The two main types of quantum measurements

von Neumann's postulate of the reduction of the wave function answers the question of what happens to the quantum object during the measurement. However, the reduction postulate does not answer the question of how the measuring device must be designed to realize the desired measurement. To answer this second question, one must understand the connection between the measuring device as an ordinary physical system, and the nature of the desired measurement (the quantity to be measured, the desired precision, and the choice of how the measurement is to perturb the quantum object).

The Schrödinger equation cannot tell us the connection between the design of the measuring device and the nature of the measurement, because the Schrödinger equation neither governs nor describes the process of measurement. More specifically: The evolution of the wave function as described by the Schrödinger equation has two key features: it is a reversible evolution (from the final state in principle one can always evolve back to the initial one), and it is a deterministic evolution (the final state is determined uniquely by the initial one). By contrast, the reduction of the wave function in a measurement is irreversible (once the measurement is finished and the information has been extracted from it, it is impossible to return to the pre-measurement state), and it is nondeterministic (one cannot predict, before the measurement, the post-measurement state of the measured object).

My association with Erwin Schrödinger was not a close one, although I spent the summer of 1927 in Zürich, with the stated purpose (stated in my letters to the John Simon Guggenheim Memorial Foundation) of working under his supervision. In fact, I spent most of my time in my room, trying to solve the Schrödinger equation for a system consisting of two helium atoms. I did not have very much success, except that, as was mentioned later by John C. Slater, I formulated a determinant of the several spin-orbital functions of the individual electrons as a way of ensuring that the wave function is antisymmetric. This was a device that Slater made much use of in discussing the electronic structure of atoms and also of molecules in 1929 and 1931.

Walter Heitler and Fritz London were also in Zürich that summer, also with the plan of working with Schrödinger. They told me that they had talked with Schrödinger several times about their work, while walking through the woods. I did not have even that much contact with him, because he was working so hard on his own problems.

It might be possible to put theoretical physicists on a scale ranging from one extreme, those who deal with ideas, to the other, those who deal with mathematics. Wolfgang Pauli is an example of a theoretical physicist near the mathematics end.

The prehistory to Schrödinger's activity in unifed field theory

Unification is one of those long-standing quests of science. A superior point of view allows one to recognize connections and to uncover common roots. In the theory of general relativity Einstein succeeded in achieving a superior point of view in an especially impressive manner. By extension of the theory of special relativity he was able to comprehend gravitation in the geometrization of the space-time-continuum. After the success of this process of geometrization, the inclusion and unification of the electromagnetic field and possibly other fields could be considered to be a particularly important goal.

A few years after the discovery of general relativity, Weyl (1918) had already tried a fundamental extension of the framework of the theory in order to include electromagnetism as well. His attempt was based on the idea of gauge in variance–a concept which was to emerge 30 years later as a cornerstone of the modern theories of unification. Nevertheless, no agreement with observed facts could be reached by his concept of the path-dependence of a displaced length.

Whereas Weyl's generalization consisted in placing a connection of lengths beside the connection of directions given by the metric, Eddington (1923) followed a different course in considering the connection of directions as an a priori property of the manifold. In this case the metric becomes a deduced quantity.

Introduction

Erwin Schrödinger and those in his scientific generation – men like Werner Heisenberg and P. A. M. Dirac – were interested in physics because it provides us with the basic understanding of the laws of nature. Thus, even though their work has provided us with the basis of most of modern high technology, particularly through the emergence of the quantum theory of the solid state, their pursuit of physics was not motivated by this. In this sense we, in our generation, look upon Erwin Schrödinger's situation and that of his colleagues with a degree of envy.

In keeping with the tradition of Erwin Schrödinger, I have pleasure in presenting this overview of particle physics as in early 1986. I am sure this overview will be outdated by the time it sees print – but that is the fate of anything one may write in so fast moving a subject.

Physics is an incredibly rich discipline: it not only provides us with the basic understnading of the laws of nature, it is also the basis of most of modern high technology. This remark is relevant to our developing countries. A fine example of this synthesis of a basic understanding of nature with high technology is provided by liquid-crystal physics which was worked out at Bangalore by Prof. S. Chandrasekhar and his group. In this context, one may note that, because of this connection with high technology and materials’ exploitation, physics is the ‘science of wealth creation’ par excellence.

