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arXiv:1606.06474 · Symmetry Integrability and Geometry Methods and Applications

Born-Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator

Universit&, Giovanni Rastelli, #224, di Torino, Italy

Published Aug 17, 2016• 1 citation
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Abstract

We apply the Born-Jordan and Weyl quantization formulas for polynomials in canonical coordinates to the constants of motion of some examples of the superintegrable 2D anisotropic harmonic oscillator.Our aim is to study the behaviour of the algebra of the constants of motion after the different quantization procedures.In the examples considered, we have that the Weyl formula always preserves the original superintegrable structure of the system, while the Born-Jordan formula, when producing different operators than the Weyl's one, does not.

DOI
10.3842/sigma.2016.081

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We apply the Born-Jordan and Weyl quantization formulas for polynomials in canonical coordinates to the constants of motion of some examples of the superintegrable 2D anisotropic harmonic oscillator.Our aim is to study the behaviour of the algebra of the constants of motion after the different quantization procedures.In the examples considered, we have that the Weyl formula always preserves the original superintegrable structure of the system, while the Born-Jordan formula, when producing different operators than the Weyl's one, does not.
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Quantization of the 1-D harmonic oscillator with variable mass using the operators v^\hat vv^ and p^\hat pp^​
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The aim of the famous Born and Jordan 1925 paper was to put Heisenberg's matrix mechanics on a firm mathematical basis. Born and Jordan showed that if one wants to ensure energy conservation in Heisenberg's theory it is necessary and sufficient to quantize observables following a certain ordering rule. One apparently unnoticed consequence of this fact is that Schrödinger's wave mechanics cannot be equivalent to Heisenberg's more physically motivated matrix mechanics unless its observables are quantized using this rule, and not the more symmetric prescription proposed by Weyl in 1926, which has become the standard procedure in quantum mechanics. This observation confirms the superiority of Born-Jordan quantization, as already suggested by Kauffmann. We also show how to explicitly determine the Born--Jordan quantization of arbitrary classical variables, and discuss the conceptual advantages in using this quantization scheme. We finally suggest that it might be possible to determine the correct quantization scheme by using the results of weak measurement experiments.