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Effective description of the dissipative interaction between simple model‐systems and their environment
Author(s) -
Schuch Dieter
Publication year - 1999
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/(sici)1097-461x(1999)72:6<537::aid-qua1>3.0.co;2-q
Subject(s) - dissipative system , nonlinear system , degrees of freedom (physics and chemistry) , hamiltonian (control theory) , harmonic oscillator , logarithm , quantum , classical mechanics , mathematics , statistical physics , physics , mathematical analysis , quantum mechanics , mathematical optimization
Chemical reactions are generally connected with energy transfer between reactands and their environment. There are different ways of taking this environment into account. The traditional system‐plus‐reservoir (S+R) approach couples the relevant system to a large number of environmental degrees of freedom, represented, e.g., by harmonic oscillators. However, this has the disadvantage—from a computational point of view—that a system of many coupled differential equations has to be solved due to the large number of degrees of freedom of the reservoir. The number of degrees of freedom can be drastically reduced if an effective description of the interaction between system and reservoir is applied. On the quantum mechanical level, the effect of the environment can be included, e.g., by the use of explicitly time‐dependent Hamiltonians or nonlinear additions to the Schrödinger equation (SE). An advantage of the former method is the preservation of the canonical formalism and the linearity of the theory as well as the fact that its operators can be derived directly from the S+R approach. Therefore, it seems to be even more of a paradox that this method apparently violates the uncertainty principle. This violation does not occur in the nonlinear approaches but the nonlinearity seems, at first sight, to be problematic. However, it can be shown that a particular logarithmic nonlinear SE is equivalent to the approach using the time‐dependent Hamiltonian and can be transformed into this. The fact that this transformation is nonunitary provides the key for the solution of the paradox and also allows one to show the equivalence of all three of the above‐mentioned approaches. In investigating the classical counterpart, a noncanonical transformation is used between the dissipative physical level and the canonical formal level, which makes it even possible to transform the physical interacting system into a formal isolated system that can be treated in the conventional way. As a result, the effect of the interaction with the environment can be taken into account classically by noncanonical transformations and quantum mechanically by nonunitary transformations. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 537–547, 1999