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International Journal of Modern Physics B

ON THE CORRESPONDENCE BETWEEN CLASSICAL AND QUANTUM MECHANICS IN DEFINING ATOMS AND MOLECULES IN CONDENSED SYSTEMS

Luigi Delle Site

Published Jul 20, 2001• 0 citations
Publisher

Abstract

We analyze and discuss the concept of "proper quantum subsystem" (PQS). In particular we focus the attention on the theory of "quantum mechanics of a subspace" developed by R. F. W. Bader and coworkers whose definition of a PQS is based on the Schwinger's Quantum Action Principle; we illustrate some properties of this definition and propose a problem whose solution could be relevant in formulating a correct physical interpretation of the concept above.

DOI
10.1142/s0217979201006586

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We analyze and discuss the concept of "proper quantum subsystem" (PQS). In particular we focus the attention on the theory of "quantum mechanics of a subspace" developed by R. F. W. Bader and coworkers whose definition of a PQS is based on the Schwinger's Quantum Action Principle; we illustrate some properties of this definition and propose a problem whose solution could be relevant in formulating a correct physical interpretation of the concept above.
In this paper we propose a new realist, non-collapse interpretation of quantum mechanics, which moves away from the prevailing trend in the subject by paying special attention to the physical relevance of the interpretation. In particular, our proposal endows the Hamiltonian of the system, systematically ignored in the traditional interpretations, with a central role: it distinguishes between systems and subsystems and is the main ingredient in the selection of the definite-valued observables. We show how this interpretation solves the measurement problem, both in the ideal and in the non-ideal version, and we argue for the physical relevance of the new definite-value assignment.
We formulate a general principle that supplants a Boolean σ-algebra of intrinsic properties of a classical system by a σ-complex (a union of σ-algebras) of extrinsic properties of a quantum system that are elicited by interactions with other systems. We apply the canonical definitions of the concepts of classical physics of observables, states, combined systems, symmetries, and dynamics to derive the standard quantum characterizations, including the Schrödinger equation and the von Neumann-Lüder's Projection Rule. This manifestly consistent reconstruction of quantum mechanics is then used to discuss and resolve the dilemmas of the orthodox interpretation.
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.
A realistic physical axiomatic formulation of the relativistic quantum theory of fields is presented. Based on the action principle of Schwinger, the approach is developed in an aaabstract ( it does not depend on any particular representation of field operators) and covariant way for fields satisfying global gauge invariance. The field equations and the explicit expressions for the generators are deduced without using the "correspondence principle". Unlike other proofs, crossing symmetry arises in an independent way of the causality requirement. By contrast to the mathematical systems of axioms, particular emphasis on the problem of attaching a physical meaning to the formalism is handled. Arguments for a realistic interpretation in terms of "quantum fields" are advanced. Our realistic view is illustrated by regarding quantum fields as "real things" with a number of properties (e.g., interaction, symmetry properties, etc.). As a result of this conception, the fundamental variables in terms of which our axiomatics is constructed are field operators, not particles states. The advantages of our axiomatic approach are finally discussed.
Dividing the world into subsystems is an important component of the scientific method. The choice of subsystems, however, is not defined a priori. Typically, it is dictated by experimental capabilities, which may be different for different agents. Here, we propose a way to define subsystems in general physical theories, including theories beyond quantum and classical mechanics. Our construction associates every agent A with a subsystem S A , equipped with its set of states and its set of transformations. In quantum theory, this construction accommodates the notion of subsystems as factors of a tensor product, as well as the notion of subsystems associated with a subalgebra of operators. Classical systems can be interpreted as subsystems of quantum systems in different ways, by applying our construction to agents who have access to different sets of operations, including multiphase covariant channels and certain sets of free operations arising in the resource theory of quantum coherence. After illustrating the basic definitions, we restrict our attention to closed systems, that is, systems where all physical transformations act invertibly and where all states can be generated from a fixed initial state. For closed systems, we show that all the states of all subsystems admit a canonical purification. This result extends the purification principle to a broader setting, in which coherent superpositions can be interpreted as purifications of incoherent mixtures.