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arXiv:1207.1477

Bohr-Sommerfeld-Heisenberg Quantization of the 2-dimensional Harmonic Oscillator

Richard H. Cushman, Jędrzej Śniatycki

Published Jul 5, 2012• 5 citations
arXivHTMLPDF

Abstract

We study geometric quantization of the harmonic oscillator in terms of a singular real polarization given by fibres of the energy momentum map.
DOI
10.48550/arxiv.1207.1477

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In the framework of geometric quantization we extend the Bohr-Sommerfeld rules to a full quantization theory which resembles Heisenberg's matrix theory. This extension is possible because Bohr-Sommerfeld rules not only provide an orthogonal basis in the space of quantum states, but also give a lattice structure to this basis. This permits the definition of appropriate shifting operators. As examples, we discuss the 1-dimensional harmonic oscillator and the coadjoint orbits of the rotation group.
An Introduction to the geometric quantization procedure which is the attempt at finding a procedure that maps classical observables to quantum observables.\n\nFrom this idea, we made a more mathematical approach to some very interesting physics.\nWe derived a mapping and while analysing it. It becomes clear that more conditions have to be imposed for this to produce physically correct answers. \nFrom geometric quantization, we produced the canonical quantization and the operator form of the Hamiltonian of the harmonic oscillator.\nAlong the way, we also found some physical consequences of the math.