Abstract The problem of the collision of two particles which act upon each other with forces varying as the inverse square of the distance between them has been solved exactly on the basis of the new quantum mechanics, and the solution is the same as that given by classical mechanics. In a recent paper, however, Mott has pointed out that this agreement between the predictions of classical mechanics and wave mechanics depends upon the dissimilarity of the colliding particles; if the particles are identical the scattering laws given by the wave mechanics will be very different from those of classical theory. Mott has considered the two types of collision between similar particles, (1) in which the particles possess spin, such as the collisions of electrons with electrons or protons with hydrogen nuclei, and (2) in which the particles have no spin, such as the collisions of α-particles with helium nuclei.
Abstract The problem of the collision of two particles which act upon each other with forces varying as the inverse square of the distance between them has been solved exactly on the basis of the new quantum mechanics, and the solution is the same as that given by classical mechanics. In a recent paper, however, Mott has pointed out that this agreement between the predictions of classical mechanics and wave mechanics depends upon the dissimilarity of the colliding particles; if the particles are identical the scattering laws given by the wave mechanics will be very different from those of classical theory. Mott has considered the two types of collision between similar particles, (1) in which the particles possess spin, such as the collisions of electrons with electrons or protons with hydrogen nuclei, and (2) in which the particles have no spin, such as the collisions of α-particles with helium nuclei.
Abstract It has been shown by Mott on the basis of the wave mechanics, that in the case of collisions between identical particles the scattered particles should interfere with the projected particles travelling in the same direction. When α-particles are scattered in helium, if the scattered α-particles and projected helium nuclei of similar velocity are identical in all respects, there will be interference between the two streams of particles. For collisions in which the particles act upon each other with forces varying as the inverse square of the distance between them, the interference results in the scattering intensity varying above and below the classical value and rising to double the classical numbers at 45º. At small angles the scattering predicted by the quantum mechanics does not differ greatly from that given by the classical theory. An experiment carried out by Chadwick showed quite definitely that for sufficiently slow α-particles the amount of scattering at 45º was double that of the classical theory. For these α-particles of low velocity the results showed that the forces varied very little from Coulomb forces; hence it was evident that the discrepancy was due to the failure of the classical theory. The scattering of slow α-particles by helium has also been investigated by Blackett and Champion by means of an expansion chamber. The observed scattering was in good agreement with the wave mechanical scattering. These experiments verify the assumption upon which Mott s theory is based, namely, that it is impossible to distinguish between an α-particle and a nucleus of helium travelling at the same velocity. Thus the helium nucleus has no spin or vector quantity associated with it; its field of force is perfectly spherical.
Abstract 1. Gordon has proved that the scattering of particles by an inverse square field is the same on the wave mechanics as on the classical theory. Mott has however shown, from consideration of the symmetry of the wave functions, that the scattering is quite different from the classical when the scattering and scattered particles are identical. The scattering of alpha particles by helium is typical of the case of the collision of identical particles that have no pin and obey the Einstein-Bose statistics. For this case, the expression given by Mott for the number of alpha particles scattered between angles θ, and θ +dθ, in going a distance dr in a gas containing n atoms per unit volume, is N dθ dr = 4πnN0e4/m2v4 {cosec4 θ + sec4 θ + 2 cosec2 θ sec2 θ cos u} dθ dr, (1) where u = 8/137 c/v log cot θ, (2) and N0 is the total number of incident particles, v, m and e, their initial velocity, mass and charge and c the velocity of light.
