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Mathematics
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Time and Time Dependence in Quantum Mechanics
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Time and Time Dependence in Quantum Mechanics

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (30 April 2019) | Viewed by 15561

Special Issue Editors

Prof. Dr. J. Gonzalo Muga

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Guest Editor
Department of Physical Chemistry, Universidad del Pais Vasco/Euskal Herriko Unibertsitatea (UPV/EHU), Apdo. 644 Bilbao, Spain
Interests: quantum technologies; shortcuts to adiabaticity; nonhermitian physics; trapped ions and cold atoms; time in quantum mechanics
Special Issues, Collections and Topics in MDPI journals
Dr. Andreas Ruschhaupt

E-Mail Website
Guest Editor
Department of Physics, University College Cork, T12 YN60 Cork, Ireland
Interests: quantum control; shortcuts to adiabaticity; quantum optics; time in quantum mechanics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Time enables us to describe changes and order events in daily life and in the laboratories. Physics takes charge of defining the official time by measuring quantum processes in time-frequency metrology and yet, paradoxically, we keep wondering about fundamental questions such as how the times of events such as the arrival of particles at a detector should be described in quantum theory, the meaning of time operators and time-energy uncertainty principles, or the emergence of irreversibility from time-symmetrical laws. Time is also the basic running parameter to narrate quantum dynamics. Time-dependent quantum phenomena are ubiquitous, often strange compared to classical counterparts, and have to be well understood, in particular to control them and develop quantum-based technologies:

Geometric phases, quantum transients, the Zeno effect, or short and long deviations from exponential decay are concepts and phenomena used to comprehend and possibly modify dynamics. Dynamical control is often implemented based on adiabatic and sudden approximations and also via shortcuts to adiabaticity. This Special Issue of Mathematics aims at offering a broad perspective of recent work to fathom time in quantum theory and harness the evolution of quantum systems.

Prof. Dr. J. Gonzalo Muga
Dr. Andreas Ruschhaupt
Guest Editors

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Keywords

  • time in quantum mechanics, tunnelling times, arrival times, times of events, energy-time uncertainty relations
  • berry phase, aharonov-anandan phase, lewis-riesenfeld phase
  • short and long time deviations from exponential decay, zeno time
  • moshinsky shutter and quantum transients
  • adiabatic and sudden approximations
  • quantum control with shortcuts to adiabaticity
  • time reversal invariance
  • time and frequency quantum metrology
  • quantum time’s arrow
  • quantum speed limits, optimal control theory for time minimization

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Published Papers (4 papers)

