Živković, Tomislav
(2004)
On the interaction of an isolated state with the known infinitedimensional quantum system.
Croatica Chemica Acta, 77
(12).
pp. 279293.
ISSN 00111643
Abstract
The interaction of an isolated state Theta> with a known infinitedimensional quantum system Sinfinity(b) containing multiple eigenvalue bands is considered. An example is the interaction of an isolated molecular state with the electromagnetic field. This problem is the main subject of spectroscopy. Another example is the interaction of an isolated molecular state of a molecule situated on the surface of a solid with this solid. This is the subject of the surface state physics. Such problems are usually treated within the formalism of the perturbation expansion. Recently a new mathematical method for the treatment of such problems was suggested. This method produces correct solution of the combined system, however strong the interaction between the state Theta> and the system Sinfinity(b). Key quantities in this method are characteristic functions f(v)(epsilon) and functions omega(v)(epsilon). Functions f(v)(epsilon) are constructed from the quantities describing unperturbed system Sinfinity(b) and from the known interaction of this system with the state Theta>. Functions omega(v)(epsilon) are then expressed in terms of the characteristic functions f(v)(epsilon). An important mathematical problem is the calculation of the functions omega(v)(epsilon), once functions f(v)(epsilon) are known. A general mathematical solution to this problem is obtained for the case when characteristic functions f(v)(epsilon) are polynomials. Since each function which is continuous in a finite interval can be approximated (to any desired degree of accuracy) with some polynomial, this solution presents an important mathematical tool for the practical application of the suggested method. Obtained results are illustrated with few model examples of the interaction of an isolated state with an infinite quantum system.
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