Abstract In the present work the wave mechanics are applied to the discussion of a collision problem of a more complicated type than has, I think, hitherto been done. Though there is nothing very new in the process it is rather interesting to see how a wave function behaves which involves two bodies, and is therefore outside our ordinary space intuitions. But the work had a much more general aim, as an attempt to see whether a sharper line of demarcation could be drawn between the particle-like and the wave-like properties of matter. Put briefly, I take a problem which would be regarded at first sight as irreconcilable with a pure wave theory, but thoroughly typical of the behaviour of particles, and show how in fact the correct result arises naturally from the consideration of wave alone. I have been led in this way to certain not very original speculation on the basis of the quantum theory and these are given at the end of the paper. These are not in the least intended to be a general formulation of the theory, but rather as an examination of a particular type of attempted formulation. I fear that the views expressed will be found partly obvious and perhaps partly inconsistent or even erroneous. My excuse for giving them is that the quantum theory seem already sufficiently developed for us to attempt something more than a mathematical formulation of it, and that a generally accepted physical understanding will be more quickly reached by agreement or disagreement with expressed opinions, than by waiting for the arrival of a full-grown theory. It has been an extraordinarily difficult task to express the ideas I intended, and I wish to thank Prof. D. R. Hartree for many valuable criticisms of my first attempts to do so, though he must be exonerated from all blame if the final result is unintelligible or erroneous. 1. The quantum mechanics have been expressed in a variety of ways, which place the emphasis differently, but it has been shown that they are all exactly equivalent, so what may choose which we will. I shall use the following description, limiting myself to the simpler cases which do not involve considerations of relativity. We take a partial differential equation of the wave type, set down according to certain rules depending on the particular system considered. We solve this subject to certain boundary conditions, expressive of all the limitations imposed by silts, mirrors, diffraction, gratings, moving shutters, etc., that may occur in the proposed system, and obtain as result the wave function Ψ, a function of all the co-ordinates of the system and of the time. From this we from a set of functions, of which the simplest is |Ψ|2; this process may be called taking the intensities. We next have to prepare for the particle aspect, by applying certain rules so as to normalise the intensities, a process entirely foreign to the idea of waves. Finally, we make the interpretation, in which we say that the normalised intensities measure the probability of the particles having such and such positions, momenta, energies, etc., and these are the results verified by observation.
Abstract In the present work the wave mechanics are applied to the discussion of a collision problem of a more complicated type than has, I think, hitherto been done. Though there is nothing very new in the process it is rather interesting to see how a wave function behaves which involves two bodies, and is therefore outside our ordinary space intuitions. But the work had a much more general aim, as an attempt to see whether a sharper line of demarcation could be drawn between the particle-like and the wave-like properties of matter. Put briefly, I take a problem which would be regarded at first sight as irreconcilable with a pure wave theory, but thoroughly typical of the behaviour of particles, and show how in fact the correct result arises naturally from the consideration of wave alone. I have been led in this way to certain not very original speculation on the basis of the quantum theory and these are given at the end of the paper. These are not in the least intended to be a general formulation of the theory, but rather as an examination of a particular type of attempted formulation. I fear that the views expressed will be found partly obvious and perhaps partly inconsistent or even erroneous. My excuse for giving them is that the quantum theory seem already sufficiently developed for us to attempt something more than a mathematical formulation of it, and that a generally accepted physical understanding will be more quickly reached by agreement or disagreement with expressed opinions, than by waiting for the arrival of a full-grown theory. It has been an extraordinarily difficult task to express the ideas I intended, and I wish to thank Prof. D. R. Hartree for many valuable criticisms of my first attempts to do so, though he must be exonerated from all blame if the final result is unintelligible or erroneous. 1. The quantum mechanics have been expressed in a variety of ways, which place the emphasis differently, but it has been shown that they are all exactly equivalent, so what may choose which we will. I shall use the following description, limiting myself to the simpler cases which do not involve considerations of relativity. We take a partial differential equation of the wave type, set down according to certain rules depending on the particular system considered. We solve this subject to certain boundary conditions, expressive of all the limitations imposed by silts, mirrors, diffraction, gratings, moving shutters, etc., that may occur in the proposed system, and obtain as result the wave function Ψ, a function of all the co-ordinates of the system and of the time. From this we from a set of functions, of which the simplest is |Ψ|2; this process may be called taking the intensities. We next have to prepare for the particle aspect, by applying certain rules so as to normalise the intensities, a process entirely foreign to the idea of waves. Finally, we make the interpretation, in which we say that the normalised intensities measure the probability of the particles having such and such positions, momenta, energies, etc., and these are the results verified by observation.
Abstract It will probably be agreed that among all the recent developments of the quantum theory, one of the least satisfactory is the theory of radiation. The present paper is intended as a preliminary to a new line of attack on the subject. It was begun some time ago, but owing to lack of success in carrying it to a conclusion, its publication has been much delayed. In the meantime other papers have appeared, which in some respects follow the same train of thought. The authors of these works have carried their methods further in some directions than I have attempted, but there is still perhaps room for the discussion of a number of questions from the rather different point of view adopted here. 1. The main principle of the present work is the idea that, since matter and light both possess the dual characters of particle and wave, a similar mathematical treatment ought to be applied to both, and that this has not yet bee done as fully as should be possible. Whereas we have a fairly complete calculus for dealing with the behaviour of any number of electrons or atoms, for photons the existing processes are much less satisfactory. The central difficulty, which makes it hard to apply the ordinary methods of wave mechanics to light, is the fact that (at least according to our present ideas) photons can be created and annihilated, and to represent this in a wave system we have to be able to think of a medium suddenly coming into existence and then going out again, when the light that it was carrying is absorbed. Such behaviour is a grave difficulty in the way of allowing us to think of the photon as a wave, and tends to make us think with more favour of its particle aspect, until we recall that after all it is quite unlike any known particle to come into existence and later to disappear without trace. The theories at present current, such as that of Heisenberg and Pauli, avoid these difficulties because they are mainly formal generalisations of the classical theory; this frees them from the above difficulties, but they pay for it in being highly abstract, and, as it has turned out, rather unsuccessful.
