Application Of Quantum Mechanics Research Articles

Investigating the fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation: an iterative framework for nonlinear wave dynamics

Abstract This paper addresses the solution of the fractional Caudrey-Dodd-Gibbon-Sawada-Kotera (CDGSK) equation using the Conformable Laplace Decomposition Method (CLDM). The CDGSK equation, a fundamental model in wave dynamics and fluid mechanics, is explored for its applications in quantum mechanics and nonlinear optics. By employing fractional calculus, we demonstrate how fractional derivatives influence the physical characteristics of wave propagation in both optical and quantum systems. The exact solutions obtained provide insight into soliton behavior, essential for understanding wave-particle interactions in quantum fields and light–matter interactions in optics. The fractional nature of the equation allows for more accurate modeling of non-integer order dynamics commonly found in optical fibers and quantum waveguides. The CLDM method proves to be highly effective, providing approximate solutions with minimal computational effort. These findings offer significant contributions to the fields of quantum mechanics and nonlinear optics, where the fractional CDGSK equation can be applied to solve complex wave equations with great accuracy.

On the Lanczos Method for Computing Some Matrix Functions

The study of matrix functions is highly significant and has important applications in control theory, quantum mechanics, signal processing, and machine learning. Previous work has mainly focused on how to use the Krylov-type method to efficiently calculate matrix functions f(A)β and βTf(A)β when A is symmetric. In this paper, we mainly illustrate the convergence using the polynomial approximation theory for the case where A is symmetric positive definite. Numerical results illustrate the effectiveness of our theoretical results.

OQuPy: A Python package to efficiently simulate non-Markovian open quantum systems with process tensors.

Non-Markovian dynamics arising from the strong coupling of a system to a structured environment is essential in many applications of quantum mechanics and emerging technologies. Deriving an accurate description of general quantum dynamics including memory effects is, however, a demanding task, prohibitive to standard analytical or direct numerical approaches. We present a major release of our open source software package, OQuPy (Open Quantum System in Python), which provides several recently developed numerical methods that address this challenging task. It utilizes the process tensor approach to open quantum systems (OQS) in which a single map, the process tensor, captures all possible effects of an environment on the system. The representation of the process tensor in a tensor network form allows for an exact yet highly efficient description of non-Markovian OQS (NM-OQS). The OQuPy package provides methods to (1) compute the dynamics and multi-time correlations of quantum systems coupled to single and multiple environments, (2) optimize control protocols for NM-OQS, (3) simulate interacting chains of NM-OQS, and (4) compute the mean-field dynamics of an ensemble of NM-OQS coupled to a common central system. Our aim is to provide an easily accessible and extensible tool for researchers of OQS in fields such as quantum chemistry, quantum sensing, and quantum information.

𝒫𝒯 and anti-𝒫𝒯 symmetries for astrophysical waves

Context. Discrete symmetries have found numerous applications in photonics and quantum mechanics, but remain little studied in fluid mechanics, particularly in astrophysics. Aims. We aim to show how 𝒫𝒯 and anti-𝒫𝒯 symmetries determine the behaviour of linear perturbations in a wide class of astrophysical problems. They set the location of ‘exceptional points’ in the parameter space and the associated transitions to instability, and are associated with the conservation of quadratic quantities that can be determined explicitly. Methods. We study several classical local problems: the gravitational instability of isothermal spheres and thin discs, the Schwarzschild instability, the Rayleigh-Bénard instability and acoustic waves in dust–gas mixtures. We calculate the locations and the order of the exceptional points using the resultant of two univariate polynomials, as well as the conserved quantities in the different regions of the parameter space using Krein theory. Results. All problems studied here exhibit discrete symmetries, even though Hermiticity is broken by different physical processes (self-gravity, buoyancy, diffusion, and drag). This analysis provides genuine explanations for certain instabilities, and for the existence of regions in the parameter space where waves do not propagate. Those two aspects correspond to regions where 𝒫𝒯 and anti-𝒫𝒯 symmetries are broken respectively. Not all instabilities are associated to symmetry breaking (e.g. the Rayleigh-Benard instability).

