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Journal of the Physical Society of Japan

A Bracket Calculus for Adiabatic Descriptions of Quantum Composite Systems

Atushi Tanaka

Published Jan 1, 2003• 1 citation
Publisher

Abstract

A bracket calculus for adiabatic descriptions of quantum composite systems is introduced. The bracket calculus recovers a consistent and transparent way to obtain various adiabatic representations, e.g. the adiabatic nuclear momentum representation, and the adiabatic Husimi function. Furthermore, it is shown that the bracket calculus touches on an issue of the choice of subsystems (“electrons” and “nuclei” in the adiabatic description of molecules.

DOI
10.1143/jpsjs.72sc.54

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A bracket calculus for adiabatic descriptions of quantum composite systems is introduced. The bracket calculus recovers a consistent and transparent way to obtain various adiabatic representations, e.g. the adiabatic nuclear momentum representation, and the adiabatic Husimi function. Furthermore, it is shown that the bracket calculus touches on an issue of the choice of subsystems (“electrons” and “nuclei” in the adiabatic description of molecules.
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Abstract A generalization of the quantum mechanical representation transformation is presented, in this paper. It is shown that, as an important example, the Boson-expansion method, commonly employed in nuclear physics, corresponds to such a generalized transformation. Using this generalization, we were able to construct a special representation called the “Composite Particle Representation”. In the composite, particle representation, the composite particle degrees of freedom are included, as well as the original particle degrees of freedom. The former is introduced in order that the motion of certain particle clusters can be described as separate entities in a many-body system. This representation is shown to be exactly equivalent to the usual quantum mechanical representation which includes only the original particle degrees of freedom. Many applications of this theory are expected, in particular in the study of hadrons from the quark point of view and the Interacting Boson Model in nuclei.
Abstract This paper considers a Husimi representation of quantum mechanics in which the (stationary) state of a system or ensemble is described by a Husimi function and an observable is described by a phase space function or distribution such that the expectation value of the observable is given by an integral over phase space of the product of that function or distribution and the Husimi function. The density matrix, Wigner function, and Husimi function are considered to be alternative ways of describing the state of a system or ensemble, and methods of recovering the Wigner function or density matrix from the Husimi function are discussed. The classical limits of the Wigner and Husimi functions and of the relationship between them are considered. © 1993 John Wiley & Sons, Inc.
A method of constructing ’’atomic second quantization’’ representations is described, based on the introduction of redundant modes (’’ideal atom variables’’) which are then given physical content by carrying out a suitable unitary transformation, the ’’generalized Tani transformation.’’ In such a representation bound atoms or molecules are described by elementary Bose or Fermi operators, the field operators for nuclei and electrons referring only to unbound particles. The Hamiltonian thus obtained contains not only nucleus–nucleus, electron–electron, and nucleus–electron Coulomb interactions, but also atom–atom, atom–nucleus, and atom–electron Coulomb and exchange interactions, including breakup and recombination terms. All possible scattering and reaction channels are exhibited simultaneously in this transformed Hamiltonian. The method is applicable to any species or mixture of species of composite particles, but for simplicity the derivation is restricted here to the case of atomic hydrogen.
Abstract It has been demonstrated in Chapter 5 that Schrodinger’s original derivation of the wave equation can be generalized to provide a quantum description of an atom in a molecule. His derivation uses the calculus of variations as applied to an energy functional of Ψ and its extension to an atom in a molecule entails two essential steps: the introduction of a particular type of operator through its identification with the variation of the state function and a variation of the surface bounding the atomic subsystem.