Period doubling in the n+l filling rule and dynamical symmetry of the Demkov-Ostrovsky atomic model

The invariance properties of the Demkov-Ostrovsky (DO) equation are analysed. This equation models the n+l filling rule of the atomic Aufbau principle. Contrary to a claim by Ostrovsky and contrary to generally known quantum problems, the degeneracy structure of the quantum number n+l in this model is not associated with the representation of a finite-dimensional Lie algebra. It is found, however, that the degeneracy algebra does not close under the usual commutation relations, but under a generalised set of commutation rules. The properties of the new algebra are closely examined. It is shown that the characteristic 'period doubling' in the DO model follows from the structure of this algebra. It is also shown that the two-dimensional analogue of the DO equation admits an SO(3,2) dynamical group which, however, allows more states than the physically required ones. Nevertheless, by a redefinition of the quantum labellings, one obtains a mapping to the appropriate state diagram corresponding to the Aufbau chart.

[1]  V. Fock,et al.  Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems , 1930 .

[2]  H. Kleinert,et al.  Calculation of Relativistic Transition Probabilities and Form Factors from Noncompact Groups , 1967 .

[3]  O. Novaro,et al.  Approximate symmetry of the periodic table , 1972 .

[4]  A. Barut,et al.  Reduction of a Class of O(4, 2) Representations with Respect to SO(4, 1) and SO(3, 2) , 1970 .

[5]  Adolf A. Abrahamson,et al.  ONSET OF ATOMIC-SUBSHELL FILLING IN ORDINARY AND SUPERHEAVY ELEMENTS. , 1971 .

[6]  W. Meggers,et al.  Structure of Matter , 1965 .

[7]  B. G. Wybourne,et al.  Classical Groups for Physicists , 1974 .

[8]  V. Fock,et al.  Zur Theorie des Wasserstoffatoms , 1935 .

[9]  J. C. Donini.  Recent Advances in Group Theory and Their Application to Spectroscopy , 1979 .

[10]  D. Hartree.  The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods , 1928, Mathematical Proceedings of the Cambridge Philosophical Society.

[11]  Robert L. Anderson,et al.  Generalization of the Concept of Invariance of Differential Equations. Results of Applications to Some Schrödinger Equations , 1972 .

[12]  E. Fermi.  Eine statistische Methode zur Bestimmung einiger Eigenschaften des Atoms und ihre Anwendung auf die Theorie des periodischen Systems der Elemente , 1928 .

[13]  W. Pauli,et al.  Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren , 1925 .

[14]  O. Novaro,et al.  On a geometrical realization of the Aufbau scheme , 1973 .

[15]  V. Ostrovsky.  Dynamic symmetry of atomic potential , 1981 .

[16]  N. Bohr,et al.  Der Bau der Atome und die physikalischen und chemischen Eigenschaften der Elemente , 1922 .

[17]  Hagen Kleinert,et al.  Transition Probabilities of the Hydrogen Atom from Noncompact Dynamical Groups , 1967 .