Introduction

Having the privilege of writing this paper as a grandson of Boltzmann I apologize for not being a historian of science. A manifestation of Boltzmann's influence on Schrödinger is Schrödinger's enthusiastic quotation of Boltzmann's line of thought: ‘His line of thought may be called my first love in science. No other has ever thus enraptured me or will ever do so again.’ (Schrödinger, 1929; reprinted in Schrödinger, 1957, p. XII.) Even though Schrödinger had no personal contact with Boltzmann, his scientific education at the University of Vienna was in the tradition of Boltzmann. His thesis advisor and director of the second Institute for Experimental Physics at the University of Vienna, Franz Serafin Exner, was an ardent admirer of Boltzmann and so was Boltzmann's successor on the chair for theoretical physics, Friedrich Hasenöhrl. Schrödinger paid the most impressive tribute to his teacher Hasenöhrl when he received the Nobel Prize in 1933. He avowed that Hasenöhrl might have stood in his place had he not been killed in the first world war (Schrödinger, 1935, p. 87).

Later one of Schrödinger's main fields of interest was the application of Boltzmann's statistical methods which he called ‘natural statistics’ to various problems. And it was not an accident that a paper in this field ‘On Einstein's Gas Theory’ triggered the idea of wave mechanics (Schrödinger, 1926, 1984).

The Schrödinger equation is usually thought of as governing the behaviour of matter on a small scale. By a small system may be meant anything from two particles up to a whole star. Here, I want to consider a slightly larger system, the Universe. As has been remarked elsewhere, Schrödinger's equation comes into its own when classical physics breaks down. An example of breakdown on a small scale was provided by the classical model of the atom. Classical physics predicted that the electron would spiral into the nucleus and matter would collapse. Indeed, quantum mechanics and Schrödinger's equation were invented precisely to overcome this problem. There is a similar problem with the Universe. Classical physics predicts that there was a time about ten billion years ago when the density of matter would have been infinite. This is called the Big Bang singularity, and most people take it to be the beginning of the Universe. However, here I want to report some recent work which shows that, if one applies the Schrödinger equation to the whole Universe, there is no singularity. Instead one gets a wave function which corresponds in a classical limit to a Universe which starts from a minimum radius, expands in an inflationary manner at first, goes over to a matter dominated expansion, reaches a maximum radius and collapses again.

In the early 1940s Schrödinger worked at the Institute for Advanced Studies in Dublin. One day he met P. P. Ewald, another German theoretician, then a professor at the University of Belfast. Ewald, who had been a student in Göttingen before the First World War, gave Schrödinger a paper that had been published in the Nachrichten der Gesellschaft der Naturwissenschaften in Göttingen in 1935 (Yoxen, 1979). The paper was by N. W. Timoféeff-Ressovsky, K. G. Zimmer and Max Delbrück (1935), and was entitled ‘The nature of genetic mutations and the structure of the gene’. Apparently Schrödinger had been interested in that subject for some time, but the paper fascinated him so much that he made it the basis of a series of lectures at Trinity College, Dublin, in February 1943 and published them as a book in the following year, under the title What is Life? (Schrödinger, 1944). The book is written in an engaging, lively, almost poetic style (for example, ‘The probable life time of a radioactive atom is less predictable than that of a healthy sparrow’). It aroused much interest, especially among young physicists, and helped to stimulate some of them to turn to biology. I was asked by the organizers of the Schrödinger Centenary Symposium to assess its significance for the development of molecular biology.

The Timoféeff-Ressovsky, Zimmer and Delbrück paper

This paper forms the basis of Schrödinger's book. It covers 55 pages and is divided into four sections.

This book describes the many aspects of the life and scientific work of Erwin Schrödinger, who, perhaps more than anyone else, serves to represent the whole of modern physics. The contributions to the book are from many hands, and so it seemed useful to me to precede them with a short note giving an uncontroversial account of his life which will serve as a framework into which they can be fitted. Such a collection as this still leaves out many aspects of Schrödinger's thought, for it considers his philosophy only in passing, his poetry not at all and ignores his wide and deep interest in sculpture and painting and in the classics.

Schrödinger was born in Vienna on August 12, 1887, and entered the University there to read physics in 1906. He worked there as an assistant from 1910 till his war service, and again after the war. Some short term appointments at Jena, Stuttgart and Breslau led up to his appointment to the chair of theoretical physics in Zürich in 1921. He was already treating a wide range of topics, but concentrating on atomic theory, for the old quantum theory had now entered on its heroic phase of final collapse. His six papers founding wave mechanics came at the end of his Zürich years, and in 1927 he went to the chair in Berlin, to remain there till the advent of Hitler in 1933.

