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Approach to nonrelativistic scattering theory, based on the principles of causality, superposition, and unitarity

R. Kh. Gainutdinov

Published Feb 1, 1983• 0 citations
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Abstract

The possibility of constructing a nonrelativistic scattering theory without appealing to the Schroedinger formalism is investigated, the theory being based on the general physical principles of causality, superposition, and unitarity. It is proved that the proposed approach is more general than the nonrelativistic scattering theory based on the Schroedinger equation. The proposed approach is used for the construction of a model scattering theory for a system consisting of heavy nonrelativistic particles and a light relativistic particle.

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The possibility of constructing a nonrelativistic scattering theory without appealing to the Schroedinger formalism is investigated, the theory being based on the general physical principles of causality, superposition, and unitarity. It is proved that the proposed approach is more general than the nonrelativistic scattering theory based on the Schroedinger equation. The proposed approach is used for the construction of a model scattering theory for a system consisting of heavy nonrelativistic particles and a light relativistic particle.
The connection between causality and analyticity in scattering theory is formulated in terms of Hilbertspace concepts. The usual rules of nonrelativistic quantum mechanics are assumed to hold for the "in" and "out" states of the scattering system. We show that there is a (physically verifiable) causality condition which implies that each diagonal SSS-matrix element ${S}_{\ensuremath{\alpha}\ensuremath{\alpha}}(E)$ must be the limit of an analytic function of the energy EEE, regular in ImE>0\mathrm{Im}E>0ImE>0. The implications for partial-wave amplitudes and for the forward scattering amplitude in elastic two-body collisions are discussed.
It is shown that if the wave function of a nonrelativistic particle satisfies the Schr\"odinger equation with a velocity-independent potential, then its scattering amplitudes (and the SSS matrix in general) satisfy the same dispersion formulas as those derived for the scattering of light. In the present derivation, the validity of a perturbation expansion and certain integrability of the potential are assumed, and the requirement of the outgoing wave Green's function replaces the condition of strict causality for the scattering of light. The scattering amplitude for fixed momentum transfer is also shown to satisfy the dispersion relations. Together with the unitarity of the SSS matrix, the complete SSS matrix is determined by the Fourier transform of the potential through an iterative procedure using the dispersion relations. If the potential possesses bound states, then all the dispersion formulas are modified to include residue terms corresponding to singularities on the positive imaginary axis of the momentum plane. The necessity of these modifications is related to the divergence of the perturbation series for small wave numbers. Essential singularities of the SSS matrix due to exponentially damped potentials give no additional contribution to the dispersion formulas.
The most important physical aspects of quantum mechanics are interactions in bound systems and predictions relevant to particle scatterings. A special accent is placed upon the foundation of quantum scattering theory from the first principles of physics, without resorting to any free parameters. Such a complete theory, which is autonomously and consistently devised on a rigorous mathematical basis, represents a genuine modern candidate for a powerful predictor of real events in experiments. For modern scattering theory, the following two principal questions emerge: the existence and completeness of the Moller wave operators. They directly imply the unitarity of the S-matrix and the correctness of the boundary conditions. Such a general and versatile methodological concept largely surpasses the frames of scattering theory.
The objective of this series of three papers is to axiomatically derive quantum mechanics from classical mechanics and two other basic axioms. In this first paper, Schroedinger's equation for the density matrix is fist obtained and from it Schroedinger's equation for the wave functions is derived. The momentum and position operators acting upon the density matrix are defined and it is then demonstrated that they commute. Pauli's equation for the density matrix is also obtained. A statistical potential formally identical to the quantum potential of Bohm's hidden variable theory is introduced, and this quantum potential is reinterpreted through the formalism here proposed. It is shown that, for dispersion free {\\it ensembles% }, Schroedinger's equation for the density matrix is equivalent to Newton's equations. A general non-ambiguous procedure for the construction of operators which act upon the density matrix is presented. It is also shown how these operators can be reduced to those which act upon the wave functions.
In this paper, some concepts of nonrelativistic many-particle quantum mechanics (e.g., product states, density matrix) are generalized to the relativistic domain using a framework called relativistic Schr\"odinger theory (RST). By using a general ansatz, the RST framework is simplified considerably and some of its field equations are solved directly. The RST approach is then compared with nonrelativistic quantum mechanics (QM) for the case of a product state (conventional QM) and its RST analog. It is shown that relativistic wave equations can be derived from the RST formalism, so that they coincide in the nonrelativistic limit with the well-known Hartree equations.