Summary
We consider the most general case of the restricted rigid rotor, controlled by passive mechanical devices located at θ=0 and θ=π. The purpose of these devices is to restrict the particle motion to a domain of a covering space (0,Mπ), whereM is an odd integer. This system, which is not a Hamiltonian one on the physical space (0, 2π), is compared with a Hamiltonian system having delta function barriers at θ=0 and θ=π. The case ofM an even integer is also discussed by using only one mechanical device at θ=0. This non-Hamiltonian system is compared with a Hamiltonian system having a delta function barrier at θ=0. It is shown that many of the wave functions of the non-Hamiltonian systems are the same as those of the Hamiltonian ones, with an average reflection coefficient of 1/(M+1) for oddM and 2/M for evenM, which are the classical values. We show how, in the case of very largeM, the superposition principle leads to de Broglie resonances.
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References
A. Inomata andC. Bernido:Phys. Lett. A,77, 394 (1980).
H. D. Doebner, H. J. Elmers andW. F. Heidenreich:J. Math. Phys.,30, 1053 (1989).
S. M. Al-Jaber andW. C. Henneberger:J. Phys. A,23, 2939 (1990).
M. G. Benedict andL. Gy. Feher:Phys. Rev. D,39, 3194 (1989).
Y. Ohnuki:Proceedings of the 11 International Symposium on Foundation of Quantum Mechanics (Physical Society of Japan, Tokyo, 1987), p. 117.
Y. Aharonov andD. Bohm:Phys. Rev.,115, 485 (1959).
L. S. Schulman:J. Math. Phys.,12, 304 (1971).
D. Shapiro andW. C. Henneberger:J. Phys. A,22, 3605 (1989).
X. Zhu andW. C. Henneberger:J. Phys. A,23, 3983 (1990).
L. W. Jolley:Summation of Series (Dover, New York, N.Y., 1961), p. 78.
T. J. Bromwich:An Introduction to the Theory of Infinite Series (Macmillan, London, 1965), p. 211.
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Al-Jaber, S.M., Henneberger, W.C. Topological considerations in quantum theory. Nuov Cim B 107, 23–37 (1992). https://doi.org/10.1007/BF02726880
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DOI: https://doi.org/10.1007/BF02726880