scholarly journals Barrier billiard and random matrices

2021 ◽  
Author(s):  
Eugene B Bogomolny
Keyword(s):  
Quantum Mechanics ◽  
Barrier Height ◽  
Random Matrices ◽  
Low Complexity ◽  
Integrable Models ◽  
Independent Interest ◽  
Unitary Matrix ◽  
Quantum Properties ◽  
Spectral Statistics ◽  
Hopf Method

Abstract The barrier billiard is the simplest example of pseudo-integrable models with interesting and intricate classical and quantum properties. Using the Wiener-Hopf method it is demonstrated that quantum mechanics of a rectangular billiard with a barrier in the centre can be reduced to the investigation of a certain unitary matrix. Under heuristic assumptions this matrix is substituted by a special low-complexity random unitary matrix of independent interest. The main results of the paper are (i) spectral statistics of such billiards is insensitive to the barrier height and (ii) it is well described by the semi-Poisson distributions.

scholarly journals Aspects of Geodesical Motion with Fisher-Rao Metric: Classical and Quantum

2018 ◽  
Vol 25 (01) ◽  
pp. 1850005 ◽  
Author(s):  
Florio M. Ciaglia ◽  
Fabio Di Cosmo ◽  
Domenico Felice ◽  
Stefano Mancini ◽  
Giuseppe Marmo ◽  
...  
Keyword(s):  
Quantum Mechanics ◽  
Uncertainty Principle ◽  
Geometric Structure ◽  
Initial Conditions ◽  
Probability Distributions ◽  
End Points ◽  
Quantum Properties ◽  
Statistical Manifolds ◽  
Classical Setting ◽  
Qualitative Effect

The purpose of this paper is to exploit the geometric structure of quantum mechanics and of statistical manifolds to study the qualitative effect that the quantum properties have in the statistical description of a system. We show that the end points of geodesics in the classical setting coincide with the probability distributions that minimise Shannon’s entropy, i.e. with distributions of zero dispersion. In the quantum setting this happens only for particular initial conditions, which in turn correspond to classical submanifolds. This result can be interpreted as a geometric manifestation of the uncertainty principle.

scholarly journals Quantum SU(2|1) supersymmetric ℂN Smorodinsky-Winternitz system

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Evgeny Ivanov ◽  
Armen Nersessian ◽  
Stepan Sidorov
Keyword(s):  
Quantum Mechanics ◽  
Energy Levels ◽  
Wave Functions ◽  
Supersymmetric Extension ◽  
Quantum Properties ◽  
Hidden Symmetry ◽  
Euclidian Space ◽  
Symmetry Operators ◽  
Symmetry Generators ◽  
Equivalent Description

Abstract We study quantum properties of SU(2|1) supersymmetric (deformed $$ \mathcal{N} $$ N = 4, d = 1 supersymmetric) extension of the superintegrable Smorodinsky-Winternitz system on a complex Euclidian space ℂN. The full set of wave functions is constructed and the energy spectrum is calculated. It is shown that SU(2|1) supersymmetry implies the bosonic and fermionic states to belong to separate energy levels, thus exhibiting the “even-odd” splitting of the spectra. The superextended hidden symmetry operators are also defined and their action on SU(2|1) multiplets of the wave functions is given. An equivalent description of the same system in terms of superconformal SU(2|1, 1) quantum mechanics is considered and a new representation of the hidden symmetry generators in terms of the SU(2|1, 1) ones is found.

scholarly journals Asymptotic Linear Spectral Statistics for Spiked Hermitian Random Matrices

2015 ◽  
Vol 160 (1) ◽  
pp. 120-150 ◽  
Author(s):  
Damien Passemier ◽  
Matthew R. McKay ◽  
Yang Chen
Keyword(s):  
Random Matrices ◽  
Spectral Statistics

Logarithmic terms in the spectral statistics of band random matrices

1993 ◽  
Vol 26 (15) ◽  
pp. 3845-3852 ◽  
Author(s):  
E Caurier ◽  
A Ramani ◽  
B Grammaticos
Keyword(s):  
Random Matrices ◽  
Spectral Statistics

scholarly journals Large violations in Kochen Specker contextuality and their applications

2021 ◽  
Author(s):  
Ravishankar Ramanathan ◽  
Yuan Liu ◽  
Pawel Horodecki
Keyword(s):  
Quantum Mechanics ◽  
Present State ◽  
Hilbert Spaces ◽  
Independent Interest ◽  
Entangled State ◽  
Maximally Entangled State ◽  
Technical Result ◽  
Orthogonal Representation ◽  
State Dependent ◽  
Minimum Dimension

Abstract It is of interest to study how contextual quantum mechanics is, in terms of the violation of Kochen Specker state-independent and state-dependent non-contextuality inequalities. We present state-independent non-contextuality inequalities with large violations, in particular, we exploit a connection between Kochen-Specker proofs and pseudo-telepathy games to show KS proofs in Hilbert spaces of dimension $d \geq 2^{17}$ with the ratio of quantum value to classical bias being $O(\sqrt{d}/\log d)$. We study the properties of this KS set and show applications of the large violation. It has been recently shown that Kochen-Specker proofs always consist of substructures of state-dependent contextuality proofs called $01$-gadgets or bugs. We show a one-to-one connection between $01$-gadgets in $\mathbb{C}^d$ and Hardy paradoxes for the maximally entangled state in $\mathbb{C}^d \otimes \mathbb{C}^d$. We use this connection to construct large violation $01$-gadgets between arbitrary vectors in $\mathbb{C}^d$, as well as novel Hardy paradoxes for the maximally entangled state in $\mathbb{C}^d \otimes \mathbb{C}^d$, and give applications of these constructions. As a technical result, we show that the minimum dimension of the faithful orthogonal representation of a graph in $\mathbb{R}^d$ is not a graph monotone, a result that may be of independent interest.

scholarly journals On Einstein’s Program and Quantum Mechanics

2015 ◽  
Vol 7 (6) ◽  
pp. 126
Author(s):  
Claude Elbaz
Keyword(s):  
Quantum Mechanics ◽  
Classical Domain ◽  
Amplitude Function ◽  
Consistent System ◽  
Quantum Properties ◽  
Space Reduction ◽  
Quantum Domain ◽  
Wave Particle Duality ◽  
The Standard Model ◽  
Classical And Quantum Mechanics

<p class="1Body">The Einstein’s program forms a consistent system for universe description, beside the standard model of particles. It is founded upon a scalar field propagating at speed of light c, which constitutes a common relativist framework for classical and quantum properties of matter and interactions. Matter corresponds to standing waves. Classical domain corresponds to geometrical optics approximation, when frequencies are infinitely high, and then hidden. Quantum domain corresponds to wave optics approximation. Adiabatic variations of frequencies yield electromagnetic interaction. They lead also to Classical and Quantum Mechanics equations, with unification of first and second quantifications for interactions and matter, and to the wave-particle duality, by space reduction of the introduced space-like amplitude function u(r,t), which completes the usual time-like function ψ(r,t).</p>

scholarly journals Local spectral statistics of the addition of random matrices

2019 ◽  
Vol 175 (1-2) ◽  
pp. 579-654 ◽  
Author(s):  
Ziliang Che ◽  
Benjamin Landon
Keyword(s):  
Random Matrices ◽  
Spectral Statistics
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