Scope of this book

Quantum mechanics is a difficult subject, and this book is intended to help the reader overcome the main difficulties in the way to understanding it. The first part of the book, Chs. 2–16, contains a systematic presentation of the basic principles of quantum theory, along with a number of examples which illustrate how these principles apply to particular quantum systems. The applications are, for the most part, limited to toy models whose simple structure allows one to see what is going on without using complicated mathematics or lengthy formulas. The principles themselves, however, are formulated in such a way that they can be applied to (almost) any nonrelativistic quantum system. In the second part of the book, Chs. 17–25, these principles are applied to quantum measurements and various quantum paradoxes, subjects which give rise to serious conceptual problems when they are not treated in a fully consistent manner.

The final chapters are of a somewhat different character. Chapter 26 on decoherence and the classical limit of quantum theory is a very sketchy introduction to these important topics along with some indication as to how the basic principles presented in the first part of the book can be used for understanding them. Chapter 27 on quantum theory and reality belongs to the interface between physics and philosophy and indicates why quantum theory is compatible with a real world whose existence is not dependent on what scientists think and believe, or the experiments they choose to carry out. The Bibliography contains references for those interested in further reading or in tracing the origin of some of the ideas presented in earlier chapters.

Introduction

Despite the fact that classical mechanics employs deterministic dynamical laws, random dynamical processes often arise in classical physics, as well as in everyday life. A stochastic or random process is one in which states-of-affairs at successive times are not related to one another by deterministic laws, and instead probability theory is employed to describe whatever regularities exist. Tossing a coin or rolling a die several times in succession are examples of stochastic processes in which the previous history is of very little help in predicting what will happen in the future. The motion of a baseball is an example of a stochastic process which is to some degree predictable using classical equations of motion that relate its acceleration to the total force acting upon it. However, a lack of information about its initial state (e.g., whether it is spinning), its precise shape, and the condition and motion of the air through which it moves limits the precision with which one can predict its trajectory.

The Brownian motion of a small particle suspended in a fluid and subject to random bombardment by the surrounding molecules of fluid is a well-studied example of a stochastic process in classical physics. Whereas the instantaneous velocity of the particle is hard to predict, there is a probabilistic correlation between successive positions, which can be predicted using stochastic dynamics and checked by experimental measurements. In particular, given the particle's position at a time t, it is possible to compute the probability that it will have moved a certain distance by the time t + Δt.

Hilbert space and inner product

In Ch. 2 it was noted that quantum wave functions form a linear space in the sense that multiplying a function by a complex number or adding two wave functions together produces another wave function. It was also pointed out that a particular quantum state can be represented either by a wave function ψ(x) which depends upon the position variable x, or by an alternative function ψ(p) of the momentum variable p. It is convenient to employ the Dirac symbol |ψ〉, known as a “ket”, to denote a quantum state without referring to the particular function used to represent it. The kets, which we shall also refer to as vectors to distinguish them from scalars, which are complex numbers, are the elements of the quantum Hilbert space H. (The real numbers form a subset of the complex numbers, so that when a scalar is referred to as a “complex number”, this includes the possibility that it might be a real number.)

If α is any scalar (complex number), the ket corresponding to the wave function ɑψ(x) is denoted by α|ψ〉, or sometimes by |ψ〉α, and the ket corresponding to ø(x)+ψ(x) is denoted by |ø〉+|ψ〉 or |ψ〉+|ø〉, and so forth. This correspondence could equally well be expressed using momentum wave functions, because the Fourier transform, (2.15) or (2.16), is a linear relationship between ψ(x) and (p), so that αø(x)+βψ(x) and αϕ(p) + βψ(p) correspond to the same quantum state α|ψ〉+β|ø〉.

Introduction

I place a tape measure with one end on the floor next to a table, read the height of the table from the tape, and record the result in a notebook. What are the essential features of this measurement process? The key point is the establishment of a correlation between a physical property (the height) of a measured system (the table) and a suitable record (in the notebook), which is itself a physical property of some other system. It will be convenient in what follows to think of this record as part of the measuring apparatus, which consists of everything essential to the measuring process apart from the measured system. Human beings are not essential to the measuring process. The height of a table could be measured by a robot. In the modern laboratory, measurements are often carried out by automated equipment, and the results stored in a computer memory or on magnetic tape, etc. While scientific progress requires that human beings pay attention to the resulting data, this may occur a long time after the measurements are completed.

In this and the next chapter we consider measurements as physical processes in which a property of some quantum system, which we shall usually think of as some sort of “particle”, becomes correlated with the outcome of the measurement, itself a property of another quantum system, the “apparatus”. Both the measured system and the apparatus which carries out the measurement are to be thought of as parts of a single closed quantum mechanical system. This makes it possible to apply the principles of quantum theory developed in earlier chapters.

The problem of quantum state reduction

The fundamental problem of quantum mechanics, as that theory is presently understood, is to make sense of the reduction of the state vector (i.e. collapse of the wave function), denoted here by R. This issue is usually addressed in terms of the ‘quantum measurement problem’, which is to comprehend how, upon measurement of a quantum system, this (seemingly) discontinuous R-process can come about. A measurement, after all, merely consists of the quantum state under consideration becoming entangled with a more extended part of the physical universe, e.g. with a measuring apparatus. This measuring apparatus – together with the observing physicist and their common environment – should, according to conventional understanding, all also have some quantum description. Accordingly, there should be a quantum description of this entire quantum state, involving not only the original system under consideration but also the apparatus, physicist, and remaining environment – and this entire state would be expected to evolve continuously, solely according to the Schrödinger equation (unitary evolution), here denoted by the symbol U.

Numerous different attitudes to R have been expressed over many years, ever since quantum mechanics was first clearly formulated. The most influential viewpoint has been the ‘Copenhagen interpretation’ of Niels Bohr, according to which the state vector │ψ〉 is not to be taken seriously as describing a quantum-level physical reality, but is to be regarded as merely referring to our (maximal) ‘knowledge’ of a physical system, and whose ultimate role is simply to provide us with a means to calculate probabilities when a measurement is performed on the system.

Introduction

Physicists who work on canonical quantum gravity will sometimes remark that the general covariance of general relativity is responsible for many of the thorniest technical and conceptual problems in their field. In particular, it is sometimes alleged that one can trace to this single source a variety of deep puzzles about the nature of time in quantum gravity, deep disagreements surrounding the notion of ‘observable’ in classical and quantum gravity, and deep questions about the nature of the existence of spacetime in general relativity.

Philosophers who think about these things are sometimes sceptical about such claims. We have all learned that Kretschmann was quite correct to urge against Einstein that the ‘General Theory of Relativity’ was no such thing, since any theory could be cast in a generally covariant form, and hence that the general covariance of general relativity could not have any physical content, let alone bear the kind of weight that Einstein expected it to. Friedman's assessment is widely accepted: ‘As Kretschmann first pointed out in 1917, the principle of general covariance has no physical content whatever: it specifies no particular physical theory; rather, it merely expresses our commitment to a certain style of formulating physical theories’ (Friedman 1983, p. 44). Such considerations suggest that general covariance, as a technically crucial but physically contentless feature of general relativity, simply cannot be the source of any significant conceptual or physical problems.