{"title":"从经典的 Frenet-Serret 装置到量子力学演化的曲率和扭转。第二部分.非稳态哈密顿","authors":"Paul M. Alsing, Carlo Cafaro","doi":"10.1142/s0219887824501512","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we present a geometric perspective on how to quantify the bending and the twisting of quantum curves traced by state vectors evolving under nonstationary Hamiltonians. Specifically, relying on the existing geometric viewpoint for stationary Hamiltonians, we discuss the generalization of our theoretical construct to time-dependent quantum-mechanical scenarios where both time-varying curvature and torsion coefficients play a key role. Specifically, we present a quantum version of the Frenet–Serret apparatus for a quantum trajectory in projective Hilbert space traced out by a parallel-transported pure quantum state evolving unitarily under a time-dependent Hamiltonian specifying the Schrödinger evolution equation. The time-varying curvature coefficient is specified by the magnitude squared of the covariant derivative of the tangent vector <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mo>|</mo><mi>T</mi><mo stretchy=\"false\">〉</mo></math></span><span></span> to the state vector <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mo>|</mo><mi mathvariant=\"normal\">Ψ</mi><mo stretchy=\"false\">〉</mo></math></span><span></span> and measures the bending of the quantum curve. The time-varying torsion coefficient, instead, is given by the magnitude squared of the projection of the covariant derivative of the tangent vector <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mo>|</mo><mi>T</mi><mo stretchy=\"false\">〉</mo></math></span><span></span> to the state vector <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mo>|</mo><mi mathvariant=\"normal\">Ψ</mi><mo stretchy=\"false\">〉</mo></math></span><span></span>, orthogonal to <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mo>|</mo><mi>T</mi><mo stretchy=\"false\">〉</mo></math></span><span></span> and <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mo>|</mo><mi mathvariant=\"normal\">Ψ</mi><mo stretchy=\"false\">〉</mo></math></span><span></span> and, in addition, measures the twisting of the quantum curve. We find that the time-varying setting exhibits a richer structure from a statistical standpoint. For instance, unlike the time-independent configuration, we find that the notion of generalized variance enters nontrivially in the definition of the torsion of a curve traced out by a quantum state evolving under a nonstationary Hamiltonian. To physically illustrate the significance of our construct, we apply it to an exactly soluble time-dependent two-state Rabi problem specified by a sinusoidal oscillating time-dependent potential. In this context, we show that the analytical expressions for the curvature and torsion coefficients are completely described by only two real three-dimensional vectors, the Bloch vector that specifies the quantum system and the externally applied time-varying magnetic field. Although we show that the torsion is identically zero for an arbitrary time-dependent single-qubit Hamiltonian evolution, we study the temporal behavior of the curvature coefficient in different dynamical scenarios, including off-resonance and on-resonance regimes and, in addition, strong and weak driving configurations. While our formalism applies to pure quantum states in arbitrary dimensions, the analytic derivation of associated curvatures and orbit simulations can become quite involved as the dimension increases. Thus, finally we briefly comment on the possibility of applying our geometric formalism to higher-dimensional qudit systems that evolve unitarily under a general nonstationary Hamiltonian.</p>","PeriodicalId":50320,"journal":{"name":"International Journal of Geometric Methods in Modern Physics","volume":"14 1","pages":""},"PeriodicalIF":2.1000,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"From the classical Frenet–Serret apparatus to the curvature and torsion of quantum-mechanical evolutions. Part II. Nonstationary Hamiltonians\",\"authors\":\"Paul M. Alsing, Carlo Cafaro\",\"doi\":\"10.1142/s0219887824501512\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper, we present a geometric perspective on how to quantify the bending and the twisting of quantum curves traced by state vectors evolving under nonstationary Hamiltonians. Specifically, relying on the existing geometric viewpoint for stationary Hamiltonians, we discuss the generalization of our theoretical construct to time-dependent quantum-mechanical scenarios where both time-varying curvature and torsion coefficients play a key role. Specifically, we present a quantum version of the Frenet–Serret apparatus for a quantum trajectory in projective Hilbert space traced out by a parallel-transported pure quantum state evolving unitarily under a time-dependent Hamiltonian specifying the Schrödinger evolution equation. The time-varying curvature coefficient is specified by the magnitude squared of the covariant derivative of the tangent vector <span><math altimg=\\\"eq-00001.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mo>|</mo><mi>T</mi><mo stretchy=\\\"false\\\">〉</mo></math></span><span></span> to the state vector <span><math altimg=\\\"eq-00002.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mo>|</mo><mi mathvariant=\\\"normal\\\">Ψ</mi><mo stretchy=\\\"false\\\">〉</mo></math></span><span></span> and measures the bending of the quantum curve. The time-varying torsion coefficient, instead, is given by the magnitude squared of the projection of the covariant derivative of the tangent vector <span><math altimg=\\\"eq-00003.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mo>|</mo><mi>T</mi><mo stretchy=\\\"false\\\">〉</mo></math></span><span></span> to the state vector <span><math altimg=\\\"eq-00004.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mo>|</mo><mi mathvariant=\\\"normal\\\">Ψ</mi><mo stretchy=\\\"false\\\">〉</mo></math></span><span></span>, orthogonal to <span><math altimg=\\\"eq-00005.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mo>|</mo><mi>T</mi><mo stretchy=\\\"false\\\">〉</mo></math></span><span></span> and <span><math altimg=\\\"eq-00006.