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1. Oliva, Maxime. The quantum Wigner current : a geometric approach to quantum dynamics in phase space.

Degree: PhD, 2019, University of Hertfordshire

Phase space is the unity of position and momentum configuration space. It allows for an effective description of dynamical systems and is particularly useful when it comes to studying chaos theory and statistical mechanics. After the advent of quantum physics early in the 20th century, E. Wigner [91], J. E. Moyal [62] and H. J. Groenewold [31] introduce a quantum theory in phase space. Despite the apparent added complexity of the mathematics involved in this new framework, the underlying classical and quantum equations show many similarities. The probability distribution in classical physics becomes the Wigner distribution, a probability distribution usually featuring negative values. In 2013, O. Steuernagel and D. Kakofengitis, inspired by the work of H. Bauke [7] and E. Wigner [91], identified the quantum analogue of the classical phase space flow: the Wigner current J [83]. This Wigner current allows the visualisation of quantum dynamics through a quantum fluid dynamics perspective in phase space. This thesis is written by collection of five articles. They are prefaced by an introduction into the basics of quantum phase space theory and its link with both classical phase space dynamics and the standard Schrödinger approach, followed by the articles published during this PhD. Article 1 shows the importance of the integral form of the Wigner current. We use it to derive the Ehrenfest’s theorem, as well as to refute some propositions made within the community. Article 2 shows that, using the Wigner current, an Eulerian and Lagrangian point of view do not always give the same results for the quantum case. We demonstrate that the negativities of the Wigner distribution, sign of quantumness of the system, are created by the Wigner velocity field singularities. The Wigner velocity field is the quantum analogue of the classical phase space velocity field. In Article 3, we see that even though Wigner distributions of quantum systems feature spotty structures which saturate on scales ɑZ [97], the construction of a superoscillating Wigner distribution allows one to generate much smaller structures, of the order of ɑZ /α with α a positive constant potentially very large. In Article 4, we introduce the concept of quantum shear suppression in phase space. The Wigner current features an effective quantum “viscosity”, suppressing classical dynamics fine details. This viscosity is the mechanism by which the Zurek scale is enforced dynamically onto the state in phase space. In Article 5, we apply the previous ideas to Kerr-type oscillators. Its Wigner current is derived, and using it we show that its values are conserved on a ring during the time evolution of the Kerr oscillator. The shear suppression is also studied.

Subjects/Keywords: Quantum; phase; space; theory; theoretical; physics; superoscillations; schrodinger; wigner; function; distribution; negative; probability; viscosity; fluid; mechanics; stagnation; topology; integral; moyal; groenewold; oliva; steuernagel; kakofengitis; feynman; Zurek; Planck; Structures; Scale; trajectory

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Oliva, M. (2019). The quantum Wigner current : a geometric approach to quantum dynamics in phase space. (Doctoral Dissertation). University of Hertfordshire. Retrieved from http://hdl.handle.net/2299/22363

Chicago Manual of Style (16th Edition):

Oliva, Maxime. “The quantum Wigner current : a geometric approach to quantum dynamics in phase space.” 2019. Doctoral Dissertation, University of Hertfordshire. Accessed January 26, 2021. http://hdl.handle.net/2299/22363.

MLA Handbook (7th Edition):

Oliva, Maxime. “The quantum Wigner current : a geometric approach to quantum dynamics in phase space.” 2019. Web. 26 Jan 2021.

Vancouver:

Oliva M. The quantum Wigner current : a geometric approach to quantum dynamics in phase space. [Internet] [Doctoral dissertation]. University of Hertfordshire; 2019. [cited 2021 Jan 26]. Available from: http://hdl.handle.net/2299/22363.

