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Journal of Mathematical Physics

A geometric approach to quantum scattering with group symmetry

Yihren Wu

Published Jun 1, 1987• 1 citation
Publisher

Abstract

A geometric theory for quantum scattering when the symmetry group is semisimple is presented. This theory is seen as a generalization of the partial wave analysis. As an application of this theory, the S-matrix elements for scattering in the Pöschl–Teller potential with symmetry group SO(1,2), Coulomb potential with SO(1,3), and a perturbed Coulomb potential with SO(2,3) are calculated. The last example may be considered as a model for heavy-ion scattering.

DOI
10.1063/1.527538

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A geometric theory for quantum scattering when the symmetry group is semisimple is presented. This theory is seen as a generalization of the partial wave analysis. As an application of this theory, the S-matrix elements for scattering in the Pöschl–Teller potential with symmetry group SO(1,2), Coulomb potential with SO(1,3), and a perturbed Coulomb potential with SO(2,3) are calculated. The last example may be considered as a model for heavy-ion scattering.
Coupled systems of slow and fast variables with symmetry, characterized by a semisimple Lie group G, are employed to study the effect of adiabatic decoupling of the fast degrees of freedom on the algebra of symmetry generators. The slow configuration space is assumed to be the symmetric coset space G/H, where H is a compact subgroup of G defined by the fast Hamiltonian. The induced gauge fields characterizing the effective slow dynamics are symmetric ones in the sense that the action of G on them can be compensated by an H-valued gauge transformation. The modification of the symmetry generators when such gauge fields are present can be described purely in geometric terms related to the non-Abelian geometric phase. The modified generators may be identified as the generators of the induced representation of G, where the inducing represention is the representation of H on the fast Hilbert space. This result enables us to recast the problem of exotic quantum numbers for effective quantum systems in purely algebraic terms via the Frobenius reciprocity theorem. Illustrative calculations for the symmetric spaces SO(d+1)/SO(d) approximately S d (spheres) are presented. Possible relevance of modified generators for non-compact G for obtaining scattering potentials in the framework of algebraic scattering theory is also pointed out.
The quantum dynamics of a particle in the Modified Pöschl-Teller potential is derived from the group SL(2,R)SL(2,R)SL(2,R) by applying a Group Approach to Quantization (GAQ). The explicit form of the Hamiltonian as well as the ladder operators is found in the enveloping algebra of this basic symmetry group. The present algorithm provides a physical realization of the non-unitary, finite-dimensional, irreducible representations of the SL(2,R)SL(2,R)SL(2,R) group. The non-unitarity manifests itself in that only half of the states are normalizable, in contrast with the representations of SU(2) where all the states are physical.
We consider the relativistic quantum scattering of spinless particles in one-spatial dimension using the Feshbach-Villars formalism. We construct the general form of the scattering matrix, for symmetric and non-symmetric potentials, based on the symmetry properties of the Feshbach-Villars equation. Then, since in one dimension there are only two partial waves associated with even and odd parities, we show, in a simple and comprehensive way, how to describe this transmission-reflection problem using partial-wave decomposition. As an illustration, we also discuss the special case of scattering by a symmetric square-well potential.