This book, on the interpretation of quantum mechanics and the measurement process, has evolved from lectures which I gave at the University of Turku (Finland) in 1991 and later in several improved and extended versions at the University of Cologne. In these lectures as well as in the present book I have aimed to show the intimate relations between quantum mechanics and its interpretation that are induced by the quantum mechanical measurement process. Consequently, the book is concerned both with the philosophical, metatheoretical problems of interpretation and with the more formal problems of quantum object theory.

The book is based on the idea that quantum mechanics is valid not only for microscopic objects but also for the macroscopic apparatus used for quantum mechanical measurements. We illustrate the consequences of this assumption, which turn out to be partly very promising and partly rather disappointing. On the one hand we can give a rigorous justification of some important parts of the interpretation, such as the probability interpretation, by means of object theory (chapter 3). On the other hand, the problem of the objectification of measurement results leads to inconsistencies that cannot be resolved in an obvious way (chapter 4). This open problem has far-reaching consequences for the possibility of recognising an objective reality in physics.

The manuscript of this book was carefully written in TEX by Dipl. Phys. Falko Spiller. In addition, he proposed numerous small corrections and improvements of the first version of the text.

Radiative transitions

When a system changes its energy as the result of the emission or the absorption of a photon, it is said to undergo radiative transition. There are three kinds of radiative transition. In the presence of an electromagnetic field the system can absorb a photon, so that its energy is raised to a higher level. If the system is initially in a state other than the ground state, it can emit a photon and so shed energy. This can happen without the presence of any external electromagnetic field, in which case a spontaneous emission is said to occur. On the other hand, if the excited system is placed in an electromagnetic field that varies in time with the appropriate frequency, the probability that it will emit a photon can be greatly enhanced. In this case the emission process is known as stimulated emission. Stimulated emission is the basis of maser and laser action, which we describe in this chapter.

First, we construct a simple model that allows us to investigate how the internal structure of a quantum system changes as the result of a radiative transition. Although the model is very simple, it will allow us to understand the main features of the proper, more exact treatment. We shall deal mainly with absorption and stimulated emission, the two types of radiative transition where there is an external electromagnetic field.

The concept of measurement

Basic requirements

In this chapter we give a brief account of the quantum theory of measurement. As already mentioned in the preceding chapter, the quantum theory of measurement treats the object system, as well as the measuring apparatus, as proper quantum systems. Here we restrict our considerations to a proper quantum mechanical model of the measuring process that makes use of unitary premeasurements. Furthermore, we will be mainly concerned with ordinary discrete observables of the object system that are measured by an apparatus with a pointer observable which is also assumed to be an ordinary discrete observable. These restrictive assumptions are made here and throughout the entire book in order to simplify the problems as much as possible. The remaining open problems of consistency, completeness, self-referentiality, etc. can then be discussed without unnecessary additional complications.

In order to characterize the concept of measurement in quantum mechanics, we formulate some basic requirements that must be fulfilled by any measuring process. In many situations, one can add further postulates, but these additional requirements are not essential for the concept of measurement. The basic requirements are in accordance with the most general interpretation of quantum mechanics, the minimal interpretation, which has already been mentioned in chapter 1. There is a general interplay between interpretation and the quantum theory of measurement, since the postulates that characterize a given interpretation must be compatible with, and capable of being satisfied by, a corresponding model of the measuring process.

Measurement-induced interrelations between quantum mechanics and its interpretation

The development of quantum mechanics

The formalism of quantum mechanics was developed within the very short period of a few months in 1925 and 1926 by Heisenberg [Heis 25] and by Schrodinger [Schrö 26], respectively. Together with the contributions of Born and Jordan [BoJo 26], [BHJ 26], Dirac [Dir 26] and others, the formalism of this theory was already brought in 1926 into its final form, which is still used in present-day text books. (For all details of the historical development, we refer to the monograph by M. Jammer [Jam 74].) It is a very remarkable fact that a theory which was formulated 70 years ago has never been corrected or improved and is still considered to be valid. Numerous experiments performed during this long period to test the theory have confirmed it to a very high degree of accuracy without any exception. Hence there are good reasons to believe that quantum mechanics is universally valid and can be applied to all domains of reality, i.e., to atoms, molecules, macroscopic bodies, and to the whole universe.

However, the interpretation of the new theory was at the time of its mathematical formulation still an almost open problem. Any interpretation of quantum theory should provide interrelations between the theoretical expressions of the theory and possible experimental outcomes.

