Geometry and symmetries of multi-particle systems

U. Fano, Dmitry Green, John L. Bohn, Thomas A. Heim

Published Jan 1, 1999• 21 citations

Abstract

Abstract. The quantum dynamical evolution of atomic and molecular aggregates, from their compact to their fragmented states, is parametrized by a single collective radial parameter. Treating all the remaining particle coordinates in d dimensions democratically, as a set of angles orthogonal to this collective radius or by equivalent variables, by-passes all independent-particle approximations. The invariance of the total kinetic energy under arbitrary d-dimensional transformations which preserve the radial parameter gives rise to novel quantum numbers and ladder operators interconnecting its eigenstates at each value of the radial parameter. We develop the systematics and technology of this approach, introducing the relevant mathematics tutorially, by analogy to the familiar theory of angular momentum in three dimensions. The angular basis functions so obtained are treated in a manifestly coordinate-free manner, thus serving as a flexible generalized basis for carrying out detailed studies of wave function evolution in multi-particle systems.

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Geometry and symmetries of multi-particle systems

U Fano, Dmitry Green, J L Bohn +1 more

The quantum dynamical evolution of atomic and molecular aggregates, from their compact to their fragmented states, is parametrized by a single collective radial parameter. Treating all the remaining particle coordinates in d dimensions democratically, as a set of angles orthogonal to this collective radius or by equivalent variables, bypasses all independent-particle approximations. The invariance of the total kinetic energy under arbitrary d -dimensional transformations which preserve the radial parameter gives rise to novel quantum numbers and ladder operators interconnecting its eigenstates at each value of the radial parameter. We develop the systematics and technology of this approach, introducing the relevant mathematics tutorially, by analogy to the familiar theory of angular momentum in three dimensions. The angular basis functions so obtained are treated in a manifestly coordinate-free manner, thus serving as a flexible generalized basis for carrying out detailed studies of wavefunction evolution in multi-particle systems.

arXiv:physics/9905052Journal of Physics B Atomic Molecular and Optical PhysicsJan 1, 199935 citations

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Synopsis: Direct product spaces are encountered in the generalization to higher dimensions, and the connection to wave function separability is discussed. The spherical coordinate basis is introduced in preparation for discussing spherically symmetric potentials. Angular momentum operators and properties are examined, and a complete set of compatible observables is determined. The angular momentum eigenvalues and eigenstates are determined using ladder operators, and the connection to spherical harmonics is made. The stage is set for understanding the Schrödinger radial equation.

CRC Press eBooksAug 23, 20190 citations

Generalization to Three Dimensions

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Synopsis: Direct product spaces are encountered in the generalization to higher dimensions, and the connection to wave function separability is discussed. e spherical coordinate basis is introduced in preparation for discussing spherically symmetric potentials. Angular momentum operators and properties are examined, and a complete set of compatible observables is determined. e angular momentum eigenvalues and eigenstates are determined using ladder operators and the connection to spherical harmonics is made. e stage is set for understanding the Schrödinger radial equation.

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Second-quantization and irreducible tensorial sets

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Aug 27, 20190 citations