1
|
Valtierra IF, Klimov AB. Tunneling Currents in the Hyperbolic Phase Space. ENTROPY (BASEL, SWITZERLAND) 2024; 26:639. [PMID: 39202110 PMCID: PMC11353601 DOI: 10.3390/e26080639] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 06/28/2024] [Revised: 07/23/2024] [Accepted: 07/25/2024] [Indexed: 09/03/2024]
Abstract
We introduce the quantum currents for quantum systems with an SU(1,1) dynamic symmetry group whose evolution is governed by a non-linear Hamiltonian possessing a continuous spectrum and apply them to the analysis of the tunneling dynamics on the hyperbolic phase space.
Collapse
|
2
|
Dodin A, Willard AP. State space distribution and dynamical flow for closed and open quantum systems. J Chem Phys 2019. [DOI: 10.1063/1.5100736] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/08/2023] Open
Affiliation(s)
- Amro Dodin
- Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Adam P. Willard
- Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| |
Collapse
|
3
|
Mocz P, Lancaster L, Fialkov A, Becerra F, Chavanis PH. Schrödinger-Poisson–Vlasov-Poisson correspondence. Int J Clin Exp Med 2018. [DOI: 10.1103/physrevd.97.083519] [Citation(s) in RCA: 74] [Impact Index Per Article: 12.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
|
4
|
Kopp M, Vattis K, Skordis C. Solving the Vlasov equation in two spatial dimensions with the Schrödinger method. Int J Clin Exp Med 2017. [DOI: 10.1103/physrevd.96.123532] [Citation(s) in RCA: 28] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
|
5
|
Steuernagel O, Kakofengitis D, Ritter G. Wigner flow reveals topological order in quantum phase space dynamics. PHYSICAL REVIEW LETTERS 2013; 110:030401. [PMID: 23373905 DOI: 10.1103/physrevlett.110.030401] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/12/2012] [Revised: 10/31/2012] [Indexed: 06/01/2023]
Abstract
The behavior of classical mechanical systems is characterized by their phase portraits, the collections of their trajectories. Heisenberg's uncertainty principle precludes the existence of sharply defined trajectories, which is why traditionally only the time evolution of wave functions is studied in quantum dynamics. These studies are quite insensitive to the underlying structure of quantum phase space dynamics. We identify the flow that is the quantum analog of classical particle flow along phase portrait lines. It reveals hidden features of quantum dynamics and extra complexity. Being constrained by conserved flow winding numbers, it also reveals fundamental topological order in quantum dynamics that has so far gone unnoticed.
Collapse
Affiliation(s)
- Ole Steuernagel
- School of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield, United Kingdom.
| | | | | |
Collapse
|
6
|
Liu J. Two more approaches for generating trajectory-based dynamics which conserves the canonical distribution in the phase space formulation of quantum mechanics. J Chem Phys 2011; 134:194110. [DOI: 10.1063/1.3589406] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
|
7
|
Liu J, Miller WH. An approach for generating trajectory-based dynamics which conserves the canonical distribution in the phase space formulation of quantum mechanics. I. Theories. J Chem Phys 2011; 134:104101. [DOI: 10.1063/1.3555273] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
8
|
Efthymiopoulos C, Kalapotharakos C, Contopoulos G. Origin of chaos near critical points of quantum flow. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:036203. [PMID: 19392031 DOI: 10.1103/physreve.79.036203] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/20/2008] [Indexed: 05/27/2023]
Abstract
The general theory of motion in the vicinity of a moving quantum nodal point (vortex) is studied in the framework of the de Broglie-Bohm trajectory method of quantum mechanics. Using an adiabatic approximation, we find that near any nodal point of an arbitrary wave function psi there is an unstable point (called the X point) in a frame of reference moving with the nodal point. The local phase portrait forms always a characteristic pattern called the "nodal-point- X -point complex." We find general formulas for this complex as well as necessary and sufficient conditions of validity of the adiabatic approximation. We demonstrate that chaos emerges from the consecutive scattering events of the orbits with nodal-point- X -point complexes. The scattering events are of two types (called type I and type II). A theoretical model is constructed yielding the local value of the Lyapunov characteristic numbers in scattering events of both types. The local Lyapunov characteristic number scales as an inverse power of the speed of the nodal point in the rest frame, implying that it scales proportionally to the size of the nodal-point- X -point complex. It is also an inverse power of the distance of a trajectory from the X point's stable manifold far from the complex. This distance plays the role of an effective "impact parameter." The results of detailed numerical experiments with different wave functions, possessing one, two, or three moving nodal points, are reported. Examples are given of regular and chaotic trajectories, and the statistics of the Lyapunov characteristic numbers of the orbits are found and compared to the number of encounter events of each orbit with the nodal-point- X -point complexes. The numerical results are in agreement with the theory, and various phenomena appearing at first as counterintuitive find a straightforward explanation.
