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arXiv:0707.1152

Quantum and Classical Variance in the Quantum Realm

Mario Rabinowitz

Published Jul 8, 2007• 0 citations
arXivHTMLPDF

Abstract

This paper examines the variance of quantum and classical predictions in the quantum realm, as well as unexpected presence and absence of variances. Some features are found that share an indirect commonality with the Aharonov-Bohm and Aharonov-Casher effects in that there is a quantum action in the absence of a force. Variances are also found in the presence of a force that are more subtle as they are of higher order. Significant variances related to the harmonic oscillator and particle in a box periods are found. This paper raises the question whether apparent quantum self-inconsistency may be examined internally, or must be empirically ascertained. These inherent variances may either point to inconsistencies in quantum mechanics that should be fixed, or that nature is manifestly more non-classical than expected. For the harmonic oscillator it is proven that the second spatial moment is the same in QM and CM.

DOI
10.48550/arxiv.0707.1152

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This paper examines the variance of quantum and classical predictions in the quantum realm, as well as unexpected presence and absence of variances. Some features are found that share an indirect commonality with the Aharonov-Bohm and Aharonov-Casher effects in that there is a quantum action in the absence of a force. Variances are also found in the presence of a force that are more subtle as they are of higher order. Significant variances related to the harmonic oscillator and particle in a box periods are found. This paper raises the question whether apparent quantum self-inconsistency may be examined internally, or must be empirically ascertained. These inherent variances may either point to inconsistencies in quantum mechanics that should be fixed, or that nature is manifestly more non-classical than expected. For the harmonic oscillator it is proven that the second spatial moment is the same in QM and CM.
The independence of the Aharonov-Bohm phase shift on particle velocity is one of its defining properties. The classical counterpart to this dispersionless behavior is the absence of forces along the direction of motion of the particle. A reevaluation of the experimental demonstration that forces are absent in the AB physical system is given, including previously unpublished data. It is shown that the debate on the presence or absence of forces is not settled. Experiments that measure the influence of magnetic permeability on forces and search for dispersionless quantum forces are proposed.
The independence of the Aharonov-Bohm phase shift on particle velocity is one of its defining properties. The classical counterpart to this dispersionless behavior is the absence of forces along the direction of motion of the particle. A reevaluation of the experimental demonstration that forces are absent in the AB physical system is given, including previously unpublished data. It is shown that the debate on the presence or absence of forces is not settled, and an experiment searching for dispersionless forces is proposed.
The Aharonov-Bohm (AB) effect is a purely quantum mechanical effect. The original (classified as type-I) AB-phase shift exists in experimental conditions where the electromagnetic fields and forces are zero. It is the absence of forces that makes the AB effect entirely quantum mechanical. Although the AB-phase shift has been demonstrated unambiguously, the absence of forces in type-I AB effects has never been shown. Here, we report the observation of the absence of time delays associated with forces of the magnitude needed to explain the AB-phase shift for a macroscopic system.
The Aharonov-Bohm effect is relatively little known, but it poses fundamental questions in quantum theory. It concerns physical effects on charged particles in field-free regions. The present paper provides an over-all critical review and commentary on the literature of the Aharonov-Bohm effect. The two fundamental questions raised by the effect concern the role of the force concept in quantum mechanics, and the localizability of physical effects principle. A recent letter from Professor Bohm concerning his evaluation of the significance of the effect is included in the Appendix.
Decades ago, the idea that electrons can be affected by electromagnetic potentials without experiencing force-fields changed our thinking with the advent of the Aharonov-Bohm effect. Zeilingers theorem, which casts the absence of force fields in quantum terms as the dispersionless nature of the Aharonov-Bohm effect, cemented the idea in textbooks. Surprisingly, in the late 1990s a quantum force was predicted for the Aharonov-Bohm physical system by Shelankov, and elucidated by Berry. Here we show experimentally that this force is indeed present. Nevertheless, the observation does not change the understanding that potentials can act without force-fields, as we show theoretically that Zeilingers theorem on the dispersionless nature is not generally applicable to the Aharonov-Bohm system.
Two basic ingredients in the quantum formalism are the concepts of states and observables, and apart from the average value (which includes the Born probability as a paramountly important case), the variance is the most fundamental and prominent quantity arising from the coupling of a state and an observable. Given the simplicity and ubiquity of the variance, it seems surprising that we can still exploit the variance to gain conceptually different insight into the quantum substratum. The purpose of the present work is to advocate the following idea: By inputting a general quantum state as the observable in the variance and incorporating a resolution of identity induced by coherent states, we obtain a mathematically simple and physically intuitive method for quantifying nonclassicality. We reveal its fundamental properties and highlight its connection with phase-space quantification of nonclassicality. The idea further suggests a whole family of appealing quantifiers of nonclassicality involving deep mathematical subtlety and an intriguing conjecture with physical significance. Some prototypical examples are worked out to illustrate the concept. This approach of regarding states as observables presents an opportunity for investigating quantum features from an alternative perspective of uncertainty.