Thus far we have concentrated on the postulates, on their ramifications, and on the means and methods by which they are applied. Connections with classical physics have been featured, and primary notions other than those of Chapter 5 have not been needed. We deviate from this path in the present chapter, introducing a new and extremely important concept; it is based on one of the most significant and fruitful experiments of the quantal era. This experiment was performed by O. Stern and W. Gerlach (1921) and subsequently interpreted by G. Uhlenbeck and S. Goudsmit (1925).

The experiment and its elucidation led to the new concept of “intrinsic spin” (in particular to spin “½”), a development that was one of the major turning points in modern physics. The consequences have been monumental, and include primary theoretical developments; the interpretation of data related to, and an understanding of, empirical rules concerning a wide variety of physical systems (e.g., the periodic table of the elements/atomic structure); and the generation of new experimental techniques and of innovative measurements. Our study of intrinsic spin and its correlative, two-state systems, introduces the reader to one of the most profound topics in quantum theory.

The Stern–Gerlach experiment and its interpretation

The experiment of Stern and Gerlach was designed to test the “spatial quantization” of neutral atoms, i.e., to determine whether the (orbital) angular momentum was quantized. The basic ideas are as follows. Associated with the atom's orbital angular momentum would be a magnetic moment, as in the Bohr model of hydrogen.

Quantum theory is one of the most difficult subjects in the physics curriculum. In part this is because of unfamiliar mathematics: partial differential equations, Fourier transforms, complex vector spaces with inner products. But there is also the problem of relating mathematical objects, such as wave functions, to the physical reality they are supposed to represent. In some sense this second problem is more serious than the first, for even the founding fathers of quantum theory had a great deal of difficulty understanding the subject in physical terms. The usual approach found in textbooks is to relate mathematics and physics through the concept of a measurement and an associated wave function collapse. However, this does not seem very satisfactory as the foundation for a fundamental physical theory. Most professional physicists are somewhat uncomfortable with using the concept of measurement in this way, while those who have looked into the matter in greater detail, as part of their research into the foundations of quantum mechanics, are well aware that employing measurement as one of the building blocks of the subject raises at least as many, and perhaps more, conceptual difficulties than it solves.

It is in fact not necessary to interpret quantum mechanics in terms of measurements. The primary mathematical constructs of the theory, that is to say wave functions (or, to be more precise, subspaces of the Hilbert space), can be given a direct physical interpretation whether or not any process of measurement is involved.

Some general principles

There are some important differences between quantum and classical reasoning which reflect the different mathematical structure of the two theories. The most precise classical description of a mechanical system is provided by a point in the classical phase space, while the most precise quantum description is a ray or onedimensional subspace of the Hilbert space. This in itself is not an important difference. What is more significant is the fact that two distinct points in a classical phase space represent mutually exclusive properties of the physical system: if one is a true description of the sytem, the other must be false. In quantum theory, on the other hand, properties are mutually exclusive in this sense only if the corresponding projectors are mutually orthogonal. Distinct rays in the Hilbert space need not be orthogonal to each other, and when they are not orthogonal, they do not correspond to mutually exclusive properties. As explained in Sec. 4.6, if the projectors corresponding to the two properties do not commute with one another, and are thus not orthogonal, the properties are (mutually) incompatible. The relationship of incompatibility means that the properties cannot be logically compared, a situation which does not arise in classical physics. The existence of this nonclassical relationship of incompatibility is a direct consequence of assuming (following von Neumann) that the negation of a property corresponds to the orthogonal complement of the corresponding subspace of the Hilbert space; see the discussion in Sec. 4.6.

Classical and quantum properties

We shall use the term physical property to refer to something which can be said to be either true or false for a particular physical system. Thus “the energy is between 10 and 12 μJ” or “the particle is between x1 and x2” are examples of physical properties. One must distinguish between a physical property and a physical variable, such as the position or energy or momentum of a particle. A physical variable can take on different numerical values, depending upon the state of the system, whereas a physical property is either a true or a false description of a particular physical system at a particular time. A physical variable taking on a particular value, or lying in some range of values, is an example of a physical property.

In the case of a classical mechanical system, a physical property is always associated with some subset of points in its phase space. Consider, for example, a harmonic oscillator whose phase space is the x, p plane shown in Fig. 2.1 on page 12. The property that its energy is equal to some value E0 > 0 is associated with a set of points on an ellipse centered at the origin. The property that the energy is less than E0 is associated with the set of points inside this ellipse. The property that the position x lies between x1 and x2 corresponds to the set of points inside a vertical band shown cross-hatched in this figure, and so forth.

Introduction

The conditions which define a consistent family of histories were stated in Ch. 10. The sample space must consist of a collection of mutually orthogonal projectors that add up to the history identity, and the chain operators for different members of the sample space must be mutually orthogonal, (10.20). Checking these conditions is in principle straightforward. In practice it can be rather tedious. Thus if there are n histories in the sample space, checking orthogonality involves computing n chain operators and then taking n(n − 1)/2 operator inner products to check that they are mutually orthogonal. There are a number of simple observations, some definitions, and several “tricks” which can simplify the task of constructing a sample space of a consistent family, or checking that a given sample space is consistent. These form the subject matter of the present chapter. It is probably not worthwhile trying to read through this chapter as a unit. The reader will find it easier to learn the tricks by working through examples in Ch. 12 and later chapters, and referring back to this chapter as needed.

