Quantum Systems Corresponding to Classical Free-particle Systems and the Transformations between Canonical Position and Momentum within Those Systems

Kyu Hwang Yeon

doi:10.3938/npsm.62.464

HIDDEN SCALE IN QUANTUM MECHANICS

Modern Physics Letters A ◽

10.1142/s0217732309030102 ◽

2009 ◽

Vol 24 (27) ◽

pp. 2203-2211 ◽

Cited By ~ 2

Author(s):

PULAK RANJAN GIRI

Keyword(s):

Wave Packet ◽

Quantum Particle ◽

Dimensional Space ◽

Free Particle ◽

Three Dimensional ◽

Particle Systems ◽

Quantum Mechanical ◽

Scale Invariant ◽

Mechanical Models ◽

Three Dimensional Space

We show that the intriguing localization of a free particle wave-packet is possible due to a hidden scale present in the system. Self-adjoint extensions (SAE) is responsible for introducing this scale in quantum mechanical models through the nontrivial boundary conditions. We discuss a couple of classically scale invariant free particle systems to illustrate the issue. In this context it has been shown that a free quantum particle moving on a full line may have localized wave-packet around the origin. As a generalization, it has also been shown that particles moving on a portion of a plane or on a portion of a three-dimensional space can have unusual localized wave-packet.

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Minimizing irreversible losses in quantum systems by local counterdiabatic driving

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.1619826114 ◽

2017 ◽

Vol 114 (20) ◽

pp. E3909-E3916 ◽

Cited By ~ 62

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.

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Interplay of symmetry and topology in science

Acta Crystallographica Section A Foundations and Advances ◽

10.1107/s2053273314085805 ◽

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.

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Classical particle systems. I. Galilei free particle

Journal of Physics A General Physics ◽

10.1088/0305-4470/18/11/021 ◽

1985 ◽

Vol 18 (11) ◽

pp. 1971-1984 ◽

Cited By ~ 8

Author(s):

M Rivas

Keyword(s):

Free Particle ◽

Particle Systems ◽

Classical Particle

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

Entropy ◽

10.3390/e23050607 ◽

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.

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