Abstract A hydrogen atom in the ground state scatters an electron with kinetic energy too small for inelastic collisions to occur. The wave function Ψ(r1; r2) of the system has boundary conditions at infinity which must be chosen to allow correctly for the possibilities of both direct and exchange scattering. The expansion Ψ = Σ ψ,(r1)Fy(r2) of the total wave function in y terms of a complete set of hydrogen atom wave functions ψy(r1) includes an integration over the continuous spectrum. It is si own that the integrand contains a singularity. The explicit form of this singularity and its connexion with the boundary conditions are examined in detail. The symmetrized functions Y* may be represented by expansions of the form Σ {ψy(r1) Gy±(r2) ±ψy(r2) y Gy±(r1)}, where the integrand in the continuous spectrum does not involve singularities. Finally, it is shown that because all the states ψy of the hydrogen atom are included in the expansion, the equation satisfied by F1, the coefficient of the ground state, contains a polarization potential which behaves like — a/2r4 for large r and is independent of the velocity of the incident electron.
Abstract A hydrogen atom in the ground state scatters an electron with kinetic energy too small for inelastic collisions to occur. The wave function Ψ(r1; r2) of the system has boundary conditions at infinity which must be chosen to allow correctly for the possibilities of both direct and exchange scattering. The expansion Ψ = Σ ψ,(r1)Fy(r2) of the total wave function in y terms of a complete set of hydrogen atom wave functions ψy(r1) includes an integration over the continuous spectrum. It is si own that the integrand contains a singularity. The explicit form of this singularity and its connexion with the boundary conditions are examined in detail. The symmetrized functions Y* may be represented by expansions of the form Σ {ψy(r1) Gy±(r2) ±ψy(r2) y Gy±(r1)}, where the integrand in the continuous spectrum does not involve singularities. Finally, it is shown that because all the states ψy of the hydrogen atom are included in the expansion, the equation satisfied by F1, the coefficient of the ground state, contains a polarization potential which behaves like — a/2r4 for large r and is independent of the velocity of the incident electron.
Accurate scattering lengths and effective ranges for hydrogen atomic collisions are determined from effective range theory. The precise ab initio b3Σ+u potential energy curve values of H2 calculated by Kolos and Wolniewicz and C6, C8, C10 long-range expansion coefficients are used in obtaining accurate representation of the two-body interaction potential. The s-wave scattering length obtained for this potential (A =1/33455 ao) is used to calculate the zero temperature energy as a function of density of electron spin-polarized ("spin-aligned") atomic hydrogen (H+) using previous many-body perturbation theory results for bosons. Neglecting small potential shape dependent terms, one obtains E/N (energy per particle ) as a function of density, which may be compared with similar results obtained from Monte-Carlo or variational calculations. The scattering lengths and effective ranges for the other hydrogen isotopic systems are also presented. It is shown that the first bound state for the b3Σ+u potential (within the Born-Oppenheimer approximation) only exists for a hypothetical isotopic species with mass larger than a value m = 3.258176 g/mole (where the scattering length becomes infinite). In terms of the quantum parameter, η ≡ [MATH]2/ (2µεσ2), two-body binding occurs in the b3Σ+u potential only for η < 0.17094. The second order liquid-to-gas transition (corresponding to the onset of infinite-body binding) occurs at the accurately determined critical value of the quantum parameter ηCB = 0.43891. This corresponds to zero scattering length.
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.
Scattering lengths for the collisions of ground and metastable state hydrogen atoms interacting via the B′1∑u +, EF 1∑g +, e3∑u + and h3∑g + molecular potentials are evaluated from published ab initio Born-Oppenheimer potentials with adiabatic corrections, matched to multipole expansions at large interatomic separations. It is demonstrated that the scattering lengths are very sensitive to the connections between the ab initio potentials and their asymptotic forms and the need is shown for accurate interatomic potential calculations at greater separations than are currently available. The density-dependent shift and broadening of the 1S–2S transition frequency in atomic hydrogen caused by two-body elastic collisions are presented.
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 In two recent papers, Blum and Lohmann (2016 Phys. Rev. Lett. 116 033201) and Lohmann et al (2016 Phys. Rev. A 94 032331), the possibility of continuously varying the degree of entanglement between an elastically scattered electron and the valence electron of quasi-one electron targets was discussed. Here we present results for elastic electron scattering from atomic hydrogen in the energy regime of 1−10 eV and the full range of scattering angles 0 ° − 180 ° . We confirm previous calculations at very low energies, which predicted that the hydrogen target is not a promising candidate for Bell correlations through electron collisions. This finding remains unchanged in the near-resonance regime of incident electron energies just below 10 eV. In addition to the spin-correlation parameter P , we present the angle-integrated total cross section, as well as the angle-differential cross section at a few representative collision energies.
A theoretical investigation of quasi-elastic atom-scattering is presented. The transition potential for the system, which consists of an excited atom in a neutral atom in the ground state, is derived and split into short- and long-range parts. A general expression for the leading long-range (two-photon exchange) part of the inelastic potential is obtained and the correct threshold behaviour of the partial-wave amplitude is discussed. The cross sections sigma short (nl to nl') and sigma long (nl to nl') of l-changing collisions of the Rydberg atoms are estimated for large values of n, and small as well as large angular momentum transitions. It is found that sigma short >> sigma long if n 2 q min <<1 r n>>l,l' when n to infinity .