The discussion is limited to families in which all the histories in the sample space are of the product form, that is, represented by a projector on the history space which is a tensor product of quantum properties at different times, as in (8.7). As in the remainder of this book, the “strong” consistency conditions (10.20) are used rather than the weaker (10.25).

Bohm version of the EPR paradox

Einstein, Podolsky, and Rosen (EPR) were concerned with the following issue. Given two spatially separated quantum systems A and B and an appropriate initial entangled state, a measurement of a property on system A can be an indirect measurement of B in the sense that from the outcome of the A measurement one can infer with probability 1 a property of B, because the two systems are correlated. There are cases in which either of two properties of B represented by noncommuting projectors can be measured indirectly in this manner, and EPR argued that this implied that system B could possess two incompatible properties at the same time, contrary to the principles of quantum theory.

In order to understand this argument, it is best to apply it to a specific model system, and we shall do so using Bohm's formulation of the EPR paradox in which the systems A and B are two spin-half particles a and b in two different regions of space, with their spin degrees of freedom initially in a spin singlet state (23.2). As an aid to later discussion, we write the argument in the form of a set of numbered assertions leading to a paradox: a result which seems plausible, but contradicts the basic principles of quantum theory. The assertions E1–E4 are not intended to be exact counterparts of statements in the original EPR paper, even when the latter are translated into the language of spin-half particles. However, the general idea is very similar, and the basic conundrum is the same.

Introduction

A composite system is one involving more than one particle, or a particle with internal degrees of freedom in addition to its center of mass. In classical mechanics the phase space of a composite system is a Cartesian product of the phase spaces of its constituents. The Cartesian product of two sets A and B is the set of (ordered) pairs {(a, b)}, where a is any element of A and b is any element of B. For three sets A, B, and C the Cartesian product consists of triples {(a, b, c)}, and so forth. Consider two classical particles in one dimension, with phase spaces x1, p1 and x2, p2. The phase space for the composite system consists of pairs of points from the two phase spaces, that is, it is a collection of quadruples of the form x1, p1, x2, p2, which can equally well be written in the order x1, x2, p1, p2. This is formally the same as the phase space of a single particle in two dimensions, a collection of quadruples x, y, px, py. Similarly, the six-dimensional phase space of a particle in three dimensions is formally the same as that of three one-dimensional particles.

In quantum theory the analog of a Cartesian product of classical phase spaces is a tensor product of Hilbert spaces. A particle in three dimensions has a Hilbert space which is the tensor product of three spaces, each corresponding to motion in one dimension. The Hilbert space for two particles, as long as they are not identical, is the tensor product of the two Hilbert spaces for the separate particles.

Classical sample space and event algebra

Probability theory is based upon the concept of a sample space of mutually exclusive possibilities, one and only one of which actually occurs, or is true, in any given situation. The elements of the sample space are sometimes called points or elements or events. In classical and quantum mechanics the sample space usually consists of various possible states or properties of some physical system. For example, if a coin is tossed, there are two possible outcomes: H (heads) or T (tails), and the sample space S is {H, T}. If a die is rolled, the sample space S consists of six possible outcomes: s = 1, 2, 3, 4, 5, 6. If two individuals A and B share an office, the occupancy sample space consists of four possibilities: an empty office, A present, B present, or both A and B present.

Associated with a sample space S is an event algebra B consisting of subsets of elements of the sample space. In the case of a die, “s is even” is an event in the event algebra. So are “s is odd”, “s is less than 4”, and “s is equal to 2.” It is sometimes useful to distinguish events which are elements of the sample space, such as s = 2 in the previous example, and those which correspond to more than one element of the sample space, such as “s is even”. We shall refer to the former as elementary events and to the latter as compound events.

Quantum paradoxes

The next few chapters are devoted to resolving a number of quantum paradoxes in the sense of giving a reasonable explanation of a seemingly paradoxical result in terms of the principles of quantum theory discussed earlier in this book. None of these paradoxes indicates a defect in quantum theory. Instead, when they have been properly understood, they show us that the quantum world is rather different from the world of our everyday experience and of classical physics, in a way somewhat analogous to that in which relativity theory has shown us that the laws appropriate for describing the behavior of objects moving at high speed differ in significant ways from those of pre-relativistic physics.

An inadequate theory of quantum measurements is at the root of several quantum paradoxes. In particular, the notion that wave function collapse is a physical effect produced by a measurement, rather than a method of calculation, see Sec. 18.2, has given rise to a certain amount of confusion. Smuggling rules for classical reasoning into the quantum domain where they do not belong and where they give rise to logical inconsistencies is another common source of confusion. In particular, many paradoxes involve mixing the results from incompatible quantum frameworks.

Certain quantum paradoxes have given rise to the idea that the quantum world is permeated by mysterious influences that propagate faster than the speed of light, in conflict with the theory of relativity. They are mysterious in that they cannot be used to transmit signals, which means that they are, at least in any direct sense, experimentally unobservable.