Electrons in crystals

A crystal consists of a collection of atoms arranged in a regular array, the spacing between atoms being of the same order of magnitude as the dimensions of the atoms. Each atom is more or less anchored to one point, called its site in the lattice, by the electrostatic forces produced by all the other atoms. We shall not find it necessary here to discuss the details of how this comes about; nor shall we consider the various patterns in which the atoms can be arranged. It will be sufficient to remember the essential feature that the structure of the crystal is periodic in space.

We have seen in chapter 5 that the energy of an electron bound to an atom is restricted to certain discrete values. Imagine that we can assemble a crystal of identical atoms whose spacing L can be altered at will. If L is large enough, the motion of an electron in one of the atoms will be affected to a negligible extent by the electrons and nuclei of the other atoms. Each atom then behaves as if it were isolated, with its electrons in discrete bound states. In figure 10.1 (a) we have drawn a schematic diagram of the potential V(r) in which an electron moves in this situation. Suppose that the spacing L is now reduced (figure 10.1(b)). The potential V(r) in the neighbourhood of a given atom is now affected by the presence of the nuclei and electrons of the other atoms, particularly those that are closest.

Impurities

We have so far described the properties of intrinsic semiconductors, in which the crystal lattice is perfectly regular. However, the electrical properties of semiconductors, unlike metals or semimetals, are drastically affected by the addition of small traces of impurity atoms. Observable effects occur with impurity concentrations as low as a few parts in 108, and increasing the impurity concentration to one part in 105 can increase the conductivity by as much as a factor of 103 at room temperature and 1012 at liquid-helium temperatures. The semiconductor is said to be doped with impurity atoms.

In order to study the effect of doping, we first consider a crystal consisting of a periodic array of one type of atom, except that just one of the atoms has been replaced by an atom of a different type. As a one-dimensional model of this situation, we take the same infinite chain of square wells as in chapter 10, but with one of the wells having a different depth, U1 say (figure 12.1).

We recall that for the perfectly regular crystal, the stationary state solutions are Bloch waves of the form (10.3). These have the property that when the required continuity conditions on the wave function are satisfied at the two edges of one of the potential wells, they are automatically satisfied also at the edges of all the other wells.

The concept of objectification

Weak and strong objectification

Let S be a quantum system that is prepared in a pure state φ and A a discrete nondegenerate observable with eigenvalues ai and eigenstates φai.

If the preparation φ is not an eigenstate of A, then quantum mechanics does not provide any information about the value of the observable A. The pair 〈φ,A〉 merely defines a probability distribution p(φ,ai), the experimental meaning of which is given by the statistical interpretation of quantum mechanics: the real positive number p(φ ai) is the probability of obtaining the result ai after a measurement process of the observable A. This means that if one were to perform a large series of N measurements of this kind, then the relative frequency fN(φ,ai) of the result ai would approach for almost all test series the probability p(φ, ai), for N → ∞ (cf. chapter 3). One could, however, in addition to these well-established results tentatively assume that a certain value ai or even an eigenstate φai of A pertains objectively to the system S but that this value or state is subjectively unknown to the observer, who knows only the probability p(φ, ai) and hence the distribution of possible measurement outcomes. The probability would then express the subjective ignorance (or knowledge) about an objectively decided value or eigenstate of A. The hypothetical attribution of a certain value or eigenstate of A to the system S will be called objectification.

Where is the frontier of physics? Some would say 10−33 cm, some 10−15 cm and some 10+28 cm. My vote is for 10−6 cm. Two of the greatest puzzles of our age have their origins at this interface between the macroscopic and microscopic worlds. The older mystery is the thermodynamic arrow of time, the way that (mostly) time-symmetric microscopic laws acquire a manifest asymmetry at larger scales. And then there's the superposition principle of quantum mechanics, a profound revolution of the twentieth century. When this principle is extrapolated to macroscopic scales, its predictions seem wildly at odds with ordinary experience.

This book deals with both these ‘mysteries,’ the foundations of statistical mechanics and the foundations of quantum mechanics. It is my thesis that they are related. Moreover, I have teased the reader with the word ‘foundations,’ a term that many of our hardheaded colleagues view with disdain. I think that new experimental techniques will soon subject these ‘foundations’ to the usual scrutiny, provided the right questions and tests can be formulated. Historically, it is controlled observation that transforms philosophy into science, and I am optimistic that the time has come for speculations on these two important issues to undergo that transformation.