614 research outputs found
The quantum Arnold transformation
By a quantum version of the Arnold transformation of classical mechanics, all
quantum dynamical systems whose classical equations of motion are
non-homogeneous linear second-order ordinary differential equations, including
systems with friction linear in velocity, can be related to the quantum
free-particle dynamical system. This transformation provides a basic
(Heisenberg-Weyl) algebra of quantum operators, along with well-defined
Hermitian operators which can be chosen as evolution-like observables and
complete the entire Schr\"odinger algebra. It also proves to be very helpful in
performing certain computations quickly, to obtain, for example, wave functions
and closed analytic expressions for time-evolution operators.Comment: 19 pages, minor changes, references update
Quantum Correlations and Quantum Non-Locality: A Review and a Few New Ideas
In this paper we make an extensive description of quantum non-locality, one
of the most intriguing and fascinating facets of quantum mechanics. After a
general presentation of several studies on this subject, we consider if quantum
non-locality, and the friction it carries with special relativity, can
eventually find a "solution" by considering higher dimensional spaces.Comment: 1
Symmetries in Quantum Mechanics and Statistical Physics
This book collects contributions to the Special Issue entitled "Symmetries in Quantum Mechanics and Statistical Physics" of the journal Symmetry. These contributions focus on recent advancements in the study of PT–invariance of non-Hermitian Hamiltonians, the supersymmetric quantum mechanics of relativistic and non-relativisitc systems, duality transformations for power–law potentials and conformal transformations. New aspects on the spreading of wave packets are also discussed
Bohmian quantum trajectories from coherent states
We find that real and complex Bohmian quantum trajectories resulting from well-localized Klauder coherent states in the quasi-Poissonian regime possess qualitatively the same type of trajectories as those obtained from a purely classical analysis of the corresponding Hamilton-Jacobi equation. In the complex cases we treated the quantum potential results to a constant, such that the agreement is exact. For the real cases we provide conjectures for analytical solutions for the trajectories as well as the corresponding quantum potentials. The overall qualitative behavior is governed by the Mandel parameter determining the regime in which the wave functions evolve as solitonlike structures. We demonstrate these features explicitly for the harmonic oscillator and the Pöschl-Teller potential
Quantum-spacetime effects on nonrelativistic Schr\"odinger evolution
The last three decades have witnessed the surge of quantum gravity
phenomenology in the ultraviolet regime as exemplified by the Planck-scale
accuracy of time-delay measurements from highly energetic astrophysical events.
Yet, recent advances in precision measurements and control over quantum
phenomena may usher in a new era of low-energy quantum gravity phenomenology.
In this study, we investigate relativistic modified dispersion relations (MDRs)
in curved spacetime and derive the corresponding nonrelativistic Schr\"odinger
equation using two complementary approaches. First, we take the nonrelativistic
limit, and canonically quantise the result. Second, we apply a WKB-like
expansion to an MDR-inspired deformed relativistic wave equation. Within the
area of applicability of single-particle quantum mechanics, both approaches
imply equivalent results. Surprisingly, we recognise in the generalized
uncertainty principle (GUP), the prevailing approach in nonrelativistic quantum
gravity phenomenology, the MDR which is least amenable to low-energy
experiments. Consequently, importing data from the mentioned time-delay
measurements, we constrain the linear GUP up to the Planck scale and improve on
current bounds to the quadratic one by 17 orders of magnitude. MDRs with larger
implications in the infrared, however, can be tightly constrained in the
nonrelativistic regime. We use the ensuing deviation from the equivalence
principle to bound some MDRs, for example the one customarily associated with
the bicrossproduct basis of the -Poincar\'e algebra, to up to four
orders of magnitude below the Planck scale.Comment: 34 pages, one figur
From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis
The novelty of the Jean Pierre Badiali last scientific works stems to a
quantum approach based on both (i) a return to the notion of trajectories
(Feynman paths) and (ii) an irreversibility of the quantum transitions. These
iconoclastic choices find again the Hilbertian and the von Neumann algebraic
point of view by dealing statistics over loops. This approach confers an
external thermodynamic origin to the notion of a quantum unit of time (Rovelli
Connes' thermal time). This notion, basis for quantization, appears herein as a
mere criterion of parting between the quantum regime and the thermodynamic
regime. The purpose of this note is to unfold the content of the last five
years of scientific exchanges aiming to link in a coherent scheme the Jean
Pierre's choices and works, and the works of the authors of this note based on
hyperbolic geodesics and the associated role of Riemann zeta functions. While
these options do not unveil any contradictions, nevertheless they give birth to
an intrinsic arrow of time different from the thermal time. The question of the
physical meaning of Riemann hypothesis as the basis of quantum mechanics, which
was at the heart of our last exchanges, is the backbone of this note.Comment: 13 pages, 2 figure
The Random Walk in Generalised Quantum Theory
One can view quantum mechanics as a generalization of classical probability theory that provides for pairwise interference among alternatives. Adopting this perspective, we “quantize” the classical random walk by finding, subject to a certain condition of “strong positivity”, the most general Markovian, translationally invariant “decoherence functional” with nearest neighbor transitions
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