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Physical Review

Rotational Invariance and the S Matrix in Nonrelativistic Quantum Mechanics

R. Fong, Joseph Sucher

Published Mar 15, 1963• 0 citations
Publisher

Abstract

The connection between an observable invariance property of the scattering matrix and the interaction, $V$, generating the scattering, is studied for a particlar case: rotational invariance in nonrelativistic "potential" scattering. It is shown that a determination that the scattering cross section depends only on the angle between the incident and outgoing beams by no means implies that $V$ is invariant under rotation. This is true even if coherent incident beams are used to probe the target. Some aspects of these results and their relevance for the construction of physical theories are briefly discussed.

DOI
10.1103/physrev.129.2824

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The connection between an observable invariance property of the scattering matrix and the interaction, VVV, generating the scattering, is studied for a particlar case: rotational invariance in nonrelativistic "potential" scattering. It is shown that a determination that the scattering cross section depends only on the angle between the incident and outgoing beams by no means implies that VVV is invariant under rotation. This is true even if coherent incident beams are used to probe the target. Some aspects of these results and their relevance for the construction of physical theories are briefly discussed.
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Abstract In this chapter we explain how particle scattering can be described by local observables. First we introduce the concept of the S-matrix, which is an idea common to classical and quantum mechanics. We show the construction of incoming and outgoing states which have a certain asymptotic behaviour at time −∞ and +∞, and we also show that the constructed states have an acceptable physical asymptotic behaviour. Then we derive the representation of the S-matrix given by these scattering states and discuss its properties.
We consider potential scattering theory of a nonrelativistic quantum mechanical 2-particle system in R2 with anyon statistics. Sufficient conditions are given which guarantee the existence of Møller operators and the unitarity of the S-matrix. As examples the rotationally invariant potential well and the δ-function potential are discussed in detail. In case of a general rotationally invariant potential the angular momentum decomposition leads to a theory of Jost functions. The anyon statistics parameter gives rise to an interpolation for angular momenta analogous to the Regge trajectories for complex angular momenta. Levinson’s theorem is adapted to the present context. In particular we find that in case of a zero energy resonance the statistics parameter can be determined from the scattering phase.
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.
We consider potential scattering theory of a nonrelativistic quantum mechanical 2-particle system in R"2 with anyon statistics. Sufficient conditions are given which guarantee the existence of Moeller operators and the unitarity of the S-matrix. As examples the rotationally invariant potential well and the #delta#-function potential are discussed in detail. In case of a general rotationally invariant potential the angular momentum decomposition leads to a theory of Jost functions. The anyon statistics parameter gives rise to an interpolation for angular momenta analogous to the Regge trajectories for complex angular momenta. Levinson's theorem is adapted to the present context. In particular we find that in case of a zero energy resonance the statistics parameter can be determined from the scattering phase. (orig.)
The elastic scattering of spinless particles from a spherically symmetric stationary target is considered. The main idea is to describe the incident beam of particles as a certain statistical ensemble of one-particle wave packets, and to define the differential cross section in terms of the average scattering of the particles in this ensemble. The physical significance of this mode of description is discussed, and arguments are presented as to why it should be regarded as a realistic description of typical scattering experiments. The discussion is general, and not restricted to potential scattering. It is based on the notion of the S matrix, and the most basic ideas of the S matrix are discussed. A simple derivation of the partial wave expansion is given within the framework of S matrix theory. The aim of this paper is mainly pedagogical, and the final results obtained are old and well established.