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Physical Review A

New quantum numbers in collision theory. III. Symmetries of the scattering geometry

Chun‐Woo Lee, U. Fano

Published Jul 1, 1987• 9 citations
Publisher

Abstract

Scattering geometries are represented by tensorial parameters classified according to symmetry under permutations P and Q: P changes a base set of wave functions into its Hermitian conjugate and Q interchanges the directions of incidence and scattering. These parameters interlink initial and final states of projectile and target, a linkage mediated by the angular momentum ${\mathrm{j}}_{t}$ transferred in the collision. Collision cross sections and expectation values of the observables are constructed as sums of geometrical parameters with different symmetries, weighted by dynamical parameters.

DOI
10.1103/physreva.36.66

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Scattering geometries are represented by tensorial parameters classified according to symmetry under permutations P and Q: P changes a base set of wave functions into its Hermitian conjugate and Q interchanges the directions of incidence and scattering. These parameters interlink initial and final states of projectile and target, a linkage mediated by the angular momentum jt{\mathrm{j}}_{t}jt​ transferred in the collision. Collision cross sections and expectation values of the observables are constructed as sums of geometrical parameters with different symmetries, weighted by dynamical parameters.
Abstract The new quantum mechanics consists of a scheme of equations which are very closely analogous to the equations of classical mechanics, with the funda­mental difference that the dynamical variables do not obey the commutative law of multiplication, but satisfy instead the well-known quantum conditions. It follows that one cannot suppose the dynamical variables to be ordinary numbers (c-numbers), but may call them numbers of a special type (q-numbers). The theory shows that these q-numbers can in general be represented by matrices whose elements are c-numbers (functions of a time parameter). When one has performed the calculations with the q-numbers and obtained all the matrices one wants, the question arises how one is to get physical results from the theory, i. e., how can one obtain c-numbers from the theory that one can compare with experimental values ? Hitherto this has been done with the help of a number of special assumptions. In Heisenberg’s original matrix mechanics it was assumed that the elements of the diagonal matrix that repre­sents the energy are the energy levels of the system, and the elements of the matrix that represents the total polarisation, which are periodic functions of the time, determine the frequencies and intensities of the spectral lines in analogy to the classical theory. Schrodinger’s wave representation of the quantum mechanics has provided new ways of obtaining physical results from the theory, based on the assumption that the square of the amplitude of the wave function can in certain cases be interpreted as a probability. From this assumption one can, for instance, work out the probability of a transition being produced in a system (or the number of transitions produced in an assembly of like systems) by an arbitrary external perturbing force, and can thus, by supposing the perturbation to consist of incident radiation, obtain directly Einstein’s B coefficients. Again in Born’s treatment of collision problems it is assumed that the square of the amplitude of the wave function scattered in any direction determines the probability of the colliding electron (or other body) being scattered in that direction.
The weights of the different permutational symmetry components of a nonsymmetry-adapted many-particle wave function are evaluated in terms of the expectation values of the symmetric-group class sums. This facilitates the evaluation of the weights without the construction of a complete set of symmetry adapted functions. Subspace projection operators are introduced, to be used when prior knowledge about the symmetry-species composition of a wave function is available. The permutational weight analysis of a recursively angular-momentum coupled (shell model) wave function is presented as an illustration.
The aim of this work is to investigate the dynamics of collisions between ions and Rydberg atoms by using the Classical Trajectory Monte Carlo (CTMC) method. The study is started by collisions with oriented elliptical Rydberg atom, is then continued to collisions involving hydrogen Rydberg target in an external magnetic field in the l-mixing regime, and later arrives at antihydrogen formation in an external magnetic field where Rydberg positronium is taken as target.\nThe CTMC method has been applied in the study. For different targets, the CTMC method had to be adjusted and reconstructed to properly create the initial target state for the quasi-separable one-body initial system (hydrogen target) and the two-body initial system (positronium target). Charge exchange and ionization cross sections are calculated under special consideration of different initial target states.\nThe velocity matching phenomenon and Thomas capture which appears in the upstream-downstream asymmetry are found in ion-oriented elliptical Rydberg atom collisions, reflecting the important roles of the initial electron spatial and momentum distributions. By increasing the projectile charge, the distortion of the initial states caused by the strong perturbation of approaching projectile induces pronounced changes in the momentum distribution.\nIn the collisions involving hydrogen Rydberg atoms in a magnetic field, a cross section reduction in case of an increasing magnetic field is found for multiply charged projectiles. The structure effect due to the influence of the magnetic field on the initial state distribution results in different capture cross sections for two characterized, kmax and kmin, states. The velocity matching phenomenon as well as the effect of multiply charged projectile ion are observed in the collisions.\nFurthermore, antihydrogen formation in a magnetic field has been studied. Rydberg positroniums are assumed to be the target atoms, colliding with antiprotons. The initial target positronium state is carefully constructed by means of a newly introduced conserved operator, the pseudomomentum K. Similar properties as in the previous hydrogen target atom case were investigated. It is found that the existence of the magnetic field changes the spatial distribution of positronium to irregular motion. By increasing the magnetic field, it induces a decrease of the resulting capture cross sections (i.e. the antihydrogen formation). Thomas capture which happens in the collision plane in the field-free case is foiled. A smaller positronium binding energy and a larger geometrical extension of the target positronium atom yield a larger capture cross section.
Most of our knowledge about the structure of matter and interaction between particles is derived from scattering experiments. From a theoretical point of view, scattering problems are concerned with the continuous (and positive) energy eigenvalues and unbound eigenfunctions of the Schro¨dinger equation. We have already encountered one-dimensional examples of scattering in Chapter 4, in the discussion of reflection and transmission of an incident particle with definite momentum from a potential step or barrier. In this chapter we consider scattering from a more formal point of view. We will confine our discussion to elastic scattering (scattering without loss or gain of energy by the projectile) from central potentials although many of the concepts and results are applicable to inelastic scattering.