For the next two days we will be examining the philosophical problems that arise in quantum physics, specifically those connected with the interpretation of measurement. Today's lecture will provide the necessary background and describe how measurement is understood according to the "orthodox" interpretation of quantum mechanics, and several puzzling features of that view. The second lecture will describe various "non-orthodox" interpretations of quantum mechanics that attempt to overcome these puzzles by interpreting measurement in a quite different way.

A Little Historical Background

As many of you know, the wave theory of light won out over the corpuscular theory by the beginning of the nineteenth century. By the end of that century, however, problems had emerged with the standard wave theory. The first problem came from attempts to understand the interactions of matter and light. A body at a stable temperature will radiate and absorb light. The body will be at equilibrium with the light, which has its energy distributed among various frequencies. The frequencies of light emitted by the body varies with its temperature, as is familiar from experience. However, by the end of the nineteenth century physicists found that they could not account theoretically for that frequency distribution. Two attempts to do so within the framework of Maxwell's theory of electromagnetic radiation and statistical mechanics led to two laws--Wien's law and the Rayleigh-Jeans law--that failed at lower and higher frequencies, respectively.

In 1900, Max Planck postulated a compromise law that accounted for the observed frequency distribution, but it had the odd feature of postulating that light transmitted energy in discrete packets. The packets at each frequency had energy E = hn, where h is Planck's constant and n is the frequency of the light. Planck regarded his law as merely a phenomenological law, and thought of the underlying distribution of energy as continuous. In 1905, Einstein proposed that all electromagnetic radiation consists of discrete particle-like packets of energy, each with an energy hn. He referred to these packets, which we now call photons, as quanta of light. Einstein used this proposal to explain the photoelectric effect (where light shining on an electrode causes the emission of electrons).

After it became apparent that light, a wave phenomenon, had particle-like aspects, physicists began to wonder whether particles such as electrons might also have wave-like aspects. L. de Broglie proposed that they did and predicted that under certain conditions electrons would exhibit wave-like behavior such as interference and diffraction. His prediction turned out to be correct: a diffraction pattern can be observed when electrons are reflected off a suitable crystal lattice (see Figure 1, below).

Schrödinger soon developed a formula for describing the wave associated with electrons, both in conditions where they are free and when they are bound by forces. Famously, his formula was able to account for the various possible energy levels that had already been postulated for electrons orbiting a nucleus in an atom. At the same time, Heisenberg was working on a formalism to account for the various patterns of emission and absorption of light by atoms. It soon became apparent that the discrete energy levels in Heisenberg's abstract mathematical formalism corresponded to the energy levels of the various standing waves that were possible in an atom according to Schrödinger's equation.

Thus, Schrödinger proposed that electrons could be understood as waves in physical space, governed by his equation. Unfortunately, this turned out to be a plausible interpretation only in the case of a single particle. With multiple-particle systems, the wave associated with Schrödinger's equation was represented in a multi-dimensional space (3n dimensions, where n is the number of particles in the system). In addition, the amplitudes of the waves were often complex (as opposed to real) numbers. Thus, the space in which Schrödinger's waves exist is an abstract mathematical space, not a physical space. Thus, the waves associated with electrons in Schrödinger's wave mechanics could not be interpreted generally as "electron waves" in physical space. In any case, the literal interpretation of the wave function did not cohere with the observed particle-like aspects of the electron, e.g., that electrons are always detected at particular points, never as "spread-out" waves.

The Probability Interpretation Of Schrödinger's Waves; The Projection Postulate

M. Born contributed to a solution to the problem by suggested that the square of the amplitude of Schrödinger's wave at a certain point in the abstract mathematical space of the wave represented the probability of finding the particular value (of position or momentum) associated with that point upon measurement. This only partially solves the problem, however, since the interference effects are physically real in cases such as electron diffraction and the famous two-slit experiment (see Figure 2).

If both slits are open, the probability of an electron impinging on a certain spot on the photographic plate is not the sum of the probabilities that it would impinge on that spot if it passes through slit 1 and that it would impinge on that spot if it passes through slit 2, as is illustrated by pattern (a). Instead, if both slits are opinion you get an interference pattern of the sort illustrated by pattern (b). This is the case even if the light source is so weak that you are in effect sending only one photon at a time through the slits. Eventually, an interference pattern emerges, one point at a time.

Interestingly, the interference pattern disappears if you place a detector just behind one of the slits to determine whether the photon passed through that slit or not, and you get a simple sum of the waves as illustrated by pattern (a). Thus, you can't explain the interference pattern as resulting from interaction among the many photons in the light source.

