scholarly journals Quantum Particle on Dual Weight Lattice in Weyl Alcove

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1338
Author(s):  
Adam Brus ◽  
Jiří Hrivnák ◽  
Lenka Motlochová
Keyword(s):  
Quantum Particle ◽  
Relativistic Quantum ◽  
Quantum Systems ◽  
Discrete Models ◽  
Stationary States ◽  
Finite Support ◽  
Weight Lattice ◽  
Weyl Transforms ◽  
The Matrix ◽  
Complex Valued

Families of discrete quantum models that describe a free non-relativistic quantum particle propagating on rescaled and shifted dual weight lattices inside closures of Weyl alcoves are developed. The boundary conditions of the presented discrete quantum billiards are enforced by precisely positioned Dirichlet and Neumann walls on the borders of the Weyl alcoves. The amplitudes of the particle’s propagation to neighbouring positions are determined by a complex-valued dual-weight hopping function of finite support. The discrete dual-weight Hamiltonians are obtained as the sum of specifically constructed dual-weight hopping operators. By utilising the generalised dual-weight Fourier–Weyl transforms, the solutions of the time-independent Schrödinger equation together with the eigenenergies of the quantum systems are exactly resolved. The matrix Hamiltonians, stationary states and eigenenergies of the discrete models are exemplified for the rank two cases C2 and G2.

scholarly journals Inversion of Tridiagonal Matrices Using the Dunford-Taylor’s Integral

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 870
Author(s):  
Diego Caratelli ◽  
Paolo Emilio Ricci
Keyword(s):  
Functional Analysis ◽  
Computer Algebra ◽  
Toeplitz Matrices ◽  
Test Cases ◽  
Recursive Procedure ◽  
Tridiagonal Matrices ◽  
Special Cases ◽  
The Matrix ◽  
Matrix Eigenvalues ◽  
Complex Valued

We show that using Dunford-Taylor’s integral, a classical tool of functional analysis, it is possible to derive an expression for the inverse of a general non-singular complex-valued tridiagonal matrix. The special cases of Jacobi’s symmetric and Toeplitz (in particular symmetric Toeplitz) matrices are included. The proposed method does not require the knowledge of the matrix eigenvalues and relies only on the relevant invariants which are determined, in a computationally effective way, by means of a dedicated recursive procedure. The considered technique has been validated through several test cases with the aid of the computer algebra program Mathematica©.

A matrix realignment method for recognizing entanglement

2003 ◽  
Vol 3 (3) ◽  
pp. 193-202
Author(s):  
K. Chen ◽  
L.-A. Wu
Keyword(s):  
Kronecker Product ◽  
Singular Values ◽  
Quantum Systems ◽  
Entangled States ◽  
Approximation Technique ◽  
Separable State ◽  
Simple Method ◽  
Bipartite Quantum Systems ◽  
The Matrix ◽  
Degree Of Entanglement

Motivated by the Kronecker product approximation technique, we have developed a very simple method to assess the inseparability of bipartite quantum systems, which is based on a realigned matrix constructed from the density matrix. For any separable state, the sum of the singular values of the matrix should be less than or equal to $1$. This condition provides a very simple, computable necessary criterion for separability, and shows powerful ability to identify most bound entangled states discussed in the literature. As a byproduct of the criterion, we give an estimate for the degree of entanglement of the quantum state.

scholarly journals Some effects on relativistic quantum systems due to a weak gravitational field

2005 ◽  
Vol 35 (4b) ◽  
pp. 1110-1112 ◽  
Author(s):  
Geusa de A. Marques ◽  
Sandro G. Fernandes ◽  
V. B. Bezerra
Keyword(s):  
Gravitational Field ◽  
Relativistic Quantum ◽  
Quantum Systems ◽  
Weak Gravitational Field

On hyper‐relativistic quantum systems

1987 ◽  
Vol 28 (1) ◽  
pp. 79-84
Author(s):  
Awele Maduemezia
Keyword(s):  
Relativistic Quantum ◽  
Quantum Systems

scholarly journals A Numerical Method for Computing the Roots of Non-Singular Complex-Valued Matrices

