The Dreams That Stuff is Made of

by Stephen Hawking

  The intervals in the system of the elements, in which a further development of an inner electron group takes place because of the entrance into the normal atom of electron orbits of a certain type, are designated in the figure by the horizontal lines, which are drawn between the vertical lines to which the quantum symbols are affixed. It is clear that such a development of an inner group is everywhere reflected in the curves. Particularly the course of the N- and O-curves may be regarded as a direct indication of that stage in the development of the electron groups with 4-quantum orbits of which the occurrence of the rare-earths bears witness. Although the apparent complete absence of a reflection in the X-ray spectra of the complicated relationships exhibited by most other properties of the elements was the typical and important feature of Moseley’s discovery, we can recognize, nevertheless, in the light of the progress of the last years, an intimate connexion between the X-ray spectra and the general relationships between the elements within the natural system.

  Before concluding this lecture I should like to mention one further point in which X-ray investigations have been of importance for the test of the theory. This concerns the properties of the hitherto unknown element with atomic number 72. On this question opinion has been divided in respect to the conclusions that could be drawn from the relationships within the Periodic Table, and in many representations of the table a place is left open for this element in the rare-earth family. In Julius Thomsen’s representation of the natural system, however, this hypothetical element was given a position homologous to titanium and zirconium in much the same way as in our representation in Fig. 1. Such a relationship must be considered as a necessary consequence of the theory of atomic structure developed above, and is expressed in the table (Fig. 9) by the fact that the electron configurations for titanium and zirconium show the same sort of resemblances and differences as the electron configurations for zirconium and the element with atomic number 72. A corresponding view was proposed by Bury on the basis of his above-mentioned systematic considerations of the connexion between the grouping of the electrons in the atom and the properties of the elements.

  Recently, however, a communication was published by Dauvillier announcing the observation of some weak lines in the X-ray spectrum of a preparation containing rare-earths. These were ascribed to an element with atomic number 72 assumed to be identical with an element of the rare-earth family, the existence of which in the preparation used had been presumed by Urbain many years ago. This conclusion would, however, if it could be maintained, place extraordinarily great, if not unsurmountable, difficulties in the way of the theory, since it would claim a change in the strength of the binding of the electrons with the atomic number which seems incompatible with the conditions of the quantum theory. In these circumstances Dr. Coster and Prof. Hevesy, who are both for the time working in Copenhagen, took up a short time ago the problem of testing a preparation of zircon-bearing minerals by X-ray spectroscopic analysis. These investigators have been able to establish the existence in the minerals investigated of appreciable quantities of an element with atomic number 72, the chemical properties of which show a great similarity to those of zirconium and a decided difference from those of the rare-earthsbm.

  I hope that I have succeeded in giving a summary of some of the most important results that have been attained in recent years in the field of atomic theory, and I should like, in concluding, to add a few general remarks concerning the viewpoint from which these results may be judged, and particularly concerning the question of how far, with these results, it is possible to speak of an explanation, in the ordinary sense of the word. By a theoretical explanation of natural phenomena we understand in general a classification of the observations of a certain domain with the help of analogies pertaining to other domains of observation, where one presumably has to do with simpler phenomena. The most that one can demand of a theory is that this classification can be pushed so far that it can contribute to the development of the field of observation by the prediction of new phenomena.

  When we consider the atomic theory, we are, however, in the peculiar position that there can be no question of an explanation in this last sense, since here we have to do with phenomena which from the very nature of the case are simpler than in any other field of observation, where the phenomena are always conditioned by the combined action of a large number of atoms. We are therefore obliged to be modest in our demands and content ourselves with concepts which are formal in the sense that they do not provide a visual picture of the sort one is accustomed to require of the explanations with which natural philosophy deals. Bearing this in mind I have sought to convey the impression that the results, on the other hand, fulfill, at least in some degree, the expectations that are entertained of any theory; in fact, I have attempted to show how the development of atomic theory has contributed to the classification of extensive fields of observation, and by its predictions has pointed out the way to the completion of this classification. It is scarcely necessary, however, to emphasize that the theory is yet in a very preliminary stage, and many fundamental questions still await solution.

  Chapter Three

  The following papers are some of the most central to the development of quantum mechanics, and it would be very hard to overstate their importance. In them, the beginnings of quantum theory are transformed from a group of loosely collected ideas into a fully developed theory capable of describing much of the physical world. Without the insight gained from these papers, many of the technological innovations of the last century would have been impossible. In this section, we see in the papers of Erwin Schrodinger and lectures by Werner Heisenberg the establishment of two full and independent quantum theories. This at first seemed a problem: How can two different theories describe the same reality? In 1926 Schrodinger himself demonstrated the equivalency of his and Heisenberg’s approaches.