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Research

13 pages, 341 KiB  
Article
Tunneling Time in Attosecond Experiments and Time Operator in Quantum Mechanics
by Ossama Kullie
Mathematics 2018, 6(10), 192; https://doi.org/10.3390/math6100192 - 8 Oct 2018
Cited by 1 | Viewed by 3528
Abstract
Attosecond science is of a fundamental interest in physics. The measurement of the tunneling time in attosecond experiments, offers a fruitful opportunity to understand the role of time in quantum mechanics (QM). We discuss in this paper our tunneling time model in relation [...] Read more.
Attosecond science is of a fundamental interest in physics. The measurement of the tunneling time in attosecond experiments, offers a fruitful opportunity to understand the role of time in quantum mechanics (QM). We discuss in this paper our tunneling time model in relation to two time operator definitions introduced by Bauer and Aharonov–Bohm. We found that both definitions can be generalized to the same type of time operator. Moreover, we found that the introduction of a phenomenological parameter by Bauer to fit the experimental data is unnecessary. The issue is resolved with our tunneling model by considering the correct barrier width, which avoids a misleading interpretation of the experimental data. Our analysis shows that the use of the so-called classical barrier width, to be precise, is incorrect. Full article
(This article belongs to the Special Issue Time and Time Dependence in Quantum Mechanics)
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10 pages, 237 KiB  
Article
Dirac’s Method for the Two-Dimensional Damped Harmonic Oscillator in the Extended Phase Space
by Laure Gouba
Mathematics 2018, 6(10), 180; https://doi.org/10.3390/math6100180 - 28 Sep 2018
Cited by 2 | Viewed by 3071
Abstract
The system of a two-dimensional damped harmonic oscillator is revisited in the extended phase space. It is an old problem that has already been addressed by many authors that we present here with some fresh points of view and carry on a whole [...] Read more.
The system of a two-dimensional damped harmonic oscillator is revisited in the extended phase space. It is an old problem that has already been addressed by many authors that we present here with some fresh points of view and carry on a whole discussion. We show that the system is singular. The classical Hamiltonian is proportional to the first-class constraint. We pursue with the Dirac’s canonical quantization procedure by fixing the gauge and provide a reduced phase space description of the system. As a result, the quantum system is simply modeled by the original quantum Hamiltonian. Full article
(This article belongs to the Special Issue Time and Time Dependence in Quantum Mechanics)
14 pages, 447 KiB  
Article
Time in Quantum Mechanics and the Local Non-Conservation of the Probability Current
by Giovanni Modanese
Mathematics 2018, 6(9), 155; https://doi.org/10.3390/math6090155 - 4 Sep 2018
Cited by 14 | Viewed by 4241
Abstract
In relativistic quantum field theory with local interactions, charge is locally conserved. This implies local conservation of probability for the Dirac and Klein–Gordon wavefunctions, as special cases; and in turn for non-relativistic quantum field theory and for the Schrödinger and Ginzburg–Landau equations, regarded [...] Read more.
In relativistic quantum field theory with local interactions, charge is locally conserved. This implies local conservation of probability for the Dirac and Klein–Gordon wavefunctions, as special cases; and in turn for non-relativistic quantum field theory and for the Schrödinger and Ginzburg–Landau equations, regarded as low energy limits. Quantum mechanics, however, is wider than quantum field theory, as an effective model of reality. For instance, fractional quantum mechanics and Schrödinger equations with non-local terms have been successfully employed in several applications. The non-locality of these formalisms is strictly related to the problem of time in quantum mechanics. We explicitly compute, for continuum wave packets, the terms of the fractional Schrödinger equation and the non-local Schrödinger equation by Lenzi et al. that break local current conservation. Additionally, we discuss the physical significance of these terms. The results are especially relevant for the electromagnetic coupling of these wavefunctions. A connection with the non-local Gorkov equation for superconductors and their proximity effect is also outlined. Full article
(This article belongs to the Special Issue Time and Time Dependence in Quantum Mechanics)
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8 pages, 262 KiB  
Article
Symmetries and Invariants for Non-Hermitian Hamiltonians
by Miguel Ángel Simón, Álvaro Buendía and J. G. Muga
Mathematics 2018, 6(7), 111; https://doi.org/10.3390/math6070111 - 27 Jun 2018
Cited by 14 | Viewed by 4020
Abstract
We discuss Hamiltonian symmetries and invariants for quantum systems driven by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a unitary or antiunitary transformation AHA† that leaves the Hamiltonian H unchanged represents a symmetry of the Hamiltonian, which implies the commutativity [...] Read more.
We discuss Hamiltonian symmetries and invariants for quantum systems driven by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a unitary or antiunitary transformation AHA† that leaves the Hamiltonian H unchanged represents a symmetry of the Hamiltonian, which implies the commutativity [H,A]=0 and, if A is linear and time-independent, a conservation law, namely the invariance of expectation values of A. For non-Hermitian Hamiltonians, H† comes into play as a distinct operator that complements H in generalized unitarity relations. The above description of symmetries has to be extended to include also A-pseudohermiticity relations of the form AH=H†A. A superoperator formulation of Hamiltonian symmetries is provided and exemplified for Hamiltonians of a particle moving in one-dimension considering the set of A operators that form Klein’s 4-group: parity, time-reversal, parity&time-reversal, and unity. The link between symmetry and conservation laws is discussed and shown to be richer and subtler for non-Hermitian than for Hermitian Hamiltonians. Full article
(This article belongs to the Special Issue Time and Time Dependence in Quantum Mechanics)
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