, following a precise set of probabilities, such a "confined" electron unit will suddenly exhibit a huge momentum, because such a large momentum has become mathematically possible due to the confinement. That is to say, larger and larger energies (momentums) become a possibility for the electron because of the increase in the certainty of our knowledge of its position - not because it has been bumped or energized in any way. And since these larger energies have come to be included in the range of possibilities, the electron will actually have such a large energy every so often, according to the relative likelihood of that energy level.

    Uncertainty in complementary properties that are "quantized." Recall that one distinguishing feature of quantum units is that many of their properties come in whole units and whole units only. That is, many quantum properties have an either/or quality such that there is no in between: the quantum unit must be either one way or the other. We say that these properties are "quantized," meaning that the property must be one specific value (quantity) or another, but never anything else. When the uncertainty principle is applied to two complementary properties which are themselves quantized, the result is stark. Think about it. If a property is quantized, it can only be one way or the other; therefore, if we know anything about this property, we know everything about this property.

        There are few, if any, properties in our day to day lives that can be only one way or the other, never in between. If we leave aside all quibbling, we might suggest the folk wisdom that "you can't be a little bit pregnant." A woman is either pregnant, or she is not pregnant. Therefore, if you know that the results of a reliable pregnancy test are positive, you know everything there is to know about her pregnancy property: she is pregnant.

        What would be the consequence, according to the uncertainty principle, of knowing everything about one complementary property that was quantized? Well, the consequence would be that, as a law of nature, we then would know nothing about the other complementary property. At this point in the discussion of the uncertainty principle, analogy from our day to day lives fails us. Can we imagine that, by gaining the results of a pregnancy test, we lose other knowledge? I, for one, am unable to devise a suitable illustration. The very concept is alien. By our logic and experience, gaining knowledge of the results of the pregnancy test must increase the sum total of our knowledge. We add this knowledge to all the other things we know about the woman, and our total knowledge of her condition increases. Can we imagine instead that, by learning whether a married woman is pregnant, we thereby no longer know whether she is married? Yet, this would be the effect if marital status and pregnancy were complementary properties, and we applied the uncertainty principle to knowledge of them.

        As you can see, the uncertainty principle leads to some thorny conceptual problems when we try to use our logic and experience to visualize what is going on in the quantum realm according to our intuitive "certainty principle," i.e., our general sense that once we are certain of something we can file it away as a fact and move on to the next investigation. Therefore, let us continue to talk in parables. Let us imagine that there are objects in our day-to-day, ground level lives that behave according to quantum principles. I caution you that playing out such a scenario will not help you to relate quantum phenomena to the world of your experience. At best, it will impress upon you that quantum phenomena are entirely different from the things and objects you know and love.

    The Uncertainty Problem, or, The Parable of the Quantum Grapes

    Background and definitions. The New York Times has just reported in its Tuesday Science Times section that the United States Department of Agriculture, working with the President's scientific advisors, has announced the discovery of a new kind of fruit: "quantum grapes." These (strictly hypothetical) quantum grapes are much like ordinary grapes, except that they behave in some respects like the quantum units scientists have been studying for the past 100 years or so - things like electrons and photons and neutrinos and protons and everything else of which the universe is comprised.[2]