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.

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.

Galileo's Telescope

In the summer of 1609, Galileo Galilei, professor of mathematics at the University of Padua, began constructing telescopes and using them to look at the Moon and stars. By January the next year he had seen that the Moon is not smooth, that there are far more stars than are visible to the naked eye, that the Milky Way is made of a myriad stars and that the planet Jupiter has faint “Jovian planets” (satellites) revolving about it. Galileo forthwith brought out a short book, The Starry Messenger (the Latin title was Sidereus Nuncius), to describe his discoveries, which quickly became famous. The English ambassador to the Venetian Republic reported (I quote from Nicolson's Science and Imagination):

What Are Weak Forces?

Newton, Faraday and Einstein each believed that there must be some sort of unity in the forces of nature. Faraday, having unified electricity and magnetism, conducted experiments to try to find a connection between these forces and gravity. From 1922 until his death in 1955, Einstein was trying mathematically to unify gravity with electromagnetism. In the event, the force that was eventually unified with electromagnetism is one that was unknown to Newton and Faraday, and perhaps not considered very important by Einstein. This is the so-called weak force. In previous chapters I have ignored the weak force, but now I must explain what it is.

In 1896 (a few months after the discovery of X-rays) Henri Becquerel in Paris discovered the radioactivity of uranium. Studied further by Marie and Pierre Curie, the nature of radioactivity was finally elucidated by Rutherford and Soddy in Montreal during the early 1900s.

In radioactivity, an atomic nucleus emits “rays”, unpredictably but with a definite probability per unit time, often changing into a different nucleus in the process. By a historical accident, three totally different processes are lumped under the heading “radioactivity”. One is the emission of alpha-particles, the nuclei of helium atoms, each made up of two protons and two neutrons. This happens as a result of the strong, nuclear forces. The only reason that it happens slowly (slowly compared to the time for an alpha particle to traverse the nucleus) is that the alpha particle's wave function has only a very small “tail” outside the nucleus.

I have tried to write a non-technical tour through the principles of physics. The theme running through this tour is that progress has often consisted in uncovering “hidden unities”. Let me explain what I mean by this phrase, taking the example (from Chapter 3) of electricity and magnetism. The unity here is hidden, because at first sight there seemed to be no connection between the two. The invention of the electric battery at the beginning of the nineteenth century ushered in a new period of research that showed that electricity and magnetism are interconnected when they change with time. This did not mean that electricity and magnetism are the same thing. They are certainly different, but they are two aspects of a unified whole, “electromagnetism”. In general, it makes no sense to talk about one without the other.

This pattern of unification is fairly typical. Every time such a unification is achieved, the number of “laws of nature” is reduced, so that nature looks not only more unified but also, in some sense, simpler. More and more apparently diverse phenomena are explained by fewer and fewer underlying principles. This is the message I have tried to get across.

This book has a second theme. Quite often, different branches of physics have seemed to contradict each other when taken together. The contradiction is then resolved in a new, consistent, wider theory, which includes the two branches. For example, Newton's theory of motion and of gravitation conflicted with electromagnetism, as it was understood in the nineteenth century. The resolution lay in Einstein's theories of relativity. There are several other instances of progress by resolution of contradictions in this book.