Heisenberg uncertainty relation approximation

The chemical concept of localization is based on the classical assumption that molecular electron densities are confined to finite regions of space. Although various quantum chemical approaches have been proposed to implement this concept within a theoretical framework, this is essentially a classical idea that disregards some aspects of the Heisenberg uncertainty relation. Nevertheless, the interpretation of molecular properties and chemical reactions strongly relies on the inherent assumption that both complete molecules and various, chemically identifiable, molecular pieces can be assigned to various regions of space, at least in some approximate sense. The explanation of the apparent success of this essentially classical assumption within a quantum chemical framework is a nontrivial problem. 

We now detail a second approach to the evaluation of the degree of symmetry deficiency, from a different point of view, namely the fuzzy sets approach. Fuzzy set methods are especially suitable to reflect the Heisenberg uncertainty relation and other aspects of quantum chemistry " and computational chemistry. Symmetry in molecules relies on the concepts of distance and metric. For the approximate symmetries of fuzzy electron densities, fuzzy set methods, fuzzy distance or fuzzy metric, and fuzzy symmetry are of importance. In Section 3.3 below, we compare and generalize the two approaches. 

The matter may be regarded from the point of view of the uncertainty principle. The behaviour of particles which is defined by the wave equation is equivalent to an indefiniteness in what may be known of their dynamical coordinates, lip and q are the momentum and position coordinates, Heisenberg s principle states that both cannot be known simultaneously except with a range of uncertainty given by the relation ApAg = h, approximately. In the temperature... 

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