Optimizing ZX-diagrams with deep reinforcement learning

Abstract ZX-diagrams are a powerful graphical language for the description of quantum processes with applications in fundamental quantum mechanics, quantum circuit optimization, tensor network simulation, and many more. The utility of ZX-diagrams relies on a set of local transformation rules that can be applied to them without changing the underlying quantum process they describe. These rules can be exploited to optimize the structure of ZX-diagrams for a range of applications. However, finding an optimal sequence of transformation rules is generally an open problem. In this work, we bring together ZX-diagrams with reinforcement learning, a machine learning technique designed to discover an optimal sequence of actions in a decision-making problem and show that a trained reinforcement learning agent can significantly outperform other optimization techniques like a greedy strategy, simulated annealing, and state-of-the-art hand-crafted algorithms. The use of graph neural networks to encode the policy of the agent enables generalization to diagrams much bigger than seen during the training phase.

Structural Chemistry Simulation Experiments: A Case Study on Conjugated Alkene Molecules

As a core branch of chemistry, structural chemistry presents challenges in teaching due to its theoretical constructs and complex mathematical derivations. The application of quantum mechanics is particularly difficult for students. This paper analyzes the limitations of traditional experimental teaching and proposes simulation experiments as an effective supplementary method. Simulation experiments, by reducing costs, improving safety, and enhancing intuitiveness, help students deeply understand the theoretical knowledge of structural chemistry and cultivate their critical thinking and problem-solving skills. This study designs a series of simulation experiment teaching activities aimed at enhancing students' understanding of the structural characteristics of conjugated alkene molecules and their impact on molecular properties. Through case analysis, interactive discussions, and feedback evaluation, the study demonstrates the advantages of simulation experiments in improving student engagement, knowledge mastery, and practical abilities. Additionally, it points out the shortcomings of simulation experiments in terms of authenticity and technological dependence and proposes corresponding improvement suggestions. Finally, it emphasizes the importance of combining practical experimental operations with cultivating students' autonomous learning abilities, aiming to provide a more comprehensive strategy for structural chemistry teaching.

NUMERICAL SOLUTION OF NONLINEAR SCHRODINGER EQUATION USING COLLOCATION METHOD WITH CHEBYSHEV POLYNOMIALS

The nonlinear Schrodinger equation (NSE) is a partial differential equation (PDE) with numerous applications in quantum mechanics. Analytical methods for the solution of NSE are almost impossible due to their complexity. Thus, the need for a numerical scheme to seek the approximate solution of the NSE. Hence, this paper considered the numerical solution of the NSE via the spectral collocation method (SCM) with Chebyshev orthogonal polynomials of the first kind. The scheme was efficiently constructed to solve the NSE equation with the aid of MAPLE 18, and numerical evidence compared with Adomian Decomposition Method (ADM), Homotopy Analysis Transform Method (HATM) and Residual Power Series Method (RPSM) as available in the literature. The findings show that the SCM is an effective solver for NSE with rapid convergence to the exact solution as the parameter �varies.

A numerical approach for solving nonlinear fractional Klein–Gordon equation with applications in quantum mechanics

Abstract In this paper, we propose a numerical approach for solving the nonlinear fractional Klein–Gordon equation (FKGE), a model of significant importance in simulating nonlinear waves in quantum mechanics. Our method combines the Bernoulli wavelet collocation scheme with a functional integration matrix to obtain approximate solutions for the proposed model. Initially, we transform the main problem into a system of algebraic equations, which we solve using the Newton–Raphson method to extract the unknown coefficients and achieve the desired approximate solution. To theoretically validate our method, we conduct a comprehensive convergence analysis, demonstrating its uniform convergence. We perform numerical experiments on various examples with different parameters, presenting the results through tables and figures. Our findings indicate that employing more terms in the utilized techniques enhances accuracy. Furthermore, we compare our approach with existing methods from the literature, showcasing its performance in terms of computational cost, convergence rate, and solution accuracy. These examples illustrate how our techniques yield better approximate solutions for the nonlinear model at a low computational cost, as evidenced by the calculated CPU time and absolute error. Additionally, our method consistently provides better accuracy than other methods from the literature, suggesting its potential for solving more complex problems in physics and other scientific disciplines.

