pp. 70-71

MATH IN THE COSMOS: THE QUANTUM CURTAIN

...[T]he theory uses operators on [Hilbert] spaces to represent various observable quantities of those states such as spin or momentum. The possible values of these observables are given by so-called eigenvectors, special vectors within the space that are invariant when the operators are applied to them. The various possible outcomes of an experiment have respective probabilities attached to them according to the size of the components of the eigenvectors.

The picture of fundamental particles that emerges from the mathematics is fundamentally mysterious. A photon traveling from a source to a detector exists as a wave, which amounts to a probability distribution. Where will it appear on the phosphor screen? Who knows -- until it appears? Schrödinger's major contribution to quantum mechanics, the [wave function] equation that now bears his name, describes the probability distribution of an electron in an atom. The following figure shows such a distribution for a hydrogen atom in an excited state. It looks somewhat like a pumpkin. If one measures the actual position of the electron, it might turn out to be anywhere within the pumpkin with equal probability. However, it will not appear very near either pole of the pumpkin, since the distribution does not allow it.

Since the position of the electron is unknown until it is measured, one might even say that it has no real existence until that happens. But everything in the universe (more or less) is made from atoms the properties of which depend in the most fundamental way on the shapes of these probability distributions. If we run headfirst into a stone wall composed entirely of unreal atoms, we nevertheless end by being persuaded of its reality.

THE COPENHAGEN INTERPRETATION

The central question surrounding the dynamic attributes of a fundamental particle is whether these attributes may be said to possess definite values when they're not being measured. The question reminds us of the old chestnut "If a tree falls in the forest and no one is there, does it make a sound?"

Bohr's interpretation of the quantum facts and of quantum theory led him to believe that before a dynamic attribute of an electron, photon, or any other particle/wave was measured, the attribute had no particular value. For example, until a photon manifested itself as a point of light, it had no particular position. Until its polarization was measured, it had no particular spin. This interpretation of quantum mechanics is called the "Copenhagen interpretation," after Bohr's hometown. Bohr and, ultimately, the vast majority of physicists took the view that there was simply no underlying reality. Until the polarization of a photon was measured, it had no polarization at all.

The ultimately mysterious event during which a wave becomes a point of light on a screen, or takes one slit [path] rather than another, is sometimes called "the collapse of the wave function." A wave function would appear to carry all of its potential values-as-measured simultaneously -- pregnant with possibilities, so to speak. The collapse of the wave function is essentially a birth event, the ultimate process by which our classical "lived world" becomes manifest.

pp. 79-81

THE MEASUREMENT PROBLEM

While there is little disagreement about the mathematical and operational side of quantum theory, there are several schools of thought about the underlying reality.

The best introduction to these schools takes the so-called measurement problem as its starting point. This problem centers on the question "When does the wave function collapse occur?"

The question sits exactly on the boundary between the quantum world and the "classical" world. Essentially, the quantum world involves very small objects, while the classical world consists of all the objects we can see, feel, or sense directly.

Hungarian-born John von Neumann, perhaps the greatest mathematician of the twentieth century, published a classic book on quantum mechanics in 1932. Called Die Grundlagen [Mathematische Grundlagen der Quantenmechanik or "Mathematical foundations of quantum mechanics"], von Neumann's book painted an all-quantum picture of the world. Even macroscopic objects such as your body or the Earth itself had its own proxy wave with its associated quantum numbers. On the one hand, von Neumann demonstrated that electrons and other fundamental particles are not "real" entities because they cannot be said to possess any dynamical attribute before it is measured. This seemed to buttress the philosophical approach of the Copenhagen interpretation. On the other hand, von Neumann did not endow measuring instruments with a special status, as the Copenhagen interpretation did.

Von Neumann analyzed the measurement act, breaking it down into a series of steps called the "von Neumann chain." Applied to the two-slit experiment, for example, the von Neumann chain consists of (1) the emergence of a photon from a source, (2) its passage through one of the slits, (3) the triggering of a detector, (4) the signal from the detector to a meter, (5) the movement of a needle or other registration device, (6) the light from the meter to the eye of the observer, (7) the message from the observer's retina to the observer's brain, (8) processing of the signal in the observer's brain, and (9) registration in the observer's consciousness.

Where does the collapse occur? The problem of locating the collapse is well illustrated by the story of Schrödinger's cat. A live cat, along with a photon source, a pair of slits, a detector, and a loaded revolver are placed in a sealed, soundproof, lightproof box. Inside the box, things are arranged so that if the photon passes through one slit, the revolver is triggered and the cat is killed. But if the photon passes through the other slit, the revolver is not triggered and the cat remains alive. According to the Copenhagen interpretation of quantum mechanics, the cat is neither dead nor alive until the box is opened. Is this a reasonable proposition?

Von Neumann showed that one may place the collapse at any point on the chain that one likes. Only one site, however, has anything like a privileged position in the chain: the consciousness of the observer's mind.

Many people have drawn silly conclusions from this hint that consciousness may play a special role in the collapse of the wave function. For example, one school of thought has concluded that human consciousness literally creates the world. It may be that any attempt to peer behind the quantum curtain results in silliness of one kind or another.