Classical limit of scattering in quantum mechanics—A general approach

A. N. Basu, S. Sengupta

D. Sen, A. N. Basu, S. Sengupta

William Hughes Miller

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.

Peter Taylor

Familiar formulations of classical and quantum mechanics are shown to follow from a general theory of mechanics based on pure states with an intrinsic probability structure. This theory is developed to the stage where theorems from quantum logic enable expression of the state geometry in Hilbert space. Quantum and classical mechanics are then elaborated and applied to subsystems and the measurement process. Consideration is also given to space-time geometry and the constraints this places on the dynamics. Physics and Mathematics, it is argued, are growing apart; the inadequate treatment of approximations in general and localisation in quantum mechanics in particular are seen as contributing factors. In the description of systems, the link between localisation and lack of knowledge shows that quantum mechanics should reflect the domain of applicability. Restricting the class of states provides a means of achieving this goal. Localisation is then shown to have a mathematical expression in terms of compactness, which in turn is applied to yield a topological theory of bound and scattering states. Finally, the thesis questions the validity of classical limits and quantisations in intertheoretic reduction, and demonstrates that a widely accepted classical limit does not constitute a proof of reduction. It proposes a procedure for determining whether classical and quantum mechanics are weakly equivalent over a domain of applicability, and concludes that, in this restricted sense, classical mechanics reduces to quantum mechanics.

Kenji Yajima

Kenji Yajima