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Physical Review

Bound States and the Formal Theory of Scattering

M. N. Hack

Published Oct 1, 1954• 8 citations
Publisher

Abstract

The exact expression for the transition rate for quantum-mechanical scattering problems is derived formally without recourse to adiabatically switching on the interaction or averaging the initial state in time. The equivalence is shown of the stationary definition of the $S$ matrix and an appropriate definition by the time-dependent method.

DOI
10.1103/physrev.96.196

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Progress of Theoretical Physics•Jul 1, 1957•1 citation
The exact expression for the transition rate for quantum-mechanical scattering problems is derived formally without recourse to adiabatically switching on the interaction or averaging the initial state in time. The equivalence is shown of the stationary definition of the SSS matrix and an appropriate definition by the time-dependent method.
The time-dependent theory of scattering is reformulated. In §1, it is developed on the basis of a new limiting process which is self-consistent, and the equivalence is shown of Heisenberg's S-matrix and Dyson's one, when the total Hamiltonian permits the existence of bound states. In §2, a theory of scattering of wave packet is proposed in conformity with the physical picture. The damping factor e−ε|t| is derived from the amplitude of the wave packet. In §3, the rearrangement scattering is treated on the basis of the wave packet formalism.
The problem of extending quantum-mechanical formal scattering theory to a more general class of models that also includes quantum field theories is discussed, with the aim of clarifying certain aspects of the definition of scattering states. As the strong limit is not suitable for the definition of scattering states in quantum field theory, some other limiting procedure is needed. Two possibilities are considered, the abelian limit and adiabatic switching. Formulas for the scattering states based on both methods are discussed, and it is found that generally there are significant differences between the two approaches. As an illustration of the application and the features of these formulas, S-matrix elements and energy corrections in two quantum field theoretical models are calculated using (generalized) old-fashioned perturbation theory. The two methods are found to give equivalent results.
The aim of the lecture is to briefly describe the mathematical background of scattering theory for two- and three-particle quantum systems. We discuss basic objects of the theory: wave and scattering operators and the corresponding scattering matrix and illustrate them on the example of the Schrödinger equation. Our goal is to present time-dependent and stationary approaches and to describe the underlying mathematical methods. We also give a sketch of scattering theory for three interacting quantum particles including a difficult problem of the asymptotic completeness of scattering channels. Along with traditional results, we discuss new scattering channels arising for long-range pair interactions.
We derive the conditions which give a well-defined meaning to the mathematical expressions appearing in the current formal scattering theory. These conditions remove some restrictions that are usually imposed on the spectra of H and H0. By these conditions we can systematically discuss many problems, such as the equivalence between two kinds of S-matrices given by Heisenberg and Dyson respectively, or as the adiabatic theorem in the quantum field theory. The rearrangement collisions can be investigated from this view-point.