Abstract The present note is suggested by a recent paper by Prof. Darwin,* and is intended to show how one of the most typically particle-like properties of matter can be derived from the wave mechanics. In the theory of radioactive disintegration, as presented by Gamow, the ∝-particle is represented by a spherical wave which slowly leaks out of the nucleus. On the other hand, the ∝-particle, once emerged, has particle-like properties, the most striking being the ray tracks that it forms in a Wilson cloud chamber. It is a little difficult to picture how it is that an outgoing spherical wave can produce a straight track ; we think intuitively that it should ionise atoms at random throughout space. We could consider that Gamow’s outgoing spherical wave should give the probability of disintegration, but that, when the particle is outside the nucleus, it should be represented by a wave packet moving in a definite direction, so as to produce a straight track. But it ought not to be necessary to do this. The wave mechanics unaided ought to be able to predict the possible results of any observation that we could make on a system, without invoking, until the moment at which the observation is made, the classical particle-like properties of the electrons or α-particles forming that system. If we consider the ∝-ray alone as the system under consideration, then the gas of the Wilson chamber must be considered as the means by which we observe the particle ; so in this case we must consider the ∝-ray as α-particle as soon as it is outside the nucleus, because that is the moment at which the observation is made. If, however, we consider the α-particle and the gas together as one system, then it is ionised atoms that we observe ; interpreting the wave function should give us simply the probability that such and such an atom is ionised. Until this final interpretation is made, no mention should be made of the ∝-ray being a particle at all. The difficulty that we have in picturing how it is that a spherical wave can produce a straight track arises from our tendency to picture the wave as existing in ordinary three dimensional space, whereas we are really dealing with wave functions in the multispace formed by the co-ordinates both of the α-particle and of every atom in the Wilson chamber.
Abstract The present note is suggested by a recent paper by Prof. Darwin,* and is intended to show how one of the most typically particle-like properties of matter can be derived from the wave mechanics. In the theory of radioactive disintegration, as presented by Gamow, the ∝-particle is represented by a spherical wave which slowly leaks out of the nucleus. On the other hand, the ∝-particle, once emerged, has particle-like properties, the most striking being the ray tracks that it forms in a Wilson cloud chamber. It is a little difficult to picture how it is that an outgoing spherical wave can produce a straight track ; we think intuitively that it should ionise atoms at random throughout space. We could consider that Gamow’s outgoing spherical wave should give the probability of disintegration, but that, when the particle is outside the nucleus, it should be represented by a wave packet moving in a definite direction, so as to produce a straight track. But it ought not to be necessary to do this. The wave mechanics unaided ought to be able to predict the possible results of any observation that we could make on a system, without invoking, until the moment at which the observation is made, the classical particle-like properties of the electrons or α-particles forming that system. If we consider the ∝-ray alone as the system under consideration, then the gas of the Wilson chamber must be considered as the means by which we observe the particle ; so in this case we must consider the ∝-ray as α-particle as soon as it is outside the nucleus, because that is the moment at which the observation is made. If, however, we consider the α-particle and the gas together as one system, then it is ionised atoms that we observe ; interpreting the wave function should give us simply the probability that such and such an atom is ionised. Until this final interpretation is made, no mention should be made of the ∝-ray being a particle at all. The difficulty that we have in picturing how it is that a spherical wave can produce a straight track arises from our tendency to picture the wave as existing in ordinary three dimensional space, whereas we are really dealing with wave functions in the multispace formed by the co-ordinates both of the α-particle and of every atom in the Wilson chamber.
