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Journal of Mathematical Physics

Global and Democratic Methods for Classifying N-Particle States

Jean-Marc Lévy-Leblond

Published Dec 1, 1966• 27 citations
Publisher

Abstract

The ``global method'' for describing N-particle systems (which relies on the existence of a large invariance group of the total Hamiltonian for N noninteracting particles, the ``great group'', whose Lie algebra is generated by the ``grand angular momentum tensor''), is adapted to describe systems of identical particles by means of basis states with simple symmetry properties (with respect to permutations of the particles). We are led to define and study the concept of ``democracy'' among the particles, from which we obtain the ``democratic'' subgroups of the great group. The eigenvectors of a complete set of commuting observables, consisting essentially of Casimir operators of democratic subgroups, may furnish the desired basis. Unfortunately the scheme is seen to be sufficient only in the 3- and 4-particle cases, which, however, are most important. The Appendix contains a discussion of the possible relativistic generalizations of the global method.

DOI
10.1063/1.1704908

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The global method'' for describing N-particle systems (which relies on the existence of a large invariance group of the total Hamiltonian for N noninteracting particles, the great group'', whose Lie algebra is generated by the grand angular momentum tensor''), is adapted to describe systems of identical particles by means of basis states with simple symmetry properties (with respect to permutations of the particles). We are led to define and study the concept of democracy'' among the particles, from which we obtain the ``democratic'' subgroups of the great group. The eigenvectors of a complete set of commuting observables, consisting essentially of Casimir operators of democratic subgroups, may furnish the desired basis. Unfortunately the scheme is seen to be sufficient only in the 3- and 4-particle cases, which, however, are most important. The Appendix contains a discussion of the possible relativistic generalizations of the global method.
The kinematics of a nonrelativistic three-particle system is studied with the help of the general method devised by Lévy-Leblond and Lurçat. Basis states are constructed which are eigenstates, in addition to the total momentum-energy, angular momentum, etc., of new observables; among these, the ``togetherness tensor'' describes the simultaneous localization of the three particles and therefore is of great physical interest. All of these observables arise as Casimir operators of a ``great group'' acting on the three-particle phase-space manifold in a transitive way, and of some of its subgroups. In the present case, by trying to keep all the particles on the same footing (``democracy'' arguments), we are led to choose the SU3 group as a particularly convenient ``great group''. We thus recover completely the Dragt classification of non-relativistic three-particle states. The explicit calculation of the basis functions is done in a new way, by analytical methods, solving partial derivative equations. This enables us to establish the most general form of these basis functions.
This paper is a group-theoretical study of the kinematics of n nonrelativistic particles. A systematic method is given to construct a new complete set of commuting observables. The method is based on the existence of a group (the ``great group'') which acts transitively on the phase-space manifold and preserves the phase-space volume element; the observables are then Casimir operators of the great group and of some of its subgroups, including the usual three-dimensional rotation group. Among these collective observables, in addition to the total angular momentum, the most interesting is the ``togetherness operator'' which describes the simultaneous localization of the n particles. This operator is a generalization to n > 2 of the square of orbital angular momentum; its use allows to generalize to n particles the familiar centrifugal barrier arguments.
We introduce here a new coupling scheme for three angular momenta. It relies on the properties of an operator which depends ``democratically'' upon the three individual angular momenta; this is in fact their mixed product. This operator, the total angular momentum and one of its components form a complete set of commuting observables. In the case where the three individual angular momenta are equal, the eigenstates of this set possess remarkable symmetry properties with respect to the permutation group S3.
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.
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.
General remarks As already mentioned in the Introduction, the exact solution of the main equation of quantum mechanics – the Schrödinger equation – lies beyond the potentialities of modern mathematics and computer technology. But a number of important inferences about the behaviour, structure and properties of a given quantum-mechanical many-particle system can be drawn without solving this equation, just by examining its symmetry properties. Symmetry concepts come about in physics in two ways. First, since any physical process occurs in a real space, we have to make use of one or another coordinate system. The isotropy and homogeneity of space make physically meaningful only those mathematical relationships that remain unchanged under rotations of the axes of the coordinate system, and that impose fairly rigorous constraints on the possible physical laws. Second, every physical object and process features a symmetry which should be taken into account by the physical theory. A common idea underlying particular forms of symmetry is the invariance of a system under a certain set (group) of transformations. The normally considered forms of symmetry are rotational symmetry, which is based on the equivalence of all directions in space, and permutation symmetry, which is caused by identical particles. The operations of the geometrical symmetry group are responsible for appropriate conservation laws. So, the rotational symmetry of a closed system gives rise to the law of conservation of angular momentum. The group of rotations of a three-dimensional space stands apart in atomic spectroscopy. This is mostly due to the high accuracy of the central field approximation, on which the entire modern theory of complex atoms and ions is based.