A similar sort of pattern is exhibited by electrons when you measure their spin (a two-valued quantity) in a Stern-Gerlach device. (This device creates a magnetic field that is homogeneous in all but one direction, e.g., the vertical direction; in that case, electrons are deflected either up or down when they pass through the field.) You get interference effects here just as in the two-slit experiment described above. (See Figure 3)

When a detector is placed after the second, left-right oriented Stern-Gerlach device as in (d), the final, up-down oriented Stern-Gerlach device shows that half of the electrons are spin "up" and half of the electrons are spin "down." In this case, the electron beam that impinges on the final device is a definite "mixture" of spin "left" and spin "right" electrons. On the other hand, when no detector is placed after the second, left-right oriented Stern-Gerlach device as in (e), all the electrons are measured "up" by the final Stern-Gerlach device. In that case, we say that the beam of electrons impinging on the final device is a "superposition" of spin "left" and spin "right" electrons, which happens to be equivalent to a beam consisting of all spin "up" electrons. Thus, placing a device before the final detector destroys some information present in the interference of the electron wave, just as placing a detector after a slit in the two-slit experiment destroyed the interference pattern there.

J. von Neumann generalized the formalisms of Schrödinger and Heisenberg. In von Neumann's formalism, the state of a system is represented by a vector in a complex, multi-dimensional vector space. Observable features of the world are represented by operators on this vector space, which encode the various possible values that that observable quantity can have upon measurement. Von Neumann postulated that the "state vector" evolves deterministically in a manner consistent with Schrödinger's equation, until there is a measurement, in which case there is a "collapse," which indeterministically alters the physical state of the system. This is von Neumann's famous "Projection Postulate."

Thus, von Neumann postulated that there were two kinds of change that could occur in a state of a physical system, one deterministic (Schrödinger evolution), which occurs when the system is not being measured, and one indeterministic (projection or collapse), which occurs as a result of measuring the system. The main argument that von Neumann gave for the Projection Postulate is that whatever value of a system is observed upon measurement will be found with certainty upon subsequent measurement (so long as measurement does not destroy the system, of course). Thus, von Neumann argued that the fact that the value has a stable result upon repeated measurement indicates that the system really has that value after measurement.

The Orthodox Copenhagen Interpretation (Bohr)

What about the state of the system in between measurements? Do observable quantities really have particular values that measurement reveals? Or are there real values that exist before measurement that the measurement process itself alters indeterministically? In either case, the system as described by von Neumann's state vectors (and Schrödinger's wave equation) would have to be incomplete, since the state vector is not always an "eigenvector," i.e., it does not always lie along an axis that represents a particular one of the possible values that a particular observable quantity (such as spin "up"). Heisenberg at first took the view that measurement indeterministically alters the state of the system that existed before measurement. This implies that the system was in a definite state before measurement, and that the quantum mechanical formalism gives an incomplete description of physical systems. Famously, N. Bohr proposed an interpretation of the quantum mechanical formalism that denies that the description given by the quantum mechanical formalism is incomplete. According to Bohr, it only makes sense to attribute values to observable quantities of a physical system when system is being measured in a particular way. Descriptions of physical systems therefore only make sense relative to particular contexts of measurement. (This is Bohr's solution to the puzzling wave-particle duality exhibited by entities such as photons and electrons: the "wave" and "particle" aspects of these entities are "complementary," in the sense that it is physically impossible to construct a measuring device that will measure both aspects simultaneously. Bohr concluded that from a physical standpoint it only makes sense to speak about the "wave" or "particle" aspects of quantum entities as existing relative to particular measurement procedures.) One consequence of Bohr's view is that one cannot even ask what a physical system is like between measurements, since any answer to this question would necessarily have to describe what the physical system is like, independent of any particular context of measurement.

It is important to distinguish the two views just described, and to distinguish Bohr's view from a different view that is sometimes attributed to him:

A physical system's observable properties always have definite values between measurement, but we can never know what those values are since the values can only be determined by measurement, which indeterministically disturbs the system. (Heisenberg)

It does not make sense to attribute definite values to a physical system's observable properties except relative to a particular kind of measurement procedure, and then it only makes sense when that measurement is actually being performed. (Bohr)

A physical system's observable properties have definite values between measurement, but these values are not precise, as is the case when the system's observable properties are being measured; rather, the values of the system's observable quantities before measurement are "smeared out" between the particular values that the observable quantity could have upon measurement. (Pseudo-Bohr)

Each of these views interprets a superposition differently, as a representation of our ignorance about the true state of the system (Heisenberg), as a representation of that values that the various observable quantities of the system could have upon measurement (Bohr), or as a representation of the indefinite, imprecise values that the observable quantities have between measurements (Pseudo-Bohr). Accordingly, each of these views interprets projection or collapse differently, as a reduction in our state of ignorance (Heisenberg), as the determination of a definite result obtained when a particular measurement procedure is performed (Bohr), or as an instantaneous localization of the "smeared out," imprecise values of a particular observable quantity (Pseudo-Bohr).

Next time we will discuss several problematic aspects of the Copenhagen understanding of measurement, along with several alternative views.