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29 March 2024
 
  » arxiv » quant-ph/0306059

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Quantization with maximally degenerate Poisson brackets: The harmonic oscillator!
Y. Nutku ;
Date 9 Jun 2003
Journal J.Phys. A36 (2003) 7559-7568
Subject Quantum Physics; Exactly Solvable and Integrable Systems | quant-ph hep-th nlin.SI
AbstractNambu’s construction of multi-linear brackets for super-integrable systems can be thought of as degenerate Poisson brackets with a maximal set of Casimirs in their kernel. By introducing privileged coordinates in phase space these degenerate Poisson brackets are brought to the form of Heisenberg’s equations. We propose a definition for constructing quantum operators for classical functions which enables us to turn the maximally degenerate Poisson brackets into operators. They pose a set of eigenvalue problems for a new state vector. The requirement of the single valuedness of this eigenfunction leads to quantization. The example of the harmonic oscillator is used to illustrate this general procedure for quantizing a class of maximally super-integrable systems.
Source arXiv, quant-ph/0306059
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