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Uncertainty Relations for Quantum Particles

Kechrimparis, Spyridon (2015) Uncertainty Relations for Quantum Particles. PhD thesis, University of York.

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The focus of the present investigation is uncertainty relations for quantum particles, which quantify the fundamental limitations on some of their properties due to their incompatibility. In the first, longer part, we are concerned with preparational uncertainty relations, while in the second we touch upon measurement inequalities through a generalisation of a model of joint measurement of position and momentum. Specifically, starting from a triple of canonical operators, we prove product and sum inequalities for their variances with bounds larger than those following from combining the pairwise ones. We extend these results to N observables for a quantum particle and prove uncertainty relations for the sums and products of their variances in terms of the commutators. Furthermore, we present a general theory of preparational uncertainty relations for a quantum particle in one dimension and derive conditions for a smooth function of the second moments to assume a lower bound. The Robertson-Schroedinger inequality is found to be of special significance and we geometrically study the space of second moments. We prove new uncertainty relations for various functions of the variances and covariance of position and momentum of a quantum particle. Some of our findings are shown to extend to more than one spatial degree of freedom and we derive various types of inequalities. Finally, we propose a generalisation of the Arthurs-Kelly model of joint measurement of position and momentum to incorporate the case of more than two observables and derive joint-measurement inequalities for the statistics of the probes. For the case of three canonical observables and suitable definitions of error and disturbance we obtain a number of error-error-error and error-error-disturbance inequalities and show that the lower bound is identical to the preparational one.

Item Type: Thesis (PhD)
Related URLs:
Academic Units: The University of York > Mathematics (York)
Identification Number/EthosID: uk.bl.ethos.686536
Depositing User: Mr Spyridon Kechrimparis
Date Deposited: 24 May 2016 09:50
Last Modified: 08 Sep 2016 13:34
URI: http://etheses.whiterose.ac.uk/id/eprint/13222

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