Semiclassical approximations in infinitely connected spaces

Hochschulschrift

Zusammenfassung: The aim of this dissertation is to derive, from first principles, semiclassical approximations for quantum mechanical systems whose configuration space is infinitely connected. The emergence of such spaces is intimately related to magnetic forces which are classically inaccessible to the system under consideration, but which nevertheless influence the latter’s quantum mechanical behavior. While a comprehensible analysis of the corresponding exact quantum mechanical solutions is often obstructed by their complicated form, essential features of the dynamics collapse into discontinuities in Gutzwiller’s semiclassical propagator. The intention is thus to determine the origin of these discontinuities, in anticipation of an accessible semiclassical picture.The first system under consideration is a charged particle propagating in the presence of an Aharonov-Bohm magnetic flux line. The infinite connectedness of the problem arises as a consequence of the discontinuous dependence of Dirac’s magnetic phase factor on the number of windings of any closed curve encircling the flux. A novel semiclassical limit is introduced for the propagator of the otherwise free particle. While this limit directly produces the semiclassical approximation of Gutzwiller in backward propagation direction, in forward direction it describes two half-waves which acquire Dirac’s magnetic phase as if passing above and below the flux line, respectively. This splitting of the wave mediates a smooth transition of the propagator’s phase from negative to positive values, and gives rise to an interference pattern which is entirely determined by the contribution to Hamilton’s principal function associated with the angular motion of the particle with respect to the flux line. In a subsequent semiclassical limit, in which Planck’s constant is neglected against any nonzero such angular contribution, this smooth transition is no longer resolved and Gutzwiller’s expression is obtained also in forward propagationdirection, where its phase is discontinuous.In order to describe further potentials besides the flux, our thus-obtained approximation is amended accordingly and iterated as in a path-integral approach, followed by an evaluation by the method of stationary phase. To this end, the applicability of a method existing for this purpose for Cartesian problems is extended to accommodate, for the first time, also path-integral expressions parametrized by generalized coordinates.The second setting considered here emerges from a semiclassical description of identical particles, devised so as to circumvent the symmetrization postulate. The description results from the application of Gutzwiller’s propagator in combination with a concept due to Leinaas and Myrheim, identifying those points of the Cartesian product space which, as a consequence of the particles’ indistinguishability, correspond to identical configurations. Thereby, all contributions expected from the symmetrization postulate arise as the solutions of a single set of classical equations of motion and boundary conditions, and the required exchange phases are induced by additional potentials.While specific choices for these potentials lead to bosonic or fermionic behavior, in which case the semiclassical propagator agrees with the exact quantum mechanical result if no further interaction is considered, other possibilities exist where the approach provides, at any rate, merely an approximate description. This is the case in particular for particles constrained to propagate in a two-dimensional plane. Exclusion of those points of configuration space where any two such particles coincide then renders the space of relative motion infinitely connected, which causes the exchange phase to be no longer restricted to only the bosonic or to the fermionic case. This phase may then assume any value on the complex unit circle. The particles are, consequently, called anyons, which have been shown to emerge as quasi-particles in two-dimensional electron gases subject to magnetic fields. Our newly-developed semiclassical limit is applied to the full quantum mechanical propagator of the two-anyon system, after which, once again, a further semiclassicalapproximation transforms a split-wave expression into Gutzwiller’s propagator. Thereby, the physical origin of the latter’s discontinuity is revealed, which here occurs for propagation processes where the particles, classically, exchange their position by passing directly through one another

Angaben zum Objekt

• Standort: Deutsche Nationalbibliothek Frankfurt am Main
• Umfang: Online-Ressource
• Sprache: Englisch
• Anmerkungen: Albert-Ludwigs-Universität Freiburg, Dissertation, 2015

Klassifikation und Themen

• Klassifikation: Physik

Beteiligte, Orts- und Zeitangaben

• Ereignis: Veröffentlichung
• (wo): Freiburg
• (wer): Universität
• (wann): 2016
• Urheber: Fischer, Stefan Georg
• Beteiligte Personen und Organisationen: Buchleitner, Andreas

Weitere Informationen

• DOI: 10.6094/UNIFR/10459
• URN: urn:nbn:de:bsz:25-freidok-104590
• Rechteinformation: Der Zugriff auf das Objekt ist unbeschränkt möglich.
• Letzte Aktualisierung: 25.03.2025, 13:55 MEZ