With a view to obtaining a picture for an idealized path of chemical reactions the concept of intrinsic reaction coordinate (IRC) is introduced within the space of the multidimensional potential energy function of the reacting system. The IRC uniquely determines, in a classical sense, the mode of deformation of chemically reacting molecules. On every point along the IRC an orbital analysis is possible with respect to each of the deformed molecules, the geometry of which is frozen to that in the reacting composite system. The analysis is carried out by constructing orbital pairs by respective unitary transformations of canonical molecular orbitals of the deformed molecules, so as to diagonalize the interaction matrix. Of these orbital pairs one or a few, localized in the reacting domain, play the dominant role in offering a distinct view of bond-forming processes along the IRC. In the process of these orbital analyses, the importance of electron delocalization in the chemical interaction is recognized and stressed.

Introduction

Complicated chemical phenomena, for example organic chemical reactions, are probably one of the most distant ‘lands’ from the Schrödinger equation in the vast world of its appications. In fact, the chemical reaction is the field which was forsaken by theoretical physicists. In the early stage of the development of applied quantum mechanics, many of the best theorists turned their back upon complex chemical reactions principally on account of the laboriousness involved in computations.

A conference was held from March 31 to April 3, 1987, at Imperial College London to celebrate the centenary of the birth of Erwin Schrödinger. It was supported by the Austrian Government, by the Dublin Institute for Advanced Studies and by many generous donations. A score of invited lecturers in the many scientific fields that Schrödinger had made his own were invited to survey his contributions and to discuss how it had influenced their own work. This volume contains most of these invited contributions.

The types of topological defects that can appear in gauge theories – domain walls, strings and monopoles – are described in the following. The possibility that such defects were generated at phase transitions in the very early history of the Universe is discussed, in particular the idea that cosmic strings may provide the seeds for the density perturbations from which galaxies form.

Introduction

The subject that I want to discuss did not exist in Erwin Schrödinger's time. But it forms a natural bridge between two of his major interests – the fundamental physics of the smallest and the largest scales.

As Yang describes in this volume (pp. 53–64) Schrödinger contributed very significantly to the early development of gauge theories – the fundamental framework for all our present understanding of elementary particles and their interaction. He also made influential contributions to cosmology.

At the time these two fields were essentially separate. Indeed one might say that our failure so far to reconcile the basic physical theories of the small and the large is the major outstanding problem in physics. Schrödinger was clearly aware of the lack. He was very interested in any attempt to bridge the gap, for example in Eddington's ambitious though unsuccessful fundamental theory (Schrödinger, 1938).

A final reconciliation of relativity and quantum theory still eludes us, but cosmology and particle physics are no longer poles apart.

Introduction

In a lecture in April 1970 Dirac talked about the early days of quantum mechanics (Dirac, 1972). Among other topics he discussed noncommutative algebra, and added

He then expanded on this subject and concluded

Introduction

I have borrowed the title of a characteristic paper by Schrödinger (Schrödinger, 1952). In it he contrasts the smooth evolution of the Schrödinger wavefunction with the erratic behaviour of the picture by which the wavefunction is usually supplemented, or ‘interpreted’, in the minds of most physicists. He objects in particular to the notion of ‘stationary states’, and above all to ‘quantum jumping’ between those states. He regards these concepts as hangovers from the old Bohr quantum theory, of 1913, and entirely unmotivated by anything in the mathematics of the new theory of 1926. He would like to regard the wavefunction itself as the complete picture, and completely determined by the Schrödinger equation, and so evolving smoothly without ‘quantum jumps’. Nor would he have ‘particles’ in the picture. At an early stage, he had tried to replace ‘particles’ by wavepackets (Schrödinger, 1926). But wavepackets diffuse. And the paper of 1952 ends, rather lamely, with the admission that Schrödinger does not see how, for the present, to account for particle tracks in track chambers… nor, more generally, for the definiteness, the particularity, of the world of experience, as compared with the indefiniteness, the waviness, of the wavefunction. It is the problem that he had had (Schrödinger, 1935a) with his cat. He thought that she could not be both dead and alive.