Symplectic quantization, inequivalent quantum theories, and Heisenberg's principle of uncertainty

Journal Article · Wed Sep 01 04:00:00 EDT 2004 · Physical Review. A

DOI:https://doi.org/10.1103/PhysRevA.70.032104· OSTI ID:20645996

Montesinos, Merced; Torres del Castillo, G F ^[1]

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1. Departamento de Fisica, Centro de Investigacion y de Estudios Avanzados del IPN, Avenida IPN No. 2508, 07000 Ciudad de Mexico (Mexico)

We analyze the quantum dynamics of the nonrelativistic two-dimensional isotropic harmonic oscillator in Heisenberg's picture. Such a system is taken as a toy model to analyze some of the various quantum theories that can be built from the application of Dirac's quantization rule to the various symplectic structures recently reported for this classical system. It is pointed out that that these quantum theories are inequivalent in the sense that the mean values for the operators (observables) associated with the same physical classical observable do not agree with each other. The inequivalence does not arise from ambiguities in the ordering of operators but from the fact of having several symplectic structures defined with respect to the same set of coordinates. It is also shown that the uncertainty relations between the fundamental observables depend on the particular quantum theory chosen. It is important to emphasize that these (somehow paradoxical) results emerge from the combination of two paradigms: Dirac's quantization rule and the usual Copenhagen interpretation of quantum mechanics.

• OSTI ID: 20645996
• Journal Information: Physical Review. A, Journal Name: Physical Review. A Journal Issue: 3 Vol. 70; ISSN 1050-2947; ISSN PLRAAN
• Country of Publication: United States
• Language: English

Related Subjects

74 ATOMIC AND MOLECULAR PHYSICS
COORDINATES
HARMONIC OSCILLATORS
QUANTIZATION
QUANTUM MECHANICS
QUANTUM OPERATORS
TWO-DIMENSIONAL CALCULATIONS
USES