scholarly journals Spherical-Symmetry and Spin Effects on the Uncertainty Measures of Multidimensional Quantum Systems with Central Potentials

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 607
Author(s):  
Jesús Dehesa
Keyword(s):  
Spherical Symmetry ◽  
Variational Approach ◽  
Quantum Systems ◽  
Particle Systems ◽  
Uncertainty Relations ◽  
Stationary States ◽  
Expectation Values ◽  
Spin Effects ◽  
Probability Densities ◽  
Inequality Type

The spreading of the stationary states of the multidimensional single-particle systems with a central potential is quantified by means of Heisenberg-like measures (radial and logarithmic expectation values) and entropy-like quantities (Fisher, Shannon, R\'enyi) of position and momentum probability densities. Since the potential is assumed to be analytically unknown, these dispersion and information-theoretical measures are given by means of inequality-type relations which are explicitly shown to depend on dimensionality and state's angular hyperquantum numbers. The spherical-symmetry and spin effects on these spreading properties are obtained by use of various integral inequalities (Daubechies--Thakkar, Lieb--Thirring, Redheffer--Weyl, ...) and a variational approach based on the extremization of entropy-like measures. Emphasis is placed on the uncertainty relations, upon which the essential reason of the probabilistic theory of quantum systems relies.

scholarly journals Minimizing irreversible losses in quantum systems by local counterdiabatic driving

2017 ◽  
Vol 114 (20) ◽  
pp. E3909-E3916 ◽  
Author(s):  
Dries Sels ◽  
Anatoli Polkovnikov
Keyword(s):  
Variational Approach ◽  
Chaotic Systems ◽  
Chem Phys ◽  
Quantum Systems ◽  
Particle Systems ◽  
Coupling Constant ◽  
Quantum Annealing ◽  
Physical Constraints ◽  
Adiabatic Dynamics

Counterdiabatic driving protocols have been proposed [Demirplak M, Rice SA (2003) J Chem Phys A 107:9937–9945; Berry M (2009) J Phys A Math Theor 42:365303] as a means to make fast changes in the Hamiltonian without exciting transitions. Such driving in principle allows one to realize arbitrarily fast annealing protocols or implement fast dissipationless driving, circumventing standard adiabatic limitations requiring infinitesimally slow rates. These ideas were tested and used both experimentally and theoretically in small systems, but in larger chaotic systems, it is known that exact counterdiabatic protocols do not exist. In this work, we develop a simple variational approach allowing one to find the best possible counterdiabatic protocols given physical constraints, like locality. These protocols are easy to derive and implement both experimentally and numerically. We show that, using these approximate protocols, one can drastically suppress heating and increase fidelity of quantum annealing protocols in complex many-particle systems. In the fast limit, these protocols provide an effective dual description of adiabatic dynamics, where the coupling constant plays the role of time and the counterdiabatic term plays the role of the Hamiltonian.

scholarly journals High Dimensional Atomic States of Hydrogenic Type: Heisenberg-like and Entropic Uncertainty Measures

Entropy ◽  
2021 ◽  
Vol 23 (10) ◽  
pp. 1339
Author(s):  
Jesús S. Dehesa
Keyword(s):  
Quantum Systems ◽  
High Dimensional ◽  
Uncertainty Relations ◽  
Expectation Values ◽  
Molecular Physics ◽  
Uncertainty Measures ◽  
Relevant Role ◽  
Atomic And Molecular Physics ◽  
Atomic States ◽  
Positively Charged

High dimensional atomic states play a relevant role in a broad range of quantum fields, ranging from atomic and molecular physics to quantum technologies. The D-dimensional hydrogenic system (i.e., a negatively-charged particle moving around a positively charged core under a Coulomb-like potential) is the main prototype of the physics of multidimensional quantum systems. In this work, we review the leading terms of the Heisenberg-like (radial expectation values) and entropy-like (Rényi, Shannon) uncertainty measures of this system at the limit of high D. They are given in a simple compact way in terms of the space dimensionality, the Coulomb strength and the state’s hyperquantum numbers. The associated multidimensional position–momentum uncertainty relations are also revised and compared with those of other relevant systems.

Existence and stability of static shells for the Vlasov–Poisson system

Analysis ◽  
2006 ◽  
Vol 26 (4) ◽  
Author(s):  
Achim Schulze
Keyword(s):  
Spherical Symmetry ◽  
Variational Approach ◽  
Stationary Solutions ◽  
Poisson System ◽  
Existence And Stability

We prove the existence and stability of stationary solutions to the Vlasov–Poisson System with spherical symmetry, which describe static shells, i.e., the support of their densities is bounded away from the origin. We use a variational approach which was established by Y. Guo and G. Rein.

scholarly journals Residual Entropy and Critical Behavior of Two Interacting Boson Species in a Double Well