Council of Science Editors:

Oliva M. The quantum Wigner current : a geometric approach to quantum dynamics in phase space. [Doctoral Dissertation]. University of Hertfordshire; 2019. Available from: http://hdl.handle.net/2299/22363

2. Oliva, Maxime. The quantum Wigner current : a geometric approach to quantum dynamics in phase space.

Degree: PhD, 2019, University of Hertfordshire

Phase space is the unity of position and momentum configuration space. It allows for an effective description of dynamical systems and is particularly useful when it comes to studying chaos theory and statistical mechanics. After the advent of quantum physics early in the 20th century, E. Wigner [91], J. E. Moyal [62] and H. J. Groenewold [31] introduce a quantum theory in phase space. Despite the apparent added complexity of the mathematics involved in this new framework, the underlying classical and quantum equations show many similarities. The probability distribution in classical physics becomes the Wigner distribution, a probability distribution usually featuring negative values. In 2013, O. Steuernagel and D. Kakofengitis, inspired by the work of H. Bauke [7] and E. Wigner [91], identified the quantum analogue of the classical phase space flow: the Wigner current J [83]. This Wigner current allows the visualisation of quantum dynamics through a quantum fluid dynamics perspective in phase space. This thesis is written by collection of five articles. They are prefaced by an introduction into the basics of quantum phase space theory and its link with both classical phase space dynamics and the standard Schrödinger approach, followed by the articles published during this PhD. Article 1 shows the importance of the integral form of the Wigner current. We use it to derive the Ehrenfest’s theorem, as well as to refute some propositions made within the community. Article 2 shows that, using the Wigner current, an Eulerian and Lagrangian point of view do not always give the same results for the quantum case. We demonstrate that the negativities of the Wigner distribution, sign of quantumness of the system, are created by the Wigner velocity field singularities. The Wigner velocity field is the quantum analogue of the classical phase space velocity field. In Article 3, we see that even though Wigner distributions of quantum systems feature spotty structures which saturate on scales ɑZ [97], the construction of a superoscillating Wigner distribution allows one to generate much smaller structures, of the order of ɑZ /α with α a positive constant potentially very large. In Article 4, we introduce the concept of quantum shear suppression in phase space. The Wigner current features an effective quantum “viscosity”, suppressing classical dynamics fine details. This viscosity is the mechanism by which the Zurek scale is enforced dynamically onto the state in phase space. In Article 5, we apply the previous ideas to Kerr-type oscillators. Its Wigner current is derived, and using it we show that its values are conserved on a ring during the time evolution of the Kerr oscillator. The shear suppression is also studied.

Subjects/Keywords: Quantum; phase; space; theory; theoretical; physics; superoscillations; schrodinger; wigner; function; distribution; negative; probability; viscosity; fluid; mechanics; stagnation; topology; integral; moyal; groenewold; oliva; steuernagel; kakofengitis; feynman; Zurek; Planck; Structures; Scale; trajectory

Record DetailsSimilar RecordsGoogle PlusoneFacebookTwitterCiteULikeMendeleyreddit

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Oliva, M. (2019). The quantum Wigner current : a geometric approach to quantum dynamics in phase space. (Doctoral Dissertation). University of Hertfordshire. Retrieved from https://doi.org/10.18745/th.22363 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.802947

Chicago Manual of Style (16th Edition):

Oliva, Maxime. “The quantum Wigner current : a geometric approach to quantum dynamics in phase space.” 2019. Doctoral Dissertation, University of Hertfordshire. Accessed January 26, 2021. https://doi.org/10.18745/th.22363 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.802947.

MLA Handbook (7th Edition):

Oliva, Maxime. “The quantum Wigner current : a geometric approach to quantum dynamics in phase space.” 2019. Web. 26 Jan 2021.

Vancouver:

Oliva M. The quantum Wigner current : a geometric approach to quantum dynamics in phase space. [Internet] [Doctoral dissertation]. University of Hertfordshire; 2019. [cited 2021 Jan 26]. Available from: https://doi.org/10.18745/th.22363 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.802947.

Council of Science Editors:

Oliva M. The quantum Wigner current : a geometric approach to quantum dynamics in phase space. [Doctoral Dissertation]. University of Hertfordshire; 2019. Available from: https://doi.org/10.18745/th.22363 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.802947

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