Humanities Core Course Spring 2011 Instructor: Bencivenga

LECTURE NOTES

Lecture 5.

According to the Copenhagen interpretation, a body at any one time in which it is not being observed is not in, for example, a specific position, nor does it have a specific velocity—or a specific value for any other parameter that applies to it. Its state is described, rather, by what is called a superposition of different positions or velocities, weighted as a Fourier expansion is—that is, where each individual position or velocity is accompanied by a coefficient. I, for example, might be represented right now as being 40% here, 30% in Italy, and 30% in an island of the Caribbean. The superposition evolves deterministically, that is: any of its future states can be calculated on the basis of previous states. So in a sense this evolution is classical and in a sense is not, because what evolves is not definite magnitudes but an inseparable mixture of different values for the same magnitude. Because the evolution follows the classical pattern of a continuous line, what I just presented is called the linear dynamics of a body. And that, I repeat, is supposed to describe the body’s behavior when it is not being observed.

When, on the other hand, the body is observed, and specifically when an observation is made concerning one of its parameters, say, position—to put it more simply, when one looks to see where the body is—the body will instantaneously and randomly collapse into a definite value for that parameter. Whenever I look to see where an electron is, I always find it to be some place definite. Whenever you look to see where I am, you always see me here, for example, or in Italy, or in the Caribbean—but in one of these locations to the exclusion of the others, not in some strange location which is a mixture of all of them. Because the collapse is instantaneous, it once again breaks with the classical notion of continuity: the body is found suddenly in one place, without having been nearby before. Because the collapse is random, it cannot be predicted where in fact the body will be found; you will have no way of knowing, before you make the observation, that I, say, will be seen here as opposed to anywhere else. All we can say is that there will be a definite probability that the body be found in a specific place, which probability can be calculated from the coefficient the superposition assigned to that place. To make it very simple, if we return to the superposition of my positions considered earlier, when you look to see where I am you will have 40% probability to see me here, 30% probability to see me in Italy, and 30% probability to see in the Caribbean. So we can also call the evolution of the superposition the probability function associated with, in this case, a body’s position in space—and expect from what was said before that this probability function “is completely determined by the quantum mechanical equation” (p. 23). Except that, of course, it also “does not allow a description in space and time” (p. 23) because a body is going to have a definite position in space and time only when observed, with a probability described by what is called its collapse dynamics.

There is more: when you make an observation concerning a given parameter—say, position—and as a result assign a definite value to that parameter, you thereby make it impossible to observe a different parameter—say, velocity. The more you know about where a body is, the less you know about how fast it is going. This is what Heisenberg called alternatively the Indeterminacy or the Uncertainty Principle, and notice that the choice between the two terms is a crucial one. Uncertainty is a state in which we are, when we do not know something for sure; indeterminacy is a state in which the world is, when it itself is fuzzy, when it has no definite, unique state. Here is how Heisenberg introduces his principle:

One could speak of the position and of the velocity of an electron as in Newtonian mechanics and one could observe and measure these quantities. But one could not fix both quantities simultaneously with an arbitrarily high accuracy…. Similar relations could be formulated for other experimental situations. They are usually called relations of uncertainty or principle of indeterminacy. One had learned that the old concepts fit nature only inaccurately. (pp. 16-17)

And here is Heisenberg summing up these differences between classical and quantum mechanics:

In Newtonian mechanics, for instance, we may start by measuring the position and the velocity of the planet whose motion we are going to study. The result of the observation is translated into mathematics by deriving numbers for the co-ordinates and the momenta of the planet from the observation [a body’s momentum is the product of its mass and velocity, here to remind us that classically masses too are important for making exact predictions]. Then the equations of motion are used to derive from these values of the co-ordinates and momenta at a given time the values of these co-ordinates or any other properties of the system at a later time, and in this way the astronomer can predict the properties of the system at a later time. He can, for instance, predict the exact time for an eclipse of the moon. (pp. 18-19)

