Abstract The problem of the scattering of a stream of charged particles by a Coulombian centre of force has been treated by Wentzel and by Oppenheimer, who use Born’s method of successive approximations. They have shown that, if ZZ' ε2/hv is small, the quantum theory gives the well-known Rutherford formula. Furthermore, if ZZ' ε2/hv is large, we know that the quantum theory result must tend towards the classical, and it seemed probable that the two theories would agree for intermediate values of this constant. In this paper we shall show that this is actually the case. We shall first consider the more general problem of scattering by a field V(r). The wave equation is ∇2ϕ+2m/h2 (E - V)ϕ = 0. (1)
Abstract The problem of the scattering of a stream of charged particles by a Coulombian centre of force has been treated by Wentzel and by Oppenheimer, who use Born’s method of successive approximations. They have shown that, if ZZ' ε2/hv is small, the quantum theory gives the well-known Rutherford formula. Furthermore, if ZZ' ε2/hv is large, we know that the quantum theory result must tend towards the classical, and it seemed probable that the two theories would agree for intermediate values of this constant. In this paper we shall show that this is actually the case. We shall first consider the more general problem of scattering by a field V(r). The wave equation is ∇2ϕ+2m/h2 (E - V)ϕ = 0. (1)
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.
1. It is possible to treat the excitation of an atom by an α-particle in two ways; we may either solve the Schrödinger equation for the system consisting of the α-particle and the atom, or we may, on account of the great mass of the α-particle, treat it as a moving centre of force, and solve the Schrödinger equation for the electrons in the field of the α-particle and nucleus, If the α-particle has velocity υ greater than the orbital velocities of the electrons, it is possible to obtain approximate formulae for the excitation probabilities by both methods; in the former case by the well known Born method, and in the latter case by a method first used in this connection by Gaunt*, and which is essentially the same as the method of variation of constants. The two methods give formally very different formulae for the excitation probabilities; it is the purpose of this paper to show that they are in fact identical if the ratio of the mass of the electron to that of the α-particle be considered vanishingly small.
Two different formulations of the problem of time-delay caused by scattering from an oscillating center of force are studied. In the first one, appropriate for local interactions, a semi-classical time-dependent solution of the Schrödinger equation is obtained by writing the wave function as a solution of two coupled equations one of which is the Hamilton–Jacobi equation. These equations are then solved using the narrow wave packet approximation. The classical motion of the center of this wave packet is found for two specific cases, a square well and a Morse potential. In the second formulation a time-independent many-channel solution of the Schrödinger equation is obtained for an exactly solvable quantum-mechanical problem with a potential consisting of a separable attractive part which oscillates and a fixed rigid wall. The results show that the time-delay is extremely sensitive to the values of the amplitude and the frequency of oscillations, and also to the energy of the incident particle.
Abstract An outstanding defect of the old quantum theory was its inability to treat the phenomena occurring in the collision of electrons with atoms, and Born’s theory showed how the new quantum mechanics was capable of removing this defect. The original theory of Born was a method of successive approximations to the solution of the wave equation representing the colliding systems, and could be applied to both elastic and inelastic collisions. In its actual applications, which could be worked out to the first approximation, the theory fell far short of expectations. The Ramsauer effect, the most interesting phenomenon exhibited in the elastic scattering of electrons, could not be explained, and in the field of inelastic collisions, it failed to provide a means of explaining the excitation of levels of a term system which does not combine with the ground state system, e. g., the excitation of the triplet lines of helium. These failures are, of course, failures of the theory in its first approximation. The first approximation in Born’s method of successive approximations neglected the disturbance of the electron waves by the field of the scatterer, and also the possibility of exchange of electrons between the atoms and the colliding beam. The neglect of both these effects becomes important for low velocity impacts, and it was clear that the Ramsauer effect, which is only observed for low velocity electrons, could only be explained by more accurate theory. The question was whether both effects, the distortion of the electron waves by the atomic field, and the exchange of electrons was important, or only one of them.
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.