Quantum mechanical ingredients and generalizations

Classical statistical mechanical theory is, for the most part, adequate for the solutions treated in this book (Benmore et al, 2001 Tomberli et al, 2001), as has been discussed more specifically elsewhere (Feynman and Hibbs, 1965). It is important to distinguish that issue of statistical mechanical theory from the theory, computation, and modeling involved in the interaction potential energy U N). The potential distribution theorem doesn t require specifically simplified forms for /(2V) on grounds of statistical mechanical principal simplifications can make calculations more practical, of course, but those are issues to be addressed for specific cases. 

The most basic point here is that the only specifically classical feature of Eq. (3.17), p. 40, is the n assumed in Eq. (3.14), p. 39. This feature derives from the indistinguishability of particles other than the electrons, and a more correct account of the indistinguishabihty of those heavy particles would involve exchanging identities with the proper phases (Feynman and Hibbs, 1965). But, as is well known, those exchange contributions are the least significant of quantum 

This point can be underscored by returning again to consider the unnormalized density matrix lurking underneath Eq. (2.15), p. 26  

The point to be underscored is that many available approximate solutions for p (TV, TV ) of Eq. (3.65), e.g. (Gomez and Pratt, 1998), with the Boltzmann-Gibbs treatment of heavy-particle exchange, can be applied to evaluation of the potential distribution theorem. The most important physical requirement is that such a model gracefully adapt to the classical limit, because that is the most important physical limit for molecular solutions. 

In this section we discuss quantum mechanical models that can be brought to bear on evaluation of the potential distribution theorem. These models could be tried and tested in practical calculations, but the basics of these models should be studied elsewhere - the present discussion is not about quantum mechanics for its own sake. The remainder of this section then gives a more technical discussion of current ideas for inclusion of nonexchange quantum mechanical effects. 

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This result concludes the presentation of the Heisenberg group approach as the powerful tool that allows to derive classical mechanics as a formal limit of quantum mechanics, for h —> 0. The most important ingredients that have been introduced to obtain this result are the Fourier-like representation of observables and equations of motion and the definition of the antiderivative operator. These elements will be used in section 5 to derive a similiar procedure for a mixed quantum-classical mechanics. An ansatz on the quantum-classical equations of motion will be necessary, but the subsequent application of Heisenberg group formalism will be a straightforward generalization of what has been done so far. 

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