In quantum mechanics, on the other hand,

the theoretical interpretation of an experiment requires three distinct steps: (1) the translation of the initial experimental situation into a probability function; (2) the following up of this function in the course of time; (3) the statement of a new measurement to be made of the system, the result of which can be calculated from the probability function [that is, the function lets us calculate the probability of a given result]. For the first step the fulfillment of the uncertainty relations is a necessary condition. The second step cannot be described in terms of the classical concepts; there is no description of what happens to the system between the initial observation and the next measurement [no description of, say, where an electron is between when it is observed to be at a and when later it is observed to be at b]. It is only in the third step that we change over again from the “possible” to the “actual.” (pp. 20-21)

Heisenberg himself draws a number of general philosophical morals from this situation, even comparing the current situation with a number of historical philosophical views. As Lindley notes, that he would embark on such a philosophical tour

sets Heisenberg apart from most modern physicists, who generally disdain or simply ignore philosophical thinking about their subject. But Heisenberg was educated in Germany at the beginning of the twentieth century, and had a professor of classics for his father. Being reasonably well versed in philosophy was, for Heisenberg, merely an aspect of a good general education. (pp. xiv-xv)

And it is, I might add, quite a fortune for us, as it allows him to offer analogies and contrasts that are deeply illuminating.

The first sense in which quantum mechanics is a striking revolution in the old ways of thinking and of doing science is that it makes us think that the world does not have a single, unique structure. At any one time, any one aspect of the world can be any number of different ways, and this possibility—or, if given a mathematical value, probability—is not a feature of our mental state (it is not that we don’t know which way it is); it is a feature of the world itself.

Probability in mathematics or in statistical mechanics means a statement about our degree of knowledge of the actual situation. In throwing dice we do not know the fine details of the motion of our hands which determine the fall of the dice and therefore we say that the probability for throwing a special number is just one in six. The probability wave of Bohr, Kramers, Slater, however, meant more than that: it meant a tendency for something” (pp. 14-15).

Reality is not made of definite objects behaving in definite ways—those ways and no others. Reality is made of tendencies: of things inclined (to different extents) to behave in some ways and also many others. Which also means: there is no single, coherent way of describing how the world is. For example, there are situations in which the behavior of an electron is best described as that of a particle, and other times when it is best described as that of a wave: when the electron is not being observed, it is nowhere in particular, hence one can think of it as indeed a wave of possible values for its position. The particle picture and the wave picture are incompatible, and yet Bohr claimed that we get the best picture of reality by considering them complementary pictures, each offering what the other lacks, and shifting constantly between them.

Bohr advocated the use of both pictures, which he called “complementary” to each other. The two pictures are of course mutually exclusive, because a certain thing cannot at the same time be a particle (i.e., substance confined to a very small volume) and a wave (i.e., a field spread out over a large space), but the two complement each other. By playing with both pictures, by going from the one picture to the other and back again, we finally get the right impression of the strange kind of reality behind our atomic experiments. (p. 23)

A second revolutionary aspect of quantum mechanics is the crucial physical significance it assigns to observation. That a body is observed changes its behavior drastically, to the point of turning it from indefinite to definite. The classical opposition between the objectivity of science—what science studies and discovers is what it is independently of any observer, human or otherwise—becomes here a peculiar interaction between objective and subjective. “In this way quantum theory reminds us, as Bohr has put it, of the old wisdom that when searching for harmony in life one must never forget that in the drama of existence we are ourselves both players and spectators” (p. 32).

It should be noted that both the revolutionary aspects of quantum mechanics I have mentioned so far have been criticized by a number of physicists, most notably Einstein. Whereas the Copenhagen interpretation reads Heisenberg’s principle in the stronger way (the world itself is indeterminate), these other physicists have tried to cast it in its weaker form (we are uncertain about how the world is). Typically, this resistance has been articulated in terms of so-called “hidden-variable theories,” that is, by assuming that the world contains a lot more structure than we are aware of, a lot of parameters (or “variables”) in addition to position, velocity, or what have you, which, if their values were factored in, would give a perfectly classical understanding of what we now know only very imperfectly and incompletely. However, recent experiments appear to establish that there is no possible coherent set of hidden variables that would give us that kind of classical understanding of quantum phenomena. So, for the moment at least, we are stuck with a dominant physical theory according to which “physis” (that is, nature) can be known to be anything in particular only by having the very act of knowledge contribute to making it be that way.