  Heisenberg’s method is included here as described in a series of lectures he presented in 1929 at the University of Chicago, and is summarized in his Nobel lecture in 1933. In the Heisenberg formulation, quantities that can be observed, for example a particle’s position or energy, are represented by square matrices. It is well known from mathematics that a square matrix can be characterized by a set of vectors, called the eigenvectors, and a set of numbers, called the eigenvalues. In Heisenberg’s matrix representation of quantum mechanics, the eigenvalues represent all possible values of an experimental observation of the quantity represented by that matrix. For example, the eigenvalues of the matrix corresponding to a particle’s position would be every possible location that the particle can have. When an observation is made, a certain eigenvalue is measured, and the corresponding eigenvector represents the state of the system immediately after that observation. The state of the particle is said to have “collapsed” into a state represented by that eigenvector. Thus, the act of measurement itself alters the state of the system. This is one of the most important and fundamental conclusions of quantum mechanics. No matter how cleverly we design our experiments, we will never be able to find a way to take a measurement without altering the system in some way. This is mathematically expressed in Heisenberg’s Uncertainty Principle.

  In mathematics, matrix multiplication does not always commute. This means that if A and B are matrices, A times B is not always the same as B times A. In Heisenberg’s quantum mechanics, observable quantities correspond to matrices, and their non-commutation corresponds to the idea that measuring one quantity and then a second will give a different answer than measuring the second quantity first. For example, measuring a particle’s position then its speed will result in different values than measuring its speed first then its position. We can understand this result like this: A measurement of position will alter the original state. Taking a measurement of speed afterward will be on this altered state, and so will produce a different answer than if it were taken on the original state. The more precisely we measure the particle’s speed, the more we alter the state, and conse
quently the less we can know about its original velocity. It is impossible to get a complete measurement of both simultaneously. This idea was extremely revolutionary to the physicists at the time. It required them to abandon the classical idea of a particle’s trajectory, or the path it takes when it moves. When we throw a ball, we can watch its position and measure its speed at every point along its path. But for a quantum particle this is impossible. The impossibility comes not from our inability to design a clever enough experiment, but seems is inherent in the physical laws of nature as formalized in the Heisenberg Uncertainty Principle.

  Less than a year after Heisenberg published his formulation of quantum mechanics, Schrodinger published a completely different method in a series of four papers entitled, “Quantization as an Eigenvalue Problem.” His approach was inspired by the wave-particle duality proposed by Albert Einstein and Louis de Broglie. In explaining the photoelectric effect, Einstein showed that electromagnetic waves could act as particles. De Broglie took this a step further and theorized that perhaps matter that is typically thought to be particle-like could act like waves. Schrodinger took this idea and looked for a wave equation that could describe matter. In the first of these four exceptional papers, Schrodinger made a brilliant guess as to what the proper equation should be. This wave equation is now known as the Schrodinger equation. It establishes a way to derive the time evolution of a quantum system that is evolving under the influence of an arbitrary force. It is the quantum mechanical equivalent to Newton’s laws, which are the basic equations of classical mechanics. Schrodinger recognized that he could solve for the possible states of an electron in a hydrogen atom using his equation. In the hydrogen atom the only force is the electrostatic force between the electron and the proton in the nucleus. When the solution of his equation for the electrostatic force was coupled with spin, which will be discussed in the next chapter, he was able to re-derive the hydrogen atomic spectrum that Niels Bohr had earlier explained using the planetary model. Bohr’s explanation required an assumption of energy quantization without having a good theoretical basis for doing so. Now with Schrodinger’s and Heisenberg’s quantum theories it was possible to understand why that assumption was needed. All the pieces were coming together to form a quantum theory capable of explaining much of the observable world.

  EXCERPTS FROM: THE PHYSICAL PRINCIPLES OF THE QUANTUM THEORY

  BY

  WERNER HEISENBEG

  Translated into English by Carl Eckart and Frank C. Hoyt

  FOREWORD TO THE ENGLISH EDITION

  It is an unusual pleasure to present Professor Heisenberg’s Chicago lectures on “The Physical Principles of the Quantum Theory” to a wider audience than could attend them when they were originally delivered. Professor Heinsenberg’s leading place in the development of the new quantum mechanics is well recognized by those who have been following its growth. It was in fact he who first saw clearly that in the older forms of quantum theory we were describing our spectra in terms of atomic mechanisms regarding which we could gain no definite knowledge, and who first found a way to interpret (or at least describe) spectroscopic phenomena without assuming the existence of such atomic mechanisms. Likewise, “the uncertainty principle” has become a household phrase throughout our universities, and it is especially fortunate to have this opportunity of learning its significance from one who is responsible for its formulation.

  The power of the new quantum mechanics in giving us a better understanding of events on an atomic scale is becoming increasingly evident. The structure of the helium atom, the existence of half-quantum numbers in band spectra, the continuous spatial distribution of photo-electrons, and the phenomenon of radioactive disintegration, to mention only a few examples, are achievements of the new theory which had baffled the old. While the writing of this chapter of the history of physics is doubtless not yet complete, it has progressed to such a stage that we may profitably pause and consider the significance of what has been written. As we make this survey, we are indeed fortunate to have Professor Heinsenberg to guide our thoughts.

