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Some applications of the Gel'fand–Levitan inverse method in atomic and molecular physics
Author(s) -
Hron M.,
Razavy M.
Publication year - 1983
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/qua.560240109
Subject(s) - semiclassical physics , hamiltonian (control theory) , physics , observable , bound state , wave function , quantum mechanics , homogeneous space , inverse , schrödinger equation , quantum , classical mechanics , mathematics , mathematical optimization , geometry
The Gel'fand–Levitan theory is applied to obtain the exact potential function for an atomic or a molecular system when the solution of the Schrödinger equation with a reference potential which is close to the actual potential is known. With this method one can construct bound state equivalent local potentials, or potentials which, for large quantum numbers, have spectra similar to that of the reference potential. In one‐dimensional problems the resulting potentials are not generally symmetric about the origin, even if one starts with symmetric reference potentials. This lack of symmetry is due to the presence of additional terms in the Hamiltonian which are of purely quantal origin, and therefore in any semiclassical treatment become negligible. This observation leads one to investigate the relationship between the spectrum of a given system and the symmetry of the underlying Hamiltonian. Examples are given to show that the dynamical symmetries usually associated with the spectral properties do not necessarily follow from the Gel'fand–Levitan inverse theory, but can be imposed as constraints on the solution. Additional constraints on the form of the two‐body wave function also arise from the observable transition rates for atom–molecule collisions and for the interaction of the charged particles in the system with an external electromagnetic field.