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Journal of Mathematical Physics

Adiabatic Switching in the Schrödinger Theory of Scattering

John Day Dollard

Published May 1, 1966• 48 citations
Publisher

Abstract

The time-dependent Schrödinger theory of scattering is studied rigorously with the potential Vreplaced by e−ε|l|V. Sufficient conditions are given that the Møller wave-matrices Ω± be obtainable from this theory as ε → 0. The conjecture that this theory can be used to define a reasonable S-matrix when the Ω± do not exist is shown to be false for the Coulomb potential. It is remarked that the Goldberger-Gell-Mann switching procedure also breaks down in this case.

DOI
10.1063/1.1931210

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Journal of Physics A Mathematical and General•Oct 7, 1997•21 citations
The time-dependent Schrödinger theory of scattering is studied rigorously with the potential Vreplaced by e−ε|l|V. Sufficient conditions are given that the Møller wave-matrices Ω± be obtainable from this theory as ε → 0. The conjecture that this theory can be used to define a reasonable S-matrix when the Ω± do not exist is shown to be false for the Coulomb potential. It is remarked that the Goldberger-Gell-Mann switching procedure also breaks down in this case.
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.
A general framework is set up for the description of quantum-mechanical scattering processes for the case that both the free hamiltonian H 0 (t)=H 0 +H 1 (t) and the full hamiltonian H(t)=H 0 (t)+V are time-dependent. In particular the existence of the various evolution operators is considered and the wave operators Omega +or- (s), which now depend on the initial time s, are defined. The consequences of a periodic time dependence are studied and for H 1 (t) of the type H 1 (t)=H 1 cos( omega t+ delta ) it is shown that for mod omega mod to infinity the frequency-dependent wave operators converge to the wave operators which pertain to the case that H 1 (t) is absent.
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.