Quantum mechanics proper open system

The theory of atoms in molecules192 recovers all the fundamental concepts of chemistry, of atoms and functional groups with characteristic properties, of bonds, of molecular structure and structural stability, and of electron pairs and their role in molecular geometry and reactivity. The atomic principle of stationary action extends the predictions of quantum mechanics to the atomic constituents of all matter, the proper open systems of quantum mechanics. All facets of the theory are predictive and, as a consequence, the theory can be employed in many fields of research at the atomic level, from the design and synthesis of new drugs and catalysts, to the understanding and prediction of the properties of alloys. 

If a confined system, atom or molecule, is restrained by a physical boundary rather than by a mathematically imposed boundary condition, it is an open system. This chapter is concerned with the former case, of a system bounded by the surfaces it shares with the atoms of the confining material, surfaces that enable the transfer of matter and momentum. This places the problem in the realm of the quantum mechanics of a proper open system, a system... 

The virial theorem plays a dominant role in the definition of pressure, in both classical and quantum mechanics. The following section demonstrates that the pv product for a proper open system is proportional to the surface virial, Equation (9), the virial of the Ehrenfest forces exerted by the surroundings on the open system [9,12],... 

The so-called zero-flux surface condition stated in equation (1) is the boundary condition for the definition of a proper open system whose properties are defined by quantum mechanics, as described in Section 3. 

The only subjectivity one can exercise in a strict partition of the molecule is to decide whether to adhere to the constraints imposed by physics or to ignore them and thereby abandon physics subject to one s particular needs. The constraints imposed by quantum mechanics on the definition of an atom are best stated in the form of two questions (a) Does the state function, the function that contains all of the infomiation that can be known about a system, predict a unique partitioning of the molecule into atoms and (b) Does quantum mechanics provide a complete description of the atoms so defined Affirmative answers to these two questions in effect require that quantum mechanics be extended to a uniquely defined set of open systems, termed proper open systems . 

This extension of quantum mechanics is indeed possible within the framework of the generalized action principle and, as stated in Section 1, its use has resulted in the identification of the proper open systems of quantum mechanics with the topological atom. The derivation of the quantum mechanics of a proper open system cannot be given here, but one can present an outline of the essential ideas. It is based on the reformulation of physics provided by the work of Feynman and Schwinger, an approach that makes possible the asking and answering of questions not possible within the traditional Hamiltonian framework. It is important that the reader appreciate that the alternative to the use of existing... 

The parallelism with the transferability of the electron density is found for all properties. The methyl and methylene groups of the hydrocarbons, for example, exhibit transferable volumes, polarizabilities, and magnetic susceptibilities in addition to transferable energies, in complete agreement with experiment. In all cases, given a wave function, the properties of the atom or group are determined by the quantum mechanics of a proper open system without the use of any assumptions or models. The atomic contribution to the magnetic susceptibility, for example, makes use of the atomic current theorem,... 

The recovery of molecular structure in terms of the topology of the same physical field that defines the quantum boundary condition of the atom reinforces this identification of the chemical atom with a proper open system of quantum mechanics and reinforces the obvious, that the unquestioned success of the molecular structure hypothesis demands that it be firmly rooted in physics. 

One recognizes the first term in eqn (8.136) as the variation in the quantum mechanical current density (eqns (5.94) and (5.95)). It is obtained by combining the surface term arising from the variation with respect to VT with the surface term arising from the imposition of the variational constraint, eqn (8.135). Thus the variation of the surface of the subsystem together with the restriction that the subsystem be an atom bounded by a zero-flux surface causes the quantum mechanical current density to appear in the variation of the action integral, a term whose presence is a necessary requirement for the proper description of the properties of an open system. It is now demonstrated that eqn (8.136) is the atomic equivalent of the principle of stationary action. 

Keeping the lesson of the above example in mind, we will explore three different dynamical possibilities below isolated evolution, where the system evolves without any coupling to the external world, unconditioned open ev olution, where the system evolves coupled to an external environment but where no information regarding the system is extracted from the environment, and conditioned open evolution where such information is extracted. In the third case, the evolution of the physical state is driven by the system evolution, the coupling to the external world, and by the fact that observational information regarding the state has been obtained. This last aspect - system evolution conditioned on the measurement results via Bayesian inference - leads to an intrinsically nonlinear evolution for the system state. The conditioned evolution provides, in principle, the most realistic possible description of an experiment. To the extent that quantum and classical mechanics are eventually just methodological tools to explain and predict the results of experiments, this is the proper context in which to compare them. 

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