Collapse
Affiliation(s)
- C Efthymiopoulos
- Research Center for Astronomy and Applied Mathematics, Academy of Athens, Soranou Efesiou 4, GR-115 27 Athens, Greece.
| | | | | |
Collapse
|
9
|
López H, Martens CC, Donoso A. Entangled trajectory dynamics in the Husimi representation. J Chem Phys 2006; 125:154111. [PMID: 17059243 DOI: 10.1063/1.2222368] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/28/2022] Open
Abstract
We solve quantum dynamical equations of simple systems by propagating ensembles of interacting trajectories. A scheme is proposed which uses adaptive kernel density estimation for representing probability distribution functions and their derivatives. The formulation is carried on in the Husimi representation to ensure the positiveness of the distribution functions. By comparing to previous work, the effect of changing representations is studied as well as the advantage of using adaptive kernels for the estimation of probability distributions. We found significant improvement in the accuracy of the results.
Collapse
Affiliation(s)
- Hender López
- Laboratorio de Física Estadística de Sistemas Desordenados, Centro de Física, Instituto Venezolano de Investigaciones Científicas, IVIC, Caracas 1020A, Venezuela
| | | | | |
Collapse
|
10
|
Gonzalez J, Bofill JM, Gimenez X. A Bohmian total potential view to quantum effects. I. Methodology and simple model systems. J Chem Phys 2004; 120:10961-71. [PMID: 15268126 DOI: 10.1063/1.1747869] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The coherent-state wave packet dynamics of several model systems is analyzed in terms of Bohm's total potential. The quantum dynamics has been obtained by solving the time-dependent Schrodinger equation, and a method for obtaining the total potential from it, involving just matrix algebra, has been proposed. Contrary to what one may expect, it is shown that the time- and state-dependent features of the total potential admit a rationale, classical-like description of quantum effects, leading to a unified picture of them, which is not critically dependent, as for the key features, on the classical potential. An outstanding feature is found to be the relation of the state system's density amplitude and sharpness (in its dependence with position) with quantum effects. Sharp density profiles and low densities cause the total potential to strongly depart from the classical value, in both time regimes and position ranges, which provide a clearer, more deterministic view to quantum dynamics. Free motion as well as scattering processes by square and Eckart barriers have been analyzed by means of careful inspection of several time dependent snapshots. The result is an insightful picture of processes involving tunneling and antitunneling, including their dynamical variants, as well as resonances and quantization.
Collapse
Affiliation(s)
- Javier Gonzalez
- Departament de Quimica Fisica, Universitat de Barcelona, Marti i Franques 1Centre Especial de Recerca en Quimica Teorica, Parc Cientific de Barcelona, Baldiri Reixac 10-12, 08028 Barcelona, Spain
| | | | | |
Collapse
|
11
|
Chemical Reactivity Dynamics and Quantum Chaos in Highly Excited Hydrogen Atoms in an External Field: A Quantum Potential Approach. Int J Mol Sci 2002. [DOI: 10.3390/i3040338] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022] Open
|
12
|
Widrow LM. Modeling collisionless matter in general relativity: A new numerical technique. Int J Clin Exp Med 1997. [DOI: 10.1103/physrevd.55.5997] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
|
13
|
Mo/ller KB, Jo/rgensen TG, Torres-Vega G. On coherent-state representations of quantum mechanics: Wave mechanics in phase space. J Chem Phys 1997. [DOI: 10.1063/1.473684] [Citation(s) in RCA: 36] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
14
|
Hu XG, Li QS, Tang AC. Exact treatment of reactive scattering in the Torres-Vega-Frederick quantum phase-space representation. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1995; 52:3780-3789. [PMID: 9912684 DOI: 10.1103/physreva.52.3780] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
15
|
Hu XG, Li QS, Tang AC. Inelastic atom-diatom scattering in quantum phase space representation. Chem Phys Lett 1994. [DOI: 10.1016/0009-2614(94)01159-1] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
|
16
|
Lee HW. Generalized antinormally ordered quantum phase-space distribution functions. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1994; 50:2746-2749. [PMID: 9911196 DOI: 10.1103/physreva.50.2746] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
17
|
Shin JY, Lee HW. Floquet analysis of quantum resonance in a driven nonlinear system. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:902-909. [PMID: 9962050 DOI: 10.1103/physreve.50.902] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
18
|
Torres‐Vega G, Frederick JH. A quantum mechanical representation in phase space. J Chem Phys 1993. [DOI: 10.1063/1.464085] [Citation(s) in RCA: 75] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
|
19
|
Martens CC. Coriolis interaction in polyatomic molecules: A quantum qualitative approach to a two mode model system. J Chem Phys 1992. [DOI: 10.1063/1.462088] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
20
|
Müller L, Stolze J, Leschke H, Nagel P. Classical and quantum phase-space behavior of a spin-boson system. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 44:1022-1033. [PMID: 9906052 DOI: 10.1103/physreva.44.1022] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
21
|
Torres‐Vega G, Frederick JH. Quantum mechanics in phase space: New approaches to the correspondence principle. J Chem Phys 1990. [DOI: 10.1063/1.459225] [Citation(s) in RCA: 88] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
22
|
Lin WA, Ballentine LE. Quantum tunneling and chaos in a driven anharmonic oscillator. PHYSICAL REVIEW LETTERS 1990; 65:2927-2930. [PMID: 10042735 DOI: 10.1103/physrevlett.65.2927] [Citation(s) in RCA: 80] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
|
23
|
Skodje RT, Spina A. Gaussian wave packets as probes of the destabilization of phase-space structure in the quantum standard map. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 42:6252-6255. [PMID: 9903915 DOI: 10.1103/physreva.42.6252] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|