Abstract
The finite-dimensional quantum mechanics (FDQM) based on Weyl’s form of Heisenberg’s canonical commutation relations, developed for the case of one-dimensional space, is extended to three-dimensional space. This FDQM is applicable to the physics of particles confined to move within finite regions of space and is significantly different from the current quantum mechanics in the case of atomic and subatomic particles only when the region of confinement is extremely small—of the order of nuclear or even smaller dimensions. The configuration space of such a particle has a quantized eigenstructure with a characteristic dependence on its rest mass and dimension of the region of confinement, and the current Schrödinger-Heisenberg formalism of quantum mechanics becomes an asymptotic approximation of this FDQM. As an example, a spherical harmonic oscillator with a particular radius of confinement is considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barut, A. O., and Bracken, A. J. (1980), “The Zitterbewegung and the Internal Geometry of the Electron,” Preprint, University of Colorado, Boulder, Colorado.
Dadic, I., and Pisk, K. (1979),International Journal of Theoretical Physics,18, 345.
Finkelstein, D. (1974).Physical Review D,9, 2219.
Ginsburg, V. L. (1976).Key Problems of Physics and Astrophysics, Mir. Pub. Moscow, Sec2. 12.
Jagannathan, R., Santhanam, T. S., and Vasudevan, R. (1981). “Finite-dimensional Quan tum mechanics of a Particle,”International Journal of Theoretical Physics,20, 755.
Lorente, M. (1974).International Journal of Theoretical Physics,11, 213.
Ramakrishnan, A. (Editor) (1971).Proceedings of the Conference on Clifford Algebra: Its Generalizations and Applications, Matscience Report, Madras.
Ramakrishnan, A. (1972).L-Matrix Theory or Grammar of Dirac Matrices, Tata-McGraw Hill, New Delhi.
Saavedra, I., and Utreras, C. (1981).Physics Letters,B98, 74.
Santhanam, T. S., and Tekumalla, A. R. (1976).Foundations of Physics,6, 583.
Santhanam, T. S. (1977a).Foundations of Physics,7, 121.
Santhanam, T. S. (1977b). InUncertainty Principle and Foundations of Quantum Mechanics, W. Price and S. S. Chissick, eds. John Wiley, New York, p. 227.
Santhanam, T. S. (1978). InProceedings of the 1978 International Meeting on Frontiers of Physics, K. K. Phua, C. K. Chew and Y. K. Lim, eds. The Singapore National Academy of Sciences, Singapore, p. 1167.
Schwinger, J. (1960).Proceedings of the National Academy of Sciences (U. S. A.),46, 570.
Snyder, H. (1947).Physical Review,71, 38.
Stovicek, P., and Tolar, I. (1979). “Quantum mechanics in a Discrete Space-Time,” Preprint No. Ic/79/147, International Centre for Theoretical Physics, Trieste, Italy.
Tati, T. (1980). “The Theory of Finite Degrees of Freedom,” Preprint, No. 80-8, Research Institute for Theoretical Physics, Hiroshima University, Takehara, Hiroshima, Japan.
Weyl, H. (1932).Theory of Groups and Quantum mechanics, E. P. Dutton Co. New York, (Dover, New York, 1950), Sec. 4.14.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Jagannathan, R., Santhanam, T.S. Finite-dimensional quantum mechanics of a particle. II. Int J Theor Phys 21, 351–362 (1982). https://doi.org/10.1007/BF02650236
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02650236