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Physical Review A

General formulation for direct evaluation of the local field amplitude and transition amplitude based on the Fredholm determinant

Itsuki Banno, Kazumi Fujima

Published Dec 14, 2009• 0 citations
Publisher

Abstract

We develop a determinantal method in a quantum scattering system for direct evaluation of various quantities including the local field amplitude and transition amplitude ($S$-matrix element). This method is applicable to multichannel elastic and inelastic scattering systems without rearranging the particles. The underlying principle of our formulation is that requested information can be extracted from a wave operator without solving the wave equation. The wave operator is the master operator maintaining all of the information about a system; the determinant of the inverse wave operator is just the Fredholm determinant. In the 1960s, a similar determinantal method was developed for the $S$-matrix element or equivalently the far-field amplitude of the wave function. Until now, however, no determinantal method for near-field quantities existed, although de Broglie and electromagnetic near fields are observable using scanning probe microscopes. Additionally, we prove that our formula for $S$-matrix element covers the known formula (the Le Couteur-Newton formula).

DOI
10.1103/physreva.80.062716

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We develop a determinantal method in a quantum scattering system for direct evaluation of various quantities including the local field amplitude and transition amplitude (SSS-matrix element). This method is applicable to multichannel elastic and inelastic scattering systems without rearranging the particles. The underlying principle of our formulation is that requested information can be extracted from a wave operator without solving the wave equation. The wave operator is the master operator maintaining all of the information about a system; the determinant of the inverse wave operator is just the Fredholm determinant. In the 1960s, a similar determinantal method was developed for the SSS-matrix element or equivalently the far-field amplitude of the wave function. Until now, however, no determinantal method for near-field quantities existed, although de Broglie and electromagnetic near fields are observable using scanning probe microscopes. Additionally, we prove that our formula for SSS-matrix element covers the known formula (the Le Couteur-Newton formula).
Using a classical phase space integral to approximate certain quantum mechanical operations, the classical limit of the Fredholm determinant for a general multichannel scattering system is derived. This statistical form of the classical limit of scattering is found to give exactly the same ``classical S matrix'' for the case of potential scattering as does a dynamical treatment of the classical limit. For inelastic scattering, however, the classical S matrix which results from this Fredholm determinant is only an approximation to that obtained by exact classical dynamics. Reasons for this failure of the statistical version of the classical limit for inelastic scattering are discussed.
A definition is formulated of the scattering matrix for a closed physical system with bound states which makes use throughout only of the assumed observable properties of the system. A direct product space, X, is defined in which the ingredient factor space comprises the steady states—vacuum, one-particle, and bound states—of the physical system. It is argued that the boundary conditions for a scattering experiment are suitably expressed in terms of vectors in X and that these stand in unitary correspondence, U, to the Heisenberg states. Indeed, one defines two operators U(±) to express outgoing and ingoing wave boundary conditions, and the scattering matrix is constructed from these in the usual way. A suitable Yang-Feldman formalism is then developed in which the operators in the remote past and future also describe the bound states of the system. A representation of the framework thus constructed in terms of field operators for individual fields results in the well-known formulas for S-matrix elements in terms of covariant amplitudes.
It is shown that the Fredholm-Laplace solution of the Schr\"odinger integral equation of the evolution operator in the cases of harmonic or constant external perturbations can be put into a physically meaningful determinantal quotient form. The solution exhibits the usual multiple transitions and corresponding linewidth upon expansion. The result lends itself fairly well to the setting up of correct tractable formulas in the important case of continuous spectra capable, for instance, of dealing with resonance phenomena. Furthermore, the determinantal form can be shown to obey the unitary requirement at all times and any order with respect to the applied field or the collision potential. An illustration is given in the simple model of a system submitted to a perturbation of constant matrix elements, in the spectral range of interest assumed of constant density of states.
The concept of a generalized scattering amplitude is introduced and applied to quantum potential scatterings. After an outgoing spherical wave is factored out from a wave function, the remaining wave envelope is defined as the generalized scattering amplitude. The transformed Schr\"odinger equation, in terms of the generalized scattering amplitude, can be solved numerically by using a finite-difference method over the entire scattering domain. Example problems of scatterings by a spherical potential well and the (${\mathit{e}}^{\mathrm{\ensuremath{-}}}$,H) static field interaction potential are solved to validate the theoretical formulation and numerical method. The far-field solutions for the ordinary scattering amplitude and differential cross section agree very well with those obtained from the partial-wave analysis. The radial profiles of the generalized scattering amplitude and particle density function over the entire scattering region are also presented, and their properties are discussed. These results demonstrate that the method can be used to yield a complete and accurate solution to scattering problems involving arbitrary interaction potentials.
The time-dependent form of the Lippmann–Schwinger integral equation is used as the basis for a novel wave-packet propagation scheme. The method has the advantage over a previous integral equation treatment in that it does not require extensive matrix inversions involving the potential. This feature will be important when applications are made to systems where in some degrees of freedom the potential is expressed in a basis expansion. As was the case for the previous treatment, noniterated and iterated versions of the equations are given; the iterated equations, which are much simpler in the present new scheme than in the old, eliminate a matrix inversion that is required for solving the earlier noniterated equations. In the present noniterated equations, the matrix to be inverted is a function of the kinetic energy operator and thus is diagonal in a Bessel function basis set (or a sine basis set, if the centrifugal potential operator is incorporated into an effective potential). Transition amplitudes for various orbital angular momentum quantum numbers can be obtained from: (1) Fourier transform of the amplitude density from the time to the energy domain, and (2) direct analysis of the scattered wave packet. The approach is illustrated by an application to a standard potential scattering model problem.
This chapter focuses on the scattering matrix, or S-matrix, an infinite-dimensional matrix or operator that expresses the state of a scattering system consisting of waves or particles or both in the far future in terms of its state in the remote past. In the case of electromagnetic (or acoustic) waves, the S-matrix connects the intensity, phase, and polarization of the outgoing waves in the far field at various angles to the direction and polarization of the beam pointed toward an obstacle. The chapter first considers the problem of scattering by a square well, located symmetrically with respect to the origin, before discussing bound states and a heuristic derivation of the Breit-Wigner formula. It als describes the Watson transform and Regge poles before concluding with an analysis of the time-independent radial Schrödinger equation and Levinson's theorem.