Circular Scale of Time Applied in Classifying the Quantum-Mechanical Energy Terms Entering the Framework of the Schrödinger Perturbation Theory
- 1
Abstract
The paper applies a one-to-one correspondence which exists between individual Schrödinger perturbation terms and the diagrams obtained on a circular scale of time to whole sets of the Schrödinger terms belonging to a definite perturbation order. In effect the diagram properties allowed us to derive the recurrence formulae giving the number of higher perturbative terms from the number of lower order terms. This recurrence formalism is based on a complementary property that any perturbation order N can be composed of two positive integer components N<sub>a</sub> , N<sub>b</sub> combined into N in all possible ways. Another result concerns the degeneracy of the perturbative terms. This degeneracy is shown to be only twofold and the terms having it are easily detectable on the basis of a circular scale. An analysis of this type demonstrates that the degeneracy of the perturbative terms does not exist for very low perturbative orders. But when the perturbative order exceeds five, the number of degenerate terms predominates heavily over that of nondegenerate terms.
- E. Schr?dinger, “Quantization as an Eigenvalue Problem III,” Annalen der Physik, Vol. 80, 1926, pp. 437-490.
- P. M. Morse and H. Feshbach, “Methods of Theoretical Physics, Part 2,” McGraw-Hill, New York, 1953.
- N. H. March, W. H. Young and S. Sampanthar, “The Many-Body Problem in Quantum Mechanics,” Cambridge University Press, Cambridge, 1967.
- R. D. Mattuck, “A Guide to Feynman Diagrams in the Many-Body Problem,” 2nd Edition, McGraw-Hill, New York, 1976.
- J. Killingbeck, “Quantum-Mechanical Perturbation Theory,” Reports on Progress in Physics, Vol. 40, No. 9, 1977, pp. 963-1031. doi:10.1088/0034-4885/40/9/001
- P. O. L?wdin, “Proceedings of the International Workshop on Perturbation Theory of Large Order,” International Journal of Quantum Chemistry, Vol. 21, 1982, pp. 1-353.
- G. A. Arteca, F. M. Fernandez and E. A. Castro, “Large Order Perturbation Theory and Summation Methods in Quantum Mechanics,” Springer, Berlin, 1990.
- J. C. Le Guillou and J. Zinn-Justin, “Large-Order Behaviour of Perturbation Theory,” North-Holland, Amsterdam, 1990.
- R. P. Feynman, “The Theory of Positrons,” Physical Review, Vol. 76, No. 6, 1949, pp. 749-759. doi:10.1103/PhysRev.76.749
- R. Huby, “Formulae for Rayleigh-Schr?dinger Perturbation Theory in Any Order,” Proceedings of the Physical Societ, London, Vol. 78, 1961, pp. 529-536.
- B. Y. Tong, “On Huby’s Rules for Non-Degenerate Ray- leigh-Schrodinger Perturbation Theory in Any Order,” Proceedings of the Physical Society, London, Vol. 80, 1962, pp. 1101-1104.
- S. Olszewski, “Time Scale and Its Application in the Perturbation Theory,” Zeitschrift fur Naturforschung A, Vol. 46, 1991, pp. 313-320.
- S. Olszewski, “Time Topology for Some Classical and Quantum Non-Relativistic Systems,” Studia Philosophiae Christianae, Vol. 28, 1992, pp. 119-135.
- S. Olszewski and T. Kwiatkowski, “A Topological Approach to Evaluation of Non-Degenerate Schrodinger Per- turbation Energy Based on a Circular Scale of Time,” Computation Chemistry, Vol. 22, No. 6, 1998, pp. 445- 461. doi:10.1016/S0097-8485(98)00023-0
- S. Olszewski, “Two Pathways of the Time Parameter Characteristic for the Perturbation Problem in Quantum Chemistry,” Trends in Physical Chemistry, Vol. 9, 2003, pp. 69-101.