Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Mathematical Proceedings of the Cambridge Philosophical Society

On the Mathematical Equivalence of two ways of regarding the Excitation of an Atom by a Fast Moving α-particle

J. W. Frame

Published Oct 1, 1931• 44 citations
Publisher

Abstract

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.

DOI
10.1017/s0305004100009762

Related papers

Equivalence of two alternative approaches to Schrödinger equations

B Gönül, K Köksal

Recently developed simple approach for the exact/approximate solution of Schrodinger equations with constant/position-dependent mass, in which the potential is considered as in the perturbation theory, is shown to be equivalent to the one leading to the construction of exactly solvable potentials via the solution of second-order differential equations in terms of known special functions. The formalism in the former solves difficulties encountered in the latter in revealing the corrections explicitly to the unperturbed piece of the solutions whereas the other obviate cumbersome procedures used in the calculations of the former.

arXiv:quant-ph/0512216•Physica Scripta•May 15, 2006•5 citations

Equivalence of two alternative approaches to Schrodinger equations

B. Gönül, Koray Köksal

Recently developed simple approach for the exact/approximate solution of Schrodinger equations with constant/position-dependent mass, in which the potential is considered as in the perturbation theory, is shown to be equivalent to the one leading to the construction of exactly solvable potentials via the solution of second-order differential equations in terms of known special functions. The formalism in the former solves difficulties encountered in the latter in revealing the corrections explicitly to the unperturbed piece of the solutions whereas the other obviate cumbersome procedures used in the calculations of the former.

•Dec 23, 2005•3 citations

Statistical Treatment of Electrons

Daniel D. Pollock

T he necessity for the treatment of the properties of quantum particles in terms of their most likely values is indicated in the previous chapter. Schrödinger’s equation, which provides the means to do this, is developed by examining the classical relationships describing the properties of a particle in motion and their transformation into quantum mechanical equivalents. The probable properties of quantum particles are obtained from the wavefunctions that are meaningful solutions to Schrödinger’s equation.

Routledge eBooks•May 4, 2018•0 citations

The anomalous scattering of α-particles by hydrogen and helium

Henry Martyn Taylor

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.

Proceedings of the Royal Society of London Series A Containing Papers of a Mathematical and Physical Character•Jun 1, 1932•21 citations

Atomic orbitals and classical motion

Elmer A. Ogryzlo

It is the object of this paper to describe the classical motion which is compatible with the exact distribution functions for atomic orbitals.

Journal of Chemical Education•Feb 1, 1968•0 citations

The solution of the wave equation for the scattering of particles by a coulombian centre of force

Nevill Francis Mott

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)

Proceedings of the Royal Society of London Series A Containing Papers of a Mathematical and Physical Character•Apr 2, 1928•49 citations
  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.
Recently developed simple approach for the exact/approximate solution of Schrodinger equations with constant/position-dependent mass, in which the potential is considered as in the perturbation theory, is shown to be equivalent to the one leading to the construction of exactly solvable potentials via the solution of second-order differential equations in terms of known special functions. The formalism in the former solves difficulties encountered in the latter in revealing the corrections explicitly to the unperturbed piece of the solutions whereas the other obviate cumbersome procedures used in the calculations of the former.
Recently developed simple approach for the exact/approximate solution of Schrodinger equations with constant/position-dependent mass, in which the potential is considered as in the perturbation theory, is shown to be equivalent to the one leading to the construction of exactly solvable potentials via the solution of second-order differential equations in terms of known special functions. The formalism in the former solves difficulties encountered in the latter in revealing the corrections explicitly to the unperturbed piece of the solutions whereas the other obviate cumbersome procedures used in the calculations of the former.
T he necessity for the treatment of the properties of quantum particles in terms of their most likely values is indicated in the previous chapter. Schrödinger’s equation, which provides the means to do this, is developed by examining the classical relationships describing the properties of a particle in motion and their transformation into quantum mechanical equivalents. The probable properties of quantum particles are obtained from the wavefunctions that are meaningful solutions to Schrödinger’s equation.
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 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)