The operations of many detection devices are usually explained in terms of the ionization tracks produced by classical charged particles. A wave-mechanical analysis does not seem to be available. Mott [Proc. R. Soc. London Ser. A 126, 79 (1929)] has shown that an incident \ensuremath{\alpha} wave emitted by a radioactive nucleus would be scattered by atoms primarily in directions almost forward. However, these scattered waves would appear over the entire sensitive region of the detector. The mechanism that produces a collapse of the wave function to a single track is analyzed here. The incident \ensuremath{\alpha} wave is written as a superposition of minimum-sized packets. Since it is linear, the Schr\"odinger equation for the system including an observer can be solved for each packet individually. After the minimum packets interact with the electrons in the detector, they cannot be recombined to form a plane wave. However, they can be combined in groups to form larger coherent packets interacting with the detector to form tracks. The size and shape of each coherent packet is determined by the properties of the medium that is ionized to produce its track. The track from each packet is seen by the observer and is associated with a state of his memory, recording only one track. Each such packet replaces the particle in the usual explanation. These are the packets required to predict the quantum jumps seen in the scattering of laser light from an isolated atom as I have discussed [Phys. Rev. A 45, 4925 (1992)]. The relation of the probability distribution of these tracks to the wave function is explained.
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.
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.
In (1), an outline of the derivation of the Gamow factor for alpha decay from a nucleus is given. In one dimension, the approach considers three regions in space, one with an incident and reflected free particle wavefunction, the second (within a square V barrier) with an exponentially growing and falling solution, and the third, with a single free particle wavefunction exp(-iEt)exp(ipx+phase) representing the escaped alpha particle. Alpha decay is treated as a quantum tunneling problem and the wavefunction and its first derivative d/dx must be continuous. A second step in (1), however, is to assume that one may write a time related wavefunction as exp(- g t), where d/dt partial Integral dx (region) W*W = ½ (-i W* d/dx W + i W d/dx W*). This may appear to be a quantum mechanical equation, but we argue that this is nothing more than the classical continuity equation. In fact, in (2) we argued that quantum mechanics does not exist without classical equations. In particular, the quantum free particle wavefunctions exp(-iEt) and exp(ipx) follow exp(iEt)exp(-iEt) = 1 and exp(-ipx)exp(ipx) = 1 to link them classical densities in time and space for particles moving with a constant speed. If there is a decay in time and space, then the appropriate classical equation is the continuity one. Writing d/dt partial d(t) = constant d(t) yields the familiar exp(-g t) form. We apply the ideas of the Gamow factor to a set of photons bouncing back and forth in an n1 1-dimensional cavity surrounded by n2 (n2>n1) index of refraction material. In this case, if the photons are synchronized then one may follow them classically. The catch is that the probability to reflect or refract is quantum. In general, one would not write exp(-g t) in such a scenario, but would formally write an equation of the form: fraction of remaining photons = R power Integer(t/t1), where R is the probability to reflect and integer(t/t1) is the number of cycles.. If, however, the probability to refract is very small, we show that one may in fact obtain the same result as in the case of exp(- g t) for g <<1. Alternatively, if the cavity is small in length and the speed of the particle (c in the case of light) is large, one could approximate the situation by d/dt partial d(t)= constant d(t) and then obtain exp(-g t), having the photons no longer in deterministic motion.
The behavior of monochromatic electromagnetic waves in stationary media is shown to be ruled by a frequency dependent function, which we call Wave Potential, encoded in the structure of the Helmholtz equation. Contrary to the common belief that the very concept of "ray trajectory" is reserved to the eikonal approximation, a general and exact ray-based Hamiltonian treatment, reducing to the eikonal approximation in the absence of Wave Potential, shows that its presence induces a mutual, perpendicular ray-coupling, which is the one and only cause of any typically wave-like phenomenon, such as diffraction and interference. Recalling, then, that the time-independent Schroedinger and Klein-Gordon equations (associating stationary "matter waves" to mono-energetic particles) are themselves Helmholtz-like equations, the exact, ray-based treatment developed for classical electromagnetic waves is extended - without resorting to statistical concepts - to the exact, trajectory-based Hamiltonian dynamics of mono-energetic point-like particles, both in the non-relativistic and in the relativistic case. The trajectories turn out to be perpendicularly coupled, once more, by an exact, stationary, energy-dependent Wave Potential, coinciding in the form, but not in the physical meaning, with the statistical, time-varying, energy-independent "Quantum Potential" of Bohm's theory, which views particles, just like the standard Copenhagen interpretation, as traveling wave-packets. These results, together with the connection which is shown to exist between Wave Potential and Uncertainty Principle, suggest a novel, non-probabilistic interpretation of Wave Mechanics.