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Lecture 2:  Ravens, Cats, Artistotle, and Schroedinger

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  • In everyday life we assume that either a raven is black or a raven is not black. We consider it obvious that there can not be anything between black and not-black. In fact we have been thinking this way for a long time. Aristotle formulated the Law of the Excluded Middle, which states that the proposition "either a raven is black OR a raven is not black" is true, irrespective of whether ravens are black or not.

    Notice however that things are not so obvious when we deal with less 'material' things. For example, propositions like "either you love me or you don't love me" or "either you are with us or you are against us" are more problematic. In these cases we tend to feel that some sort of middle ground may be possible. Nevertheless, we don't have any doubts that "either I am in this room or I am somewhere else" is true. Our conviction is so strong in these matters that science has adopted it too—at least, until well into the previous century.
  • It seems sensible, in the absence of contrary evidence, to assume that basic laws (such as the Law of the Excluded Middle), which are found, over and over again, to be true in the world we have access to—in the everyday world—are going to be true everywhere. That's what Newton, for example, had in mind when he labeled his law of gravitation "universal." More commonly, that's why we call certain statements, such as the Law of the Excluded Middle, "laws." Of course in science we need to be more careful: to say that a certain law is universal assumes that we have tested it everywhere—a clearly impossible feat. We can only "hypothesize" that such a law is universal. In fact, the history of science is full of cases where hypotheses so labeled turned out to be wrong. For example, for a long time it was believed (i.e. "hypothesized") that the Sun goes around the Earth…until Copernicus, Galileo, and finally Newton, showed this not to be the case. The same fate befell the Law of the Excluded Middle.
  • Consider a molecule of carbon monoxide. It is made of two atoms: one carbon atom and one oxygen atom. All atoms consist of a nucleus (itself made up of a variable number of protons and neutrons) surrounded by one or more electrons. Consider now one of the electrons of, say, the carbon atom. Precisely as the previous sentence suggests, we are inclined to believe that such an electron is part and parcel of the carbon atom, not of the oxygen atom. After all, the Moon orbits the Earth, not the other planets or the sun. Why should the world at the astronomical level behave any different than the world at the atomic or molecular level? Yet, that's exactly what happens. The atomic world turns out to be governed by somewhat different laws. The law of the excluded middle, for example, fails there. We expected that "either an electron belongs to a certain atom or it does not belong to that atom." What we find, instead, is that the electron in question belongs to both the carbon and the oxygen. In fact that's essentially what holds the two atoms together in a molecule of carbon monoxide. In other words, an electron is not localized. It is more like a cloud with fuzzy boundaries (or no boundaries). In some places the cloud is more dense than in others, and that means that the likelihood of finding the electron in a given place is just that: a probability, not a certainty. Notice, however, that this is not just a question of ignorance on our part. Nature itself behaves this way. Such probabilistic behavior extends also to the states in which objects such as electrons find themselves in. For example, an electron, like almost everything else, can have different speeds, be in different places, spin at different rates, etc. Before making any observation, an electron is simultaneously, although with different probabilities, in all such states. Or, to use the language of Quantum Mechanics (the theory of the atomic and subatomic world), its state is a superposition of all its possible states. You can find a good account of these phenomena in Quantum Mechanics and the Probability. Now, if we make an observation, the electron will show up as being in a particular state! But we can not predict this particular state. We can only calculate how likely the electron is to be found in such a state. And there's more …
  • It seems also natural to us that an electron, just like a certain apple or that chair over there, possesses it own individuality, that we can always tell this chair from that chair, even though they are "identical." We know, for example, that even identical twins can be told apart. (And we can always "label" them with different clothes, or with a magic marker, if they seem too hard to distinguish.) This, however, is not the case in the atomic world. Two electrons, at least under certain conditions we can not discuss here, are indistinguishable.

  • One of the most famous illustrations of these properties of atoms was devised by Erwin Schroedinger in an article published in 1935. An English translation by John D Trimmer is available under the tile The Present Situation In Quantum Mechanics.


    Erwin Shroedinger

    Erwin Shroedinger


    "One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts."