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 966
Author(s):  
Diego Caratelli ◽  
Paolo Emilio Ricci
Keyword(s):  
Numerical Method ◽  
Worked Examples ◽  
Higher Order ◽  
Singular Matrix ◽  
The Matrix ◽  
General Complex ◽  
Complex Valued

A method for the computation of the n th roots of a general complex-valued r × r non-singular matrix ? is presented. The proposed procedure is based on the Dunford–Taylor integral (also ascribed to Riesz–Fantappiè) and relies, only, on the knowledge of the invariants of the matrix, so circumventing the computation of the relevant eigenvalues. Several worked examples are illustrated to validate the developed algorithm in the case of higher order matrices.

Generalized Electromagnetic Fields Associated with the Hydrogen-Like Atom Problem

2018 ◽  
Vol 74 (1) ◽  
pp. 43-50 ◽  
Author(s):  
S.A. Bruce ◽  
J.F. Diaz-Valdes
Keyword(s):  
Quantum Mechanics ◽  
Electromagnetic Coupling ◽  
Relativistic Quantum ◽  
Scalar Potential ◽  
Symmetry Algebra ◽  
Dynamical Symmetry ◽  
Canonical Commutation Relation ◽  
Quantum Systems ◽  
Minimal Coupling ◽  
Exactly Solvable

AbstractIt is known that the principle of minimal coupling in quantum mechanics determines a unique interaction form for a charged particle. By properly redefining the canonical commutation relation between (canonical) conjugate components of position and momentum of the particle, e.g. an electron, we restate the Dirac equation for the hydrogen-like atom problem incorporating a generalized minimal electromagnetic coupling. The corresponding interaction keeps the $1/\left|\mathbf{q}\right|$ dependence in both the scalar potential $V\left({\left|\mathbf{q}\right|}\right)$ and the vector potential $\mathbf{A}\left(\mathbf{q}\right)$ ($\left|{\mathbf{A}\left(\mathbf{q}\right)}\right|\sim 1/\left|\mathbf{q}\right|$). This problem turns out to be exactly solvable; moreover, the eigenstates and eigenvalues can be obtained in an elementary fashion. Some feasible models within this approach are discussed. Then we make a few remarks about the breaking of supersymmetry. Finally, we briefly comment on the possible Lie algebra (dynamical symmetry algebra) of these relativistic quantum systems.

SCHRÖDINGER EQUATION FOR QUANTUM FRACTAL SPACE–TIME OF ORDER n VIA THE COMPLEX-VALUED FRACTIONAL BROWNIAN MOTION

2001 ◽  
Vol 16 (31) ◽  
pp. 5061-5084 ◽  
Author(s):  
GUY JUMARIE
Keyword(s):  
Brownian Motion ◽  
Schrödinger Equation ◽  
Fractional Order ◽  
Schrodinger Equation ◽  
Relativistic Quantum ◽  
Relativistic Quantum Mechanics ◽  
Dimensional System ◽  
One Dimensional ◽  
Klein Gordon ◽  
Complex Valued

First remark: Feynman's discovery in accordance of which quantum trajectories are of fractal nature (continuous everywhere but nowhere differentiable) suggests describing the dynamics of such systems by explicitly introducing the Brownian motion of fractional order in their equations. The second remark is that, apparently, it is only in the complex plane that the Brownian motion of fractional order with independent increments can be generated, by using random walks defined with the complex roots of the unity; in such a manner that, as a result, the use of complex variables would be compulsory to describe quantum systems. Here one proposes a very simple set of axioms in order to expand the consequences of these remarks. Loosely speaking, a one-dimensional system with real-valued coordinate is in fact the average observation of a one-dimensional system with complex-valued coordinate: It is a strip modeling. Assuming that the system is governed by a stochastic differential equation driven by a complex valued fractional Brownian of order n, one can then obtain the explicit expression of the corresponding covariant stochastic derivative with respect to time, whereby we switch to the extension of Lagrangian mechanics. One can then derive a Schrödinger equation of order n in quite a direct way. The extension to relativistic quantum mechanics is outlined, and a generalized Klein–Gordon equation of order n is obtained. As a by-product, one so obtains a new proof of the Schrödinger equation.

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