Abstract 1. There are two problems which are fundamental to wave mechanics— that of deducing from first principles the properties of the wave field of ψ, by means of which, on modern theory, the mechanical properties of matter are to be described, and that of formulating a logical description of the properties of matter in terms of ψ. The former only of these problems is discussed in this communication. In spite of many valuable papers that have been published on the subject of Dirac’s remarkable wave equation, the derivation of the equation still seemed (to the present writer, at any rate,) to include some obscurities which could perhaps be removed. To some extent this is owing to the development of the equation, as a matter of historical necessity, having come about by successive extensions to classical mechanics; so that some difficulties have been produced by the fundamental relativity considerations having been introduced into the theory at a late stage, instead of in their proper place at the beginning. In what follows, by treating the problem from the beginning as a four-dimensional one, a deduction of the wave equation free from empirical steps is, I think, obtained, while also certain new features of the equation are brought to light. 2. The Principle of Action.—The method of four-dimensional mechanics is to assume that the motions of bodies in the world can be represented by “tracks,” or curved lines, in a “fourfold.” (This last term will be used here to mean a four-dimensional manifold with Euclidian geometry, i. e., in which ds2 = dxi2 (i = 1, . . . 4).) (1) Uniform motion in the world is represented by straight line tracks in the fourfold, and the “classical relativity” laws of motion are derived by formulating the simplest mathematical specification of curved tracks which will represent the non-uniform motions of bodies that are usually observed in the world.
Now I shall try to give you an idea of the way in which physicists at present endeavour to overcome this failure. One might term it an ‘emergency exit’, though it was not intended as such, but as a new theory. I mean, of course, wave mechanics. (Edding-ton called it ‘not a physical theory but a dodge— and a very good dodge too’.) The situation is about as follows. The observed facts (about particles and light and all sorts of radiation and their mutual interaction) appear to be repugnant to the classical ideal of a continuous description in space and time. (Let me explain myself to the physicist by hinting at one example: Bohr's famous theory of spectral lines in 1913 had to assume that the atom makes a sudden transition from one state into another state, and that in doing so it emits a train of light waves several feet long, containing hundreds of thousands of waves and requiring for its formation a considerable time. No information about the atom during this transition can be offered.) So the facts of observation are irreconcilable with a continuous description in space and time; it just seems impossible, at least in many cases. On the other hand, from an incomplete description—from a picture with gaps in space and time—one cannot draw clear and unambiguous conclusions; it leads to hazy, arbitrary, unclear thinking—and that is the thing we must avoid at all costs! What is to be done?
Now I shall try to give you an idea of the way in which physicists at present endeavour to overcome this failure. One might term it an ‘emergency exit’, though it was not intended as such, but as a new theory. I mean, of course, wave mechanics. (Eddington called it ‘not a physical theory but a dodge—and a very good dodge too’.) The situation is about as follows. The observed facts (about particles and light and all sorts of radiation and their mutual interaction) appear to be repugnant to the classical ideal of a continuous description in space and time. (Let me explain myself to the physicist by hinting at one example: Bohr's famous theory of spectral lines in 1913 had to assume that the atom makes a sudden transition from one state into another state, and that in doing so it emits a train of light waves several feet long, containing hundreds of thousands of waves and requiring for its formation a considerable time. No information about the atom during this transition can be offered.) So the facts of observation are irreconcilable with a continuous description in space and time 5 it just seems impossible, at least in many cases.
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. From the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything it is becoming more intense over the years. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.
Abstract 1. In wave-mechanics there occur in addition to the ordinary physical vectors and tensors the four-valued quantities Ψ, Ψ introduced by Dirac. In this combination certain relations of invariance in regard to transformation appear which had escaped the ordinary tensor calculus. If we call the new type of quantity involving Ψ and its combinations Ψ-tensor, the position is that we cannot reach Ψ-tensors from the calculus of ordinary space tensors, but we can reach space tensors from a calculus of Ψ-tensors. I have shown that although Ψ-vectors cannot be expressed in terms of space vectors, mixed Ψ-tensors can be immediately resolved into space vectors. The present paper contains a simplification and systematisation of my earlier work on the Ψ-tensor calculus as well as a number of new results. The particular point round which the new results centre is as follows. The Ψ-tensors occurring in wave-mechanics are the product of two Ψ-vectors (Ψ, Ψ), and apparently the primary reason for introducing the unobservable quantity Ψ rather than working with physical tensors throughout is to impose this condition. I therefore examine the question: If we impose the condition that a wave tensor is the product of two Ψ-vectors, what is the corresponding restriction on the physical tensors equivalent to it ? The answer is (§7) that whatever is described by two Ψ-vectors Ψ, Ψ can equivalently be described by two space vectors of equal length at right angles to one another. One of these is the momentum vector; the other (generally ignored in current investigations) presumably represents positional relations (co-ordinates or distance); or rather I would regard it as the source of positional relations, which can only become explicit in more complicated developments involving many particles. The ordinary wave equation for one particle is obtained as an identity.