Croatica Chemica Acta, Vol. 77 No. 1-2, 2004.
Original scientific paper
On the Interaction of an Isolated State with the Known Infinite-dimensional Quantum System
Tomislav P. Živković
; Ruđer Bošković Institute, P. O. Box 180, HR-10002 Zagreb, Croatia
Abstract
The interaction of an isolated state |Θ〉 with a known infinite-dimensional quantum system S∞ 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 |Θ〉 and the system S∞ b . Key quantities in this method are characteristic functions fv(ε) and functions ωv(ε). Functions fv(ε) are constructed from the quantities describing unperturbed system S∞ b and from the known interaction of this system with the state |Θ〉. Functions ωv(ε) are then expressed in terms of the characteristic functions fv(ε). An important mathematical problem is the calculation of the functions ωv(ε), once functions fv(ε) are known. A general mathematical solution to this problem is obtained for the case when characteristic functions fv(ε) 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.
Keywords
interaction of quantum systems; perturbation; eigenvalues; eigenstates
Hrčak ID:
102676
URI
Publication date:
31.5.2004.
Visits: 922 *