    Here is the … English translation:

    "[Schroedinger] imagined a box containing an atom having a 50 percent likelihood of decaying in an hour, a radiation detector, a flask containing poison gas and a cat. When or if the atom decays, the Geiger counter will trigger a switch that causes a hammer to smash the flask, releasing the gas and killing the cat. When the experimenter opens the lid of the box and peers inside after an hour has passed, he or she will find the atom either intact or decayed and the cat either alive or dead. But according to quantum mechanics, during the period before the lid is opened, the cat exists in two superposed states: both dead and alive." [ from John Horgan's article Schroedinger's Cation: Physicists Prove That an Atom Can Be in Two Different Places at Once published in the June 1995 issue of Scientific American ]
  • Schroedinger's Cat is of course only an illustration. The basic reason is that steel chambers, radiation counters, vials full of poison, and cats (but radioactive atoms) are macroscopic objects, are part of our everyday world, not of the atomic world per se, where different laws apply. But experimental confirmations of the Schroedinger's idea have been numerous. See for example, the article NIST Scientists Cross the Bridge between Atomic and Real Worlds.

    "In 1996, researchers at the Boulder, Colo., laboratories of the Commerce Departmentís National Institute of Standards and Technology confirmed the belief that a single atom can be simultaneously located in two separated places (a physical state termed a quantum-mechanical superposition). Now, almost four years later, they have performed experiments to help explain why this works only at the atomic level and not in our world of everyday experience."
  • It is important to ask whether, say, the subatomic world, and our everyday world obey laws that are the negation of each other. After all, what I said above seems to suggest such incompatibility. Things are actually a bit subtler than they may first appear. It turns out that the laws of the atomic world do apply to our everyday world, but in this case their effects are completely negligible. The reason is that the everyday world is made of such a huge number of atoms that the probabilistic effects cancel out. Another example, as we shall see later, is the apparent incompatibility of Einstein's Theory of Relativity and Newtonian mechanics. The former applies to the cosmos at large (great distances, extreme speeds, etc.), while the latter describes very well our everyday experience. If we try to apply Relativity to the fall of the apple from Newton's tree, we discover that the results are essentially undistinguishable from those predicted by Newton. In other words, there are not true contradictions between these different ways of understanding and representing the world. Here is a useful analogy: psychology helps us understand the behavior of individual persons, but it is of little use when we deal with a crowd, even though a crowd is composed of individual persons. The sum of their behaviors is impossibly complicated, and we must focus on a crowd as a sort of new creature, which follows a different set of rules (which are, in turn, of little significance in the study of individual persons).

Readings, Resources and Questions

  • Just to get a sense of what an actual scientific article looks like, browse through the English translation of Schroedinger's original paper, The Present Situation In Quantum Mechanics, and compare his description of the idea to popular accounts of it, including the present one. Apart from the use of mathematical formulas, how do the two types of account differ? As a further aid in discussing this question, read Phoebe.com's Schrödinger's Cat.
  • If you have some background in formal logic, and are interested in a more rigorous treatment of the Law of the Excluded Middle, read Peter Suber's short article Non-Contradiction and Excluded Middle.
  • The apparent paradoxes of Quantum Mechanics are not unique. There are many cases, in various areas of science, where what we expect, on the basis of our everyday experience, is simply not the case. Consider for example the 'obvious' statement that the whole is greater than its parts. Think now of all the positive integers 1, 2, 3, … Some of them are even, some are odd. How many of each? It turns out that counting the number of positive even (or odd) integers is equivalent to counting the number of positive integers. In other words, in this case the whole is exactly equal to its parts! Here is how you can visualize this claim.
    Set of positive integers:              1  2  3  4  5  6  7  …
    Multiply each positive integer by 2:   2  4  6  8 10  12 14 …
    In other words, to each and every even (or odd) integer there corresponds a unique integer, and viceversa; and all positive integers are used in the counting process.
  • The fact that different laws hold true in different places in, or at different levels of, the universe raises serious questions. How do such different laws mesh together? Is there a gradual or sudden transition from one to the other? etc. Perhaps the most interesting reflection of this type of dilemma is the concerted effort that 20th-century science has been engaged in: the belief that there may be a truly unified and universal way of looking at the whole of reality. But the jury is still out on this enterprise.
  • For the lighter side of the Schroedinger's Cat paradox, read The Story of Schroedinger's Cat (An Epic Poem)
  • The subatomic world of Quantum Mechanics is not the only instance of a level of reality where laws are different from those of the everyday world we are familiar with. Albert Einstein's Theory of Relativity is another.


© Copyright Luigi M Bianchi 2003-2005
Picture Credits: U of St Andrews, Scotland
Last Modification Date: 07 November 2004