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.

Introduction

Two of the great men of our times are remembered here. Each possessed a wondrous range of endowments. They were men of courageous vision. The fruits of their endeavour and their vision have much significance in diverse ways for ourselves and our successors. Both were obviously of extraordinary independence of mind and given to much solitary work, yet both owed a great deal to their living among contemporaries who were themselves of outstanding distinction and dedication to their ideals.

Eamon de Valera, statesman and leader, visionary, natural scholar and devotee of mathematics, was born in New York in 1882 of somewhat obscure parentage of Spanish and Irish descent. A British court sentenced him to death for his part in the Irish uprising of 1916, although this was later commuted. In 1921 he became Chancellor of the National University of Ireland; in 1932 he was made head of the government of the Irish Free State and the President of the Council of the League of Nations; and in 1938 he became President of the Assembly of that body. In 1940 he established the Dublin Institute for Advanced Studies with its Schools of Celtic Studies, Theoretical Physics and, in 1947, its School of Cosmic Physics. He was President of the Republic of Ireland between 1959 and 1973. The Royal Society of London elected him a Fellow in 1968. He died in 1975, aged nearly 93.

Schrödinger's original interpretation of the Schrödinger equation had many attractive features lost in later interpretations of the quantum theory. But that interpretation runs into a number of formidable well-known and not so well-known objections. I argue, following the methodological precepts of Paul Feyerabend, that we need not regard any of these objections as fatal, provided we are prepared to opt for a number of bold and rather radical mathematical and theoretical conjectures. These would amount jointly to the conjecture that a fully time-symmetric consistently classically interpreted non-second-quantized analogue of existing quantum field theory would (pace Jaynes, Tomonaga, Bell, and others) ultimately prove predictively equivalent to orthodox second-quantized theory.

Introduction

Schrödinger initially proposed his equation as a classical theory of matter waves directly analogous to Maxwell's theory of electromagnetic waves. |ψ|2 represented a classical charge density functioning in the ordinary classical way as a source of electromagnetic fields, and acted on by these fields via the potential term in the matter–wave equation. This is a theory of coupled classical fields with no probabilities entering into its interpretation, and from a modern point of view it can be thought of in terms of the coupled Dirac and Maxwell fields, without second-quantization and interpreted in a purely classical manner.

Quantum mechanics is crucial to an understanding of chemistry. As Linus Pauling (1985) put it: ‘Chemistry is a quantum phenomenon, or, rather, a great collection of quantum phenomena’. The name of Schrödinger is as familiar to present-day chemistry undergraduates as is that of Faraday, Kekulé or Mendeléev. There exists a thriving branch of the subject known as quantum chemistry; it is concerned with approximate solutions of the Schrödinger equation ℋΨn = EnΨn and it provides descriptions of atoms, molecules and clusters that are of interest to chemists. But there are still a number of scientists – and great chemists among them – who are of the opinion that they do not need a knowledge of quantum mechanics. They believe that the results of quantum-chemical computations represent a simplified description of reality that may miss the essential truth. It has sometimes been suggested that results obtained by computation are lacking in elegance and are less important than those deduced by pure reasoning (Coulson, 1960). Hirschfelder (1983) wrote that ‘scientists in the 1980s get so immersed in a maze of computational detail that they lose sight of the simple, elegant theories.’

It is true that computations may have a short life; we have been going through a phase where ab initio calculations on small molecules may be improved annually!

But through computation, quantum chemists have created (Davidson, 1984) ‘a quantitative model of the chemical bond which is beautiful to those who understand it and which is likely to be permanent.

Schrödinger's paper (1926) on Bose–Einstein condensation was submitted for publication on 15 December, 1925, immediately preceding the first of his papers on wave mechanics. Historians of science (Hanle, 1977; Klein, 1964) have seen ‘an organic connection’ between the two. It came between the three papers of Einstein (1924, 1925a, b) on the boson gas and Uhlenbeck's doctoral thesis (Uhlenbeck, 1927), and did not receive the attention it deserved, although, with hindsight, it is seen to contain the gem of the idea which was later to resolve the conflict between Einstein and Uhlenbeck. Einstein based his prediction (Einstein, 1925a) of condensation in the free boson gas on a combination of Bose–Einstein statistics and the classical density of states in phase-space; this leads to an expression for the mean particle number density as a function of kinetic energy which saturates as the total density increases so that there is a maximum possible density of particles having nonzero kinetic energy. Einstein claimed that, if the particle density is increased above this maximum, the excess goes into a state with zero kinetic energy. This is the phenomenon of condensation. Uhlenbeck (1927) objected that the result holds only ‘when the quantization of translational motion is neglected’. The situation remained unclear until the Amsterdam meeting to celebrate the centenary (23 November 1937) of the birth of Johannes van der Waals; here Kramers pointed out the importance of the thermodynamic limit for the sharp manifestatioon of phase transitions.