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Path representations of the quantum-mechanical energy-level density

Published online by Cambridge University Press:  24 October 2008

Viktor Bezák
Viktor Bezák
Affiliation:
Electrotechnical Institute, Slovak Academy of Sciences, Bratislava, Czechoslovakia
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Abstract

The quantum-mechanical energy-level density g(E) is given as a functional of the quantum-mechanical kernel K(q″, q′, t″ −t′). On taking the kernel K in the Feynman's form, one obtains the function g(E), without solving a Schrödinger equation. As an example, the embedding of a particle in the one-dimensional square well with infinitely high walls is analysed. The functions K(x″, x′, t−t′) and g(E) are represented as sums of terms corresponding to classical paths of different types. By an adequate choice of some terms due to the ‘most important’ paths, one may construct partial sums giving approximations of the function g(E). The utilization of such approximations for estimation of energy levels is demonstrated.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 64 , Issue 3 , July 1968 , pp. 787 - 794
Copyright
Copyright © Cambridge Philosophical Society 1968

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References

REFERENCES

(1)Feynman, R. P.Rev. Modern Phys. 20 (1948), 367–387.CrossRefGoogle Scholar
(2)Feynman, R. P. and Hibbs, A. R.Quantum mechanics and path integrals (McGraw-Hill; New York, 1965).Google Scholar
(3)Kilmister, C. W.Proc. Cambridge Philos. Soc. 54 (1958), 302–304.CrossRefGoogle Scholar
(4)Erdogan, F. On the Evaluation of Feynman Path Integrals (unpublished, Parke Mathematical Laboratories Report AFCRL-TN-60–1109; Carlisle, Mass., 1960).Google Scholar
(5)Brush, S. G.Rev. Modern Phys. 33 (1961), 79–92.CrossRefGoogle Scholar
(6)Beauregard, L. A.Amer. J. Phys, 34 (1966), 324–332.CrossRefGoogle Scholar
(7)Clutton-Brock, M.Proc. Cambridge Philos. Soc. 61 (1965), 201–205.CrossRefGoogle Scholar
(8)March, N. H., Murray, A. M.Phys. Rev. 120 (1960), 830–836.CrossRefGoogle Scholar
(9)Gel'fand, I. M. and Shilov, G. E.Generalized functions, volume 1 (in Russian, Moscow, 1959, p. 49; English translation: Academic Press, London, New York, 1964).Google Scholar