Analyzing the Non-compositionality of Conceptual Combinations

Objectives: This research is first of its kind to investigate the application of quantum mechanics and related mathematical models to cognitive science, specifically focusing on the challenge of modeling how the human mind assigns meaning to combinations of words. The primary objective is to classify whether given word combinations are compositional or non-compositional. Methods: The author introduces the "WWW method," which leverages the World Wide Web (WWW) as an experimental tool in place of human subjects, which makes this method cheaper and handy in comparison to the earlier methods which used human subjects for spoken and sign language studies and robots as non-human subjects. The "WWW method" involves checking the adherence to Bell inequalities and marginal selectivity. The term "meaning bound" is introduced to quantify the extent to which two words influence each other's meaning within a combination. This concept proves instrumental in devising a method for assessing compositionality using the WWW method. Findings: When the WWW serves as the information space, the verification of whether a word combination adheres to Bell inequalities entails determining if the meaning bound for the word combination (AB & C) exceeds that of its constituent parts (A & C, B & C) within the conceptual combination AB. Marginal selectivity is evaluated by establishing a vector model for grammatically framed word combinations, utilizing the WWW to collect data. If the word combination 'A then B' yields an equivalent number of hits as 'B then A,' it signifies that the word combination AB satisfies marginal selectivity. Novelty: Consequently, this research presents an accessible approach to identifying entangled pairs of word combinations using the WWW method, eliminating the need for human subject experiments. The earlier studies which include human subjects are costly and cannot be replicated anytime-anywhere, for obvious reasons. Keywords: Quantum cognition, Marginal selectivity, Compositionality, WWW method, CHSH inequality

Invariant measures on p-adic Lie groups: the p-adic quaternion algebra and the Haar integral on the p-adic rotation groups

We provide a general expression of the Haar measure—that is, the essentially unique translation-invariant measure—on a p-adic Lie group. We then argue that this measure can be regarded as the measure naturally induced by the invariant volume form on the group, as it happens for a standard Lie group over the reals. As an important application, we next consider the problem of determining the Haar measure on the p-adic special orthogonal groups in dimension two, three and four (for every prime number p). In particular, the Haar measure on SO(2,Qp)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {SO}(2,\\mathbb {Q}_p)$$\\end{document} is obtained by a direct application of our general formula. As for SO(3,Qp)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {SO}(3,\\mathbb {Q}_p)$$\\end{document} and SO(4,Qp)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {SO}(4,\\mathbb {Q}_p)$$\\end{document}, instead, we show that Haar integrals on these two groups can conveniently be lifted to Haar integrals on certain p-adic Lie groups from which the special orthogonal groups are obtained as quotients. This construction involves a suitable quaternion algebra over the field Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_p$$\\end{document} and is reminiscent of the quaternionic realization of the real rotation groups. Our results should pave the way to the development of harmonic analysis on the p-adic special orthogonal groups, with potential applications in p-adic quantum mechanics and in the recently proposed p-adic quantum information theory.

Strained diamond for quantum sensing applications

Apart from being an extraordinary optical and electronic material, diamond has also found applications in quantum mechanics especially in quantum sensing with the discovery and research development of various color centers. Elastic strain engineering (ESE), as a powerful modulation method, can tune the quantum properties and improve the performance of diamond quantum sensors. In recent years, deep ESE (DESE, when >5% elastic strain, or >σ ideal/2 is achieved) has been realized in micro/nano-fabricated diamond and shows a great potential for tuning the quantum mechanical properties of diamond substantially. In this perspective, we briefly review the quantum properties of diamond and some of the corresponding sensing applications carried out with ESE, and look at how DESE could be applied for further tuning the quantum sensing properties of diamond with desired applications and what the critical challenges are.

Solve third order Lane–Emden–Fowler equation by Adomian decomposition method and quartic trigonometric B-spline method

In this work, we study the Lane–Emden–Fowler equation which has a wide range of applications in astrophysics, classical and quantum mechanics. Firstly, we convert this nonlinear differential equation into an equivalent integral equation. We employ the powerful Adomian decomposition method to derive an analytical solution that will also lead to numerical solutions. Moreover, we exhibit two theorems to ensure the uniqueness of the solution of the problem and to confirm the convergence of the proposed scheme. In addition, we use quartic trigonometric B-spline as a powerful numerical method to achieve numerical solutions. Finally we present a comparison between the proposed methods to show their accuracy performance. Proper graphs are also provided to illustrate the obtained solutions.