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mo>|</mo><mi mathvariant=\\\"normal\\\">Ψ</mi><mo stretchy=\\\"false\\\">〉</mo></math></span><span></span> and, in addition, measures the twisting of the quantum curve. We find that the time-varying setting exhibits a richer structure from a statistical standpoint. For instance, unlike the time-independent configuration, we find that the notion of generalized variance enters nontrivially in the definition of the torsion of a curve traced out by a quantum state evolving under a nonstationary Hamiltonian. To physically illustrate the significance of our construct, we apply it to an exactly soluble time-dependent two-state Rabi problem specified by a sinusoidal oscillating time-dependent potential. In this context, we show that the analytical expressions for the curvature and torsion coefficients are completely described by only two real three-dimensional vectors, the Bloch vector that specifies the quantum system and the externally applied time-varying magnetic field. Although we show that the torsion is identically zero for an arbitrary time-dependent single-qubit Hamiltonian evolution, we study the temporal behavior of the curvature coefficient in different dynamical scenarios, including off-resonance and on-resonance regimes and, in addition, strong and weak driving configurations. While our formalism applies to pure quantum states in arbitrary dimensions, the analytic derivation of associated curvatures and orbit simulations can become quite involved as the dimension increases. Thus, finally we briefly comment on the possibility of applying our geometric formalism to higher-dimensional qudit systems that evolve unitarily under a general nonstationary Hamiltonian.</p>\",\"PeriodicalId\":50320,\"journal\":{\"name\":\"International Journal of Geometric Methods in Modern Physics\",\"volume\":\"14 1\",\"pages\":\"\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2024-03-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"International Journal of Geometric Methods in Modern Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1142/s0219887824501512\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Geometric Methods in Modern Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1142/s0219887824501512","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
From the classical Frenet–Serret apparatus to the curvature and torsion of quantum-mechanical evolutions. Part II. Nonstationary Hamiltonians
In this paper, we present a geometric perspective on how to quantify the bending and the twisting of quantum curves traced by state vectors evolving under nonstationary Hamiltonians. Specifically, relying on the existing geometric viewpoint for stationary Hamiltonians, we discuss the generalization of our theoretical construct to time-dependent quantum-mechanical scenarios where both time-varying curvature and torsion coefficients play a key role. Specifically, we present a quantum version of the Frenet–Serret apparatus for a quantum trajectory in projective Hilbert space traced out by a parallel-transported pure quantum state evolving unitarily under a time-dependent Hamiltonian specifying the Schrödinger evolution equation. The time-varying curvature coefficient is specified by the magnitude squared of the covariant derivative of the tangent vector to the state vector and measures the bending of the quantum curve. The time-varying torsion coefficient, instead, is given by the magnitude squared of the projection of the covariant derivative of the tangent vector to the state vector , orthogonal to and and, in addition, measures the twisting of the quantum curve. We find that the time-varying setting exhibits a richer structure from a statistical standpoint. For instance, unlike the time-independent configuration, we find that the notion of generalized variance enters nontrivially in the definition of the torsion of a curve traced out by a quantum state evolving under a nonstationary Hamiltonian. To physically illustrate the significance of our construct, we apply it to an exactly soluble time-dependent two-state Rabi problem specified by a sinusoidal oscillating time-dependent potential. In this context, we show that the analytical expressions for the curvature and torsion coefficients are completely described by only two real three-dimensional vectors, the Bloch vector that specifies the quantum system and the externally applied time-varying magnetic field. Although we show that the torsion is identically zero for an arbitrary time-dependent single-qubit Hamiltonian evolution, we study the temporal behavior of the curvature coefficient in different dynamical scenarios, including off-resonance and on-resonance regimes and, in addition, strong and weak driving configurations. While our formalism applies to pure quantum states in arbitrary dimensions, the analytic derivation of associated curvatures and orbit simulations can become quite involved as the dimension increases. Thus, finally we briefly comment on the possibility of applying our geometric formalism to higher-dimensional qudit systems that evolve unitarily under a general nonstationary Hamiltonian.
期刊介绍:
This journal publishes short communications, research and review articles devoted to all applications of geometric methods (including commutative and non-commutative Differential Geometry, Riemannian Geometry, Finsler Geometry, Complex Geometry, Lie Groups and Lie Algebras, Bundle Theory, Homology an Cohomology, Algebraic Geometry, Global Analysis, Category Theory, Operator Algebra and Topology) in all fields of Mathematical and Theoretical Physics, including in particular: Classical Mechanics (Lagrangian, Hamiltonian, Poisson formulations); Quantum Mechanics (also semi-classical approximations); Hamiltonian Systems of ODE''s and PDE''s and Integrability; Variational Structures of Physics and Conservation Laws; Thermodynamics of Systems and Continua (also Quantum Thermodynamics and Statistical Physics); General Relativity and other Geometric Theories of Gravitation; geometric models for Particle Physics; Supergravity and Supersymmetric Field Theories; Classical and Quantum Field Theory (also quantization over curved backgrounds); Gauge Theories; Topological Field Theories; Strings, Branes and Extended Objects Theory; Holography; Quantum Gravity, Loop Quantum Gravity and Quantum Cosmology; applications of Quantum Groups; Quantum Computation; Control Theory; Geometry of Chaos.