Some Algebraic Aspects of the Relationship Between Classical and Quantum Mechanics .

Enrico G. Beltrametti, Slawomir Bugajski

Jan J. Sławianowski, Vasyl Kovalchuk, Agnieszka Martens +2 more

Discussed are some geometric aspects of the phase space formalism in quantum mechanics in the sense of Weyl, Wigner, Moyal, and Ville. We analyze the relationship between this formalism and geometry of the Galilei group, classical momentum mapping, theory of unitary projective representations of groups, and theory of groups algebras. Later on, we present some generalization to quantum mechanics on locally compact Abelian groups. It is based on Pontryagin duality. Indicated are certain physical aspects in quantum dynamics of crystal lattices, including the phenomenon of ‘Umklapp–Prozessen’. Copyright © 2011 John Wiley & Sons, Ltd.

Anton Amann, Werner Gans

Abstract Quantum mechanical and classical aspects are of equal importance in chemistry, and theoretical chemistry must be able to deal with this situation. Classical quantities with which a chemist has to deal include, for example, the temperature and the chemical potential of a substance, and chirality, knot‐type and tertiary structures of molecules. Classical concepts of this type are no more consequences of traditional quantum chemistry than is the theory of complex chemical processes (for example adsorption, heterogeneous catalysis) and macroscopic systems. Algebraic quantum mechanics affords a general framework for the discussion of classical quantities and large systems. There are other independent approaches for the theoretical treatment of chemical problems; these can be summarized by using the keywords “fractals”, “chaotic motion”, “quasicrystals” and “knot theory”. The relationships between these four research topics and their connection with algebraic quantum mechanics, as well as their importance for theoretical chemistry, were discussed at a NATO Workshop in 1987 (see Ref. [1]). In an extended theoretical chemistry, the notion of “molecules” will not play a role as central as it does in quantum chemistry. In the theory of large systems in particular, the introduction of complementary approaches appears to be appropriate; the term “molecule” as used by the chemist does not appear either in the description of disordered systems by means of fractals or in the thermodynamic limit of algebraic statistical mechanics.

A. Horzela

Jarmo Hietarinta

The author shows that in quantum mechanics there can be more commuting constants of motion than the dimension suggests. There will be an algebraic relationship between the invariants, but when this relationship is non-trivial it can be used to reduce the Schrodinger equation to quadratures. The Lame equation and some of its generalisations fall into the class of potentials for which this method works.

Emile Grgin, Aage Petersen