2017 ◽  
Author(s):  
Fabio Lingua ◽  
Andrea Richaud ◽  
Vittorio Penna
Keyword(s):  
Coherent States ◽  
Variational Approach ◽  
Entanglement Entropy ◽  
Quantum Systems ◽  
Residual Entropy ◽  
Zero Temperature ◽  
Temperature Scenario ◽  
Physical Insight ◽  
Insight Into ◽  
Specific Quantum

Motivated by the importance of entanglement and correlation indicators in the analysis of quantum systems, we study the equilibrium and the residual entropy in a two-species Bose Hubbard dimer when the spatial phase separation of the two species takes place. We consider both the zero and non-zero-temperature regime. We present different kinds of residual entropies (each one associated to a different way of partitioning the system), and we show that they strictly depend on the specific quantum phase characterizing the two species (supermixed, mixed or demixed) even at finite temperature. To provide a deeper physical insight into the zero-temperature scenario, we apply the fully-analytical variational approach based on su(2) coherent states and provide a considerbly good approximation of the entanglement entropy. Finally, we show that the effectiveness of residual entropy as a critical indicator at non-zero temperature is unchanged when considering a restricted combination of energy eigenstates.

scholarly journals Interplay of symmetry and topology in science

2014 ◽  
Vol 70 (a1) ◽  
pp. C1419-C1419
Author(s):  
Boris Zhilinskii
Keyword(s):  
Classical System ◽  
Quantum Systems ◽  
Qualitative Theory ◽  
Topological Invariant ◽  
Particle Systems ◽  
Qualitative Description ◽  
Energy Bands ◽  
And Topology ◽  
Finite Particle

Qualitative methods in natural science are based mainly on simultaneous use of symmetry and topology arguments. The idea of the present talk is to demonstrate how the corresponding mathematical tools (based on symmetry and topology arguments) initially applied to describe classification of different phases of matter and transitions between them are extended to construct qualitative theory of finite particle systems and more general dynamical systems. I start with reminding basic notions and tools associated with application of group action ideas to physics as initiated and developed by Louis Michel (1923-1999) [1,2]. Then geometric combinatorial and topological ideas are used to give qualitative description of singularities of dynamical integrable classical system and their quantum analogs. Quantum monodromy and its various generalizations as well as description of energy bands of isolated finite particle quantum systems in terms of topological invariant, Chern number [3], will be discussed on concrete molecular and atomic examples.

scholarly journals Quantum Mechanics Needs No Interpretation

2005 ◽  
Vol 70 (5) ◽  
pp. 621-637 ◽  
Author(s):  
Lubomír Skála ◽  
Vojtěch Kapsa
Keyword(s):  
Quantum Mechanics ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Mathematical Structure ◽  
Uncertainty Relations ◽  
Mean Values ◽  
Vector Potentials ◽  
Probability Densities ◽  
Klein Gordon ◽  
Definition Of

Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that the basic mathematical structure of quantum mechanics like the probability amplitudes, Born rule, probability density current, commutation and uncertainty relations, momentum operator, rules for including scalar and vector potentials and antiparticles can be derived from the definition of the mean values of powers of space coordinates and time. Equations of motion of quantum mechanics, the Klein-Gordon equation, Schrödinger equation and Dirac equation are obtained from the requirement of the relativistic invariance of the theory. The limit case of localized probability densities leads to the Hamilton-Jacobi equation of classical mechanics. Many-particle systems are also discussed.

scholarly journals Massively Parallel Classical Logic via Coherent Dynamics of an Ensemble of Quantum Systems with Dispersion in Size

2020 ◽  
Author(s):  
Hugo Gattuso ◽  
Raphael D. Levine ◽  
Francoise Remacle
Keyword(s):  
Quantum System ◽  
Quantum Dynamics ◽  
Algebraic Theory ◽  
Laser Pulses ◽  
Quantum Systems ◽  
Individual Variability ◽  
Expectation Values ◽  
Charge Transfer Excitons ◽  
Simultaneous Reading ◽  
Size Dispersion

Quantum parallelism can be implemented on a classical ensemble of discrete level quantum systems. The nano systems are not quite identical and the ensemble represents their individual variability. An underlying Lie algebraic theory is developed using the closure of the algebra to demonstrate the parallel information processing at the level of the ensemble. The ensemble is addressed by a sequence of laser pulses. In the Heisenberg picture of quantum dynamics the coherence between the N levels of a given quantum system can be handled as an observable. Thereby there are N2 logic variables per N level system. This is how massive parallelism is achieved in that there are N2 potential outputs for a quantum system of N levels. The use of an ensemble allows simultaneous reading of such outputs. Due to size dispersion the expectation values of the observables can differ somewhat from system to system. We show that for a moderate variability of the systems one can average the N2 expectation values over the ensemble while retaining closure and parallelism. This allows directly propagating in time the ensemble averaged values of the observables. Results of simulations of electronic excitonic dynamics in an ensemble of quantum dot, QD, dimers are presented. The QD size and interdot distance in the dimer are used to parametrize the Hamiltonian. The dimer N levels include local and charge transfer excitons within each dimer. The well-studied physics of semi-conducting QDs suggests that the dimer coherences can be probed at room temperature

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