Abstract It is well-known that the problem of the collision between two particles interacting according to the inverse square law is exactly soluble on the wave mechanics, and that the solution yields the same scattering laws as the classical theory. If, however, the two particles are identical, e.g., two electrons or two α-particles, this is not necessarily the case; for the wave functions used must be antisymmetrical or symmetrical in the co-ordinates of the two particles; and this may affect the scattering laws. In this paper we shall discuss the collision between two particles possessing spin, such as electrons, and also between two particles without spin, such as α-particles. Assuming an inverse square law force between the particles, and neglecting the actual spin forces, we shall deduce from the symmetry properties of the wave functions a scattering law differing considerably from the classical. We shall also mention the various methods by which the effect could be observed, and give some experimental evidence in its favour. The application of the exclusion principle to collision problems has been discussed by the author in a previous paper. Suppose we wish to describe the motion of two particles interacting in any field of force. We obtain a solution w(r1 r2) of the wave equation, where r1 refers to the position of the first particle, and r2 to that of the second. If we did not use antisymmetrical wave functions, we should argue that the probability that the first particle should be at r1 and the second at r 2 would be |w (r1 r2)|2, and therefore the probability that one particle should be at r1 and the other at r2 would be |w (r1 r2)|2 + |w(r2 r1)|
Abstract 1. In a previous paper, the author has shown that the anomalous scattering of α-particles in helium can be accounted for, if one postulates the validity of the wave mechanics to describe the phenomenon, and assumes a simple spherically symmetrical form for the field between the two colliding particles. The solutions obtained were exact, no approximate methods being used, but only the scattering at large angles, 34° and 45°, was considered. The purpose of this paper is threefold. Firstly, these results are extended to the case of the scattering of α-particles by hydrogen, and good agreement with experiment is obtained. Secondly, it is shown that the scattering at small angles both in hydrogen and in helium can be explained by the same field as is used to explain the scattering at large angles. It is therefore no longer necessary to assume a “plate-like” form for the α-particle, in order to explain the scattering at small angles, as formerly according to the classical mechanics. Thirdly, a discussion is given of the extent to which the explanation here advanced for the experimental results is dependent on the particular form assumed for the potential energy of the one particle in the field of the other. It is found that, provided we assume that the potential energy, V(r), is Coulombian for distances greater than about 5 X 10-13 cm., then whatever the form of the potential for smaller distances, the ratio, R, of the scattering to that to be expected from a purely Coulombian field is given by the formula R = |e-i/ak. logcos2א + iak cos2 א (e2iK0— 1)|2, where the scattering is in hydrogen, א is angle of scattering, and ak = hv/4πє2. A similar formula is found for the scattering in helium. The formula contains only one parameter, K0, which depends on the velocity of the incident α-particles, v, and also on the field assumed, but is independent of the angle of scattering, א. Thus we may determine K0 from the observed scattering for a given value of v and of א, and deduce the scattering at other angles for the same value of the velocity. The agreement obtained with experiment is then quite independent of any special choice of a potential energy function. In order to calculate K0 and its variation with v, we should have to assume a particular form for V(r) Conversely, from the values of K0 determined by the experimental values of R, we may calculate V(r) as in the previous paper for the case of two α-particles. 2. The Scattering Formula for Hydrogen.—In the experiments on the scattering by hydrogen, it is found convenient to count the scintillations due to the hydrogen nuclei projected forward by the collision rather than those due to the deflected α-particles themselves, and therefore in this portion of the paper we shall denote by It the ratio of the number of protons found experimentally to be projected in a given direction, to the number predicted by the Rutherford law. The fact that R refers to the particles struck on from rest makes a considerable simplification in the collision relations, for if we consider the collision of two particles of mass m1 and m2, of which m1 is initially at rest, and if we denote by א the angle made by the final direction of motion of this particle with the incident direction of m2, and by θ the angle through which either particle is deflected, measured in a frame of reference moving with the velocity of the centre of gravity of the two particles, then א = ½ (π - 0), (1) independently of the ratio m1/m2. The relation between the angle θ and the angle ϕ, through which the incident particle m2 is deflected, is on the other hand dependent on the relative masses of the particles.
Abstract One of the outstanding differences between classical theory and quantum mechanics is the importance which the latter attaches to the identity of particles. Although perfectly well realised before the introduction of modern quantum theory, the fact that two electrons are experimentally indistinguishable did not introduce any novel features into the dynamics of many electron systems on Bohr’s theory. However, Heisenberg showed that these identity relations are of fundamental importance in many body problems, and was able to explain the formerly very puzzling characteristic of the spectrum of neutral helium, its division into two non-combining term systems. Owing to the identity of the particles, the wave function representing such a system must have certain symmetry properties; for the particles known in Nature this symmetry characteristic is that the function be either symmetric or anti-symmetric in the co-ordinates of the particles. For particles with a spin such as the electron and proton, only antisymmetry is allowed (Pauli’s principle), while for α-particles the wave function must be symmetrical. Since Heisenberg’s explanation of the helium spectrum the theory of anti-symmetric electron wave functions has been extended to describe the general features of the spectra of all atoms, while it appears from the work of Heitler and London that the theory is of fundamental importance in the explanation of chemical combination between atoms. Again, in furnishing a new statistics it is an essential feature of the theory of metals. As the effect of the identity relations depends on the fact that exchanging the electrons does not alter the system in any way, these effects are known as exchange phenomena.
Abstract The new wave mechanics has been eminently successful in correlating and accounting for the large amount of experimental data on the periodic properties of the atom. The applications of the new theory to aperiodic phenomena, though not nearly so numerous, have been attended with no less success; indeed, it is in these experiments on free electrons, in which diffraction patterns similar to those produced by beams of X-rays and light are obtained, that the wave nature of electrons is so clearly and objectively demonstrated. Such diffraction effects have been obtained by a number of investigators by scattering beams of homogeneous electrons in crystals, thin films and by a ruled grating. Similar effects are obtained when electrons are scattered by complex molecules, owing to the symmetrically situated nuclei in the molecular structure, and when α-particles are scattered by helium owing to interference between the waves of the scattered incident particles and recoiling helium nuclei.