Phys. Rev. A 70, 032104 (2004) [8 pages]Symplectic quantization, inequivalent quantum theories, and Heisenberg’s principle of uncertaintyReceived 28 January 2004; published 14 September 2004 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. © 2004 The American Physical Society URL:
http://link.aps.org/doi/10.1103/PhysRevA.70.032104
DOI:
10.1103/PhysRevA.70.032104
PACS:
03.65.Ta, 03.65.Ca
See AlsoComment: D. C. Latimer, Comment on “Symplectic quantization, inequivalent quantum theories, and Heisenberg’s principle of uncertainty”, Phys. Rev. A 75, 066101 (2007). Comment: Merced Montesinos and G. F. Torres del Castillo, Reply to “Comment on ‘Symplectic quantization, inequivalent quantum theories, and Heisenberg’s principle of uncertainty’ ”, Phys. Rev. A 75, 066102 (2007). |
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