  ARTHUR H. COMPTON

  PREFACE

  The lectures which I gave at the University of Chicago in the spring of 1929 afforded me the opportunity of reviewing the fundamental principles of quantum theory. Since the conclusive studies of Bohr in 1927 there have been no essential changes in these principles, and many new experiments have confirmed important consequences of the theory (for example, the Raman effect). But even today the physicist more often has a kind of faith in the correctness of the new principles than a clear understanding of them. For this reason the publication of these Chicago lectures in the form of a small book seems justified.

  Since the formal mathematical apparatus of the quantum theory is already available in several excellent texts and is more familiar to many than the physical principles, I have placed it at the end of the book, in what is little more than a collection of formulas.bn In the text itself I have been at pains to use only elementary formulas and calculations, so far as this is possible.

  In the body of the text particular emphasis has been placed on the complete equivalence of the corpuscular and wave concepts, which is clearly reflected in the newer formulations of the mathematical theory. This symmetry of the book with respect to the words “particle” and “wave” shows that nothing is gained by discussing fundamental problems (such as causality) in terms of one rather than the other. I have also attempted to make the distinction between waves in space-time and the Schrödinger waves in configuration space as clear as possible.

  On the whole the book contains nothing that is not to be found in previous publications, particularly in the investigations of Bohr. The purpose of the book seems to me to be fulfilled if it contributes somewhat to the diffusion of that “Kopenhagener Geist der Quantentheorie,” if I may so express myself, which has directed the entire development of modern atomic physics.

  My thanks are due in the first place to Drs. C. Eckart and F. Hoyt, of the University of Chicago, who have taken on themselves not only the labor of preparing the English translation, but have also contributed essentially to the improvement of the book by working over several sections and giving me the benefit of their advice. I am also indebted to Dr. G. Beck for reading proof of the German edition and for valuable assistance in the preparation of the manuscript.

  W. HEISENBERG

  LEIPZIG

  March 3, 1930

  INTRODUCTORY

  § 1. THEORY AND EXPERIMENT

  The experiments of physics and their results can be described in the language of daily life. Thus if the physicist did not demand a theory to explain his results and could be content, say, with a description of the lines appearing on photographic plates, everything would be simple and there would be no need of an epistemological discussion. Difficulties arise only in the attempt to classify and synthesize the results, to establish the relation of cause and effect between them—in short, to construct a theory. This synthetic process has been applied not only to the results of scientific experiment, but, in the course of ages, also to the simplest experiences of daily life, and in this way all concepts have been formed. In the process, the solid ground of experimental proof has often been forsaken, and generalizations have been accepted uncritically, until finally contradictions between theory and experiment have become apparent. In order to avoid these contradictions, it seems necessary to demand that no concept enter a theory which has not been experimentally verified at least to the same degree of accuracy as the experiments to be explained by the theory. Unfortunately it is quite impossible to fulfil this requirement, since the commonest ideas and words would often be excluded. To avoid these insurmountable difficulties it is found advisable to introduce a great wealth of concepts into a physical theory, without attempting to justify them rigorously, and then to allow experiment to decide at what points a revision is necessary.

  Thus it was characteristic of the special theory of relativity that the concepts “measurin
g rod” and “clock” were subject to searching criticism in the light of experiment; it appeared that these ordinary concepts involved the tacit assumption that there exist (in principle, at least) signals that are propagated with an infinite velocity. When it became evident that such signals were not to be found in nature, the task of eliminating this tacit assumption from all logical deductions was undertaken, with the result that a consistent interpretation was found for facts which had seemed irreconcilable. A much more radical departure from the classical conception of the world was brought about by the general theory of relativity, in which only the concept of coincidence in space-time was accepted uncritically. According to this theory, ordinary language (i.e., classical concepts) is applicable only to the description of experiments in which both the gravitational constant and the reciprocal of the velocity of light may be regarded as negligibly small.

  Although the theory of relativity makes the greatest of demands on the ability for abstract thought, still it fulfils the traditional requirements of science in so far as it permits a division of the world into subject and object (observer and observed) and hence a clear formulation of the law of causality. This is the very point at which the difficulties of the quantum theory begin. In atomic physics, the concepts “clock” and “measuring rod” need no immediate consideration, for there is a large field of phenomena in which 1 /c is negligible. The concepts “space-time coincidence” and “observation,” on the other hand, do require a thorough revision. Particularly characteristic of the discussions to follow is the interaction between observer and object; in classical physical theories it has always been assumed either that this interaction is negligibly small, or else that its effect can be eliminated from the result by calculations based on “control” experiments. This assumption is not permissible in atomic physics; the interaction between observer and object causes uncontrollable and large changes in the system being observed, because of the discontinuous changes characteristic of atomic processes. The immediate consequence of this circumstance is that in general every experiment performed to determine some numerical quantity renders the knowledge of others illusory, since the uncontrollable perturbation of the observed system alters the values of previously determined quantities. If this perturbation be followed in its quantitative details, it appears that in many cases it is impossible to obtain an exact determination of the simultaneous values of two variables, but rather that there is a lower limit to the accuracy with which they can be known.bo