{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/89002"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/89002","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"The bound states of Dirac equation with a scalar potential","abstract":"\"We study the bound states of the 1+1 dimensional Dirac equation with a scalar potential, which can also be interpreted as a position dependent \"\"mass'', analytically as well as numerically. We derive a Prüfer-like representation for the Dirac equation, which can be used to derive a condition for the existence of bound states in terms of the fixed point of the nonlinear Prüfer equation for the angle variable. Another condition was derived by interpreting the Dirac equation as a Hamiltonian flow on the 2-dimensional Euclidean space and a shooting argument for the induced flow on the space of its Lagrangian planes following a similar calculation by Jones (Ergodic Theor Dyn Syst, 8 (1988) 119-138). The two conditions are shown to be equivalent, and used to compute the bound states analytically and numerically, as well as to derive a Calogero-like upper bound on the number of bound states. The analytic computations are also compared to the bound states computed using techniques from supersymmetric quantum mechanics.\"","abstract_html":""We study the bound states of the 1+1 dimensional Dirac equation with a scalar potential, which can also be interpreted as a position dependent ""mass'', analytically as well as numerically. We derive a Prüfer-like representation for the Dirac equation, which can be used to derive a condition for the existence of bound states in terms of the fixed point of the nonlinear Prüfer equation for the angle variable. Another condition was derived by interpreting the Dirac equation as a Hamiltonian flow on the 2-dimensional Euclidean space and a shooting argument for the induced flow on the space of its Lagrangian planes following a similar calculation by Jones (Ergodic Theor Dyn Syst, 8 (1988) 119-138). The two conditions are shown to be equivalent, and used to compute the bound states analytically and numerically, as well as to derive a Calogero-like upper bound on the number of bound states. The analytic computations are also compared to the bound states computed using techniques from supersymmetric quantum mechanics."","abstract_has_math":false,"creators":["Dwivedi, Vatsal"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"M.S.","degree_level":"Thesis","degree_discipline":"Applied Mathematics","degree_department":null,"school":null,"contributors":["Bronski, Jared"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016-03-02T19:33:48Z","date_published":"2016-03-02T19:33:48Z","updated_at":"2026-07-22T22:26:32Z","subjects":["Dirac equation","Bound states","Dynamical systems"],"languages":["en"],"rights":["Copyright 2015 Vatsal Dwivedi"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/89002","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Bronski, Jared"]},{"key":"dc:creator","label":"Author","values":["Dwivedi, Vatsal"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2016-03-02T19:33:48Z","2015-12-10","2015-12"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Applied Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["M.S."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Dirac equation","Bound states","Dynamical systems"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2015 Vatsal Dwivedi"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/89002"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"We study the bound states of the 1+1 dimensional Dirac equation with a scalar potential, which can also be interpreted as a position dependent \"\"mass'', analytically as well as numerically. We derive a Prüfer-like representation for the Dirac equation, which can be used to derive a condition for the existence of bound states in terms of the fixed point of the nonlinear Prüfer equation for the angle variable. Another condition was derived by interpreting the Dirac equation as a Hamiltonian flow on the 2-dimensional Euclidean space and a shooting argument for the induced flow on the space of its Lagrangian planes following a similar calculation by Jones (Ergodic Theor Dyn Syst, 8 (1988) 119-138). The two conditions are shown to be equivalent, and used to compute the bound states analytically and numerically, as well as to derive a Calogero-like upper bound on the number of bound states. The analytic computations are also compared to the bound states computed using techniques from supersymmetric quantum mechanics.\"","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2016-03-02 without embargo terms","The student, Vatsal Dwivedi, accepted the attached license on 2015-12-10 at 14:02.","The student, Vatsal Dwivedi, submitted this Thesis for approval on 2015-12-10 at 14:21.","This Thesis was approved for publication on 2015-12-10 at 15:05.","DSpace SAF Submission Ingestion Package generated from Vireo submission #8821 on 2016-03-02 at 12:50:19","Made available in DSpace on 2016-03-02T19:33:48Z (GMT). No. of bitstreams: 2 DWIVEDI-THESIS-2015.pdf: 1164763 bytes, checksum: 092f50ebd23ae935ac888c59b4f9056b (MD5) LICENSE.txt: 4211 bytes, checksum: 93d1fd824040131bd09c4f740eeac7d2 (MD5) Previous issue date: 2015-12-10"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["The bound states of Dirac equation with a scalar potential"]}]}],"canonical_facts":{"dc:contributor":["Bronski, Jared"],"dc:creator":["Dwivedi, Vatsal"],"dc:date":["2016-03-02T19:33:48Z","2015-12-10","2015-12"],"dc:description":["\"We study the bound states of the 1+1 dimensional Dirac equation with a scalar potential, which can also be interpreted as a position dependent \"\"mass'', analytically as well as numerically. We derive a Prüfer-like representation for the Dirac equation, which can be used to derive a condition for the existence of bound states in terms of the fixed point of the nonlinear Prüfer equation for the angle variable. Another condition was derived by interpreting the Dirac equation as a Hamiltonian flow on the 2-dimensional Euclidean space and a shooting argument for the induced flow on the space of its Lagrangian planes following a similar calculation by Jones (Ergodic Theor Dyn Syst, 8 (1988) 119-138). The two conditions are shown to be equivalent, and used to compute the bound states analytically and numerically, as well as to derive a Calogero-like upper bound on the number of bound states. The analytic computations are also compared to the bound states computed using techniques from supersymmetric quantum mechanics.\"","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2016-03-02 without embargo terms","The student, Vatsal Dwivedi, accepted the attached license on 2015-12-10 at 14:02.","The student, Vatsal Dwivedi, submitted this Thesis for approval on 2015-12-10 at 14:21.","This Thesis was approved for publication on 2015-12-10 at 15:05.","DSpace SAF Submission Ingestion Package generated from Vireo submission #8821 on 2016-03-02 at 12:50:19","Made available in DSpace on 2016-03-02T19:33:48Z (GMT). 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