Consideration of electronic mean heat transport via a low dimension system

The object of research is the complex realm of energy localization and coherent ballistic electronic transport within low-dimensional silicon quantum wires, specifically those doped with germanium atoms. Unlike their three-dimensional counterparts, low-dimensional systems exhibit unique electronic transport behaviors, necessitating novel analytical approaches for a comprehensive understanding. The core of this investigation leverages the Phase Field Matching Theory (PMFT) and the tight-binding (TB) approximation, sophisticated methodologies that enable a deep dive into the quantum mechanical nuances of these systems. Through this lens, we examine the intricate dynamics of dispersion relationships, phase factors, group velocities, and notably, the impact of defects introduced by the germanium doping. This research meticulously analyzes how these defects affect electronic and thermal conductivities, along with densities of states, offering new insights into the role of Fano resonances in the fluctuation of transmission and reflection spectra. These resonances, we find, are crucially dependent on the nature of the defects, their configuration, and the electronic parameters in their vicinity, underscoring the nuanced interplay between material composition and electronic properties in low-dimensional systems. The implications of our findings extend far beyond the theoretical. They pave the way for significant advancements in nanotechnology and the design of electronic devices, highlighting the potential for creating more efficient, high-performance components. Furthermore, this work proposes a framework for developing non-destructive testing methodologies that could revolutionize material science by enabling the precise analysis of defects in low-dimensional systems without causing damage. This is particularly critical for the ongoing development of materials with optimized properties for various applications, from electronics to energy storage. In essence, this research not only enriches our understanding of the physics governing low-dimensional systems but also offers practical insights into leveraging these properties for technological innovation. By bridging the gap between theoretical physics and material science, our study sets the stage for the next generation of electronic components and non-destructive evaluation techniques, marking a significant step forward in the application of quantum mechanics to real-world challenges.

Analytical presentation of symmetrized vibrational functions of high symmetry molecules for applications in molecular spectroscopy and chemical physics: Accurate prediction of the vibrational spectrum of GeH[formula omitted] up to 8000 cm−1 as a relevant application example

We report here the analytical description of vibrational wave functions being most important for applications in quantum mechanics, molecular and chemical physics as applied to the high symmetry XY4 (Td) molecules. A general method is discussed and, for the first time, all vibrational functions of such molecule type are presented in an analytical and symmetrized form for all polyads up to the polyad number N=4. To illustrate the relevant application of the derived results, an accurate prediction of all vibrational band centers up to 8000 cm−1 is made for five stable isotopologues MGeH4(M=76,74,73,72,70) of the germane molecule.

The Metaphysical Foundations of the Principle of Indifference

Abstract The arguments in favor of the Principle of Indifference fail to explain its fruitfulness in science. Using the recent metaphysical concept of Grounding, I devise an explanation that can justify a weak version of the principle and discuss an instance of its application in Quantum mechanics.

Quantum Evolution and Genetic Mutations

One of the important problems in quantum biology has been explained in this work. It is the main reason for genetic mutation, which is known as the origin of evolution in organisms. An approach to clarify the main reason for mutation is quantum mechanics. A brief history of genetics, new-Darwinism, and Mendelian inheritance from ancient times to the twentieth century has been presented. The meaning and location of chromosomes and genes in cells, DNA structure, the replication mechanism, the genetic code, tautomeric forms, and the application of quantum mechanics have been described. Finally, we present the importance of quantum tunnelling in playing a main role in the mutation.

Random vector representation of continuous functions and its applications in quantum mechanics

The relation between continuous functions and random vectors is revealed in the paper that the main meaning is described as, for any given continuous function, there must be a sequence of probability spaces and a sequence of random vectors where every random vector is defined on one of these probability spaces, such that the sequence of conditional mathematical expectations formed by the random vectors uniformly converges to the continuous function. This is a random vector representation of continuous functions, which is regarded as a bridge to be set up between real function theory and probability theory. By means of this conclusion, an interesting result about function approximation theory can be obtained. The random vector representation of continuous functions is of important applications in physics. Based on the conclusion, if a large proportion of certainty phenomena can be described by continuous functions and random phenomena can also be described by random variables or vectors, then any certainty phenomenon must be the limit state of a sequence of random phenomena. Then, in the approximation from a sequence of random vectors to a continuous function, the base functions are appropriately selected by us, and an important conclusion for quantum mechanics is deduced: classical mechanics and quantum mechanics are unified. Particularly, an interesting and very important conclusion is introduced as the fact that the mass point motion of a macroscopical object possesses a kind of wave characteristic curve, which is called wave-mass-point duality.

The altered Hermite matrix: implications and ramifications

<p>Matrix theory is essential for addressing practical problems and executing computational tasks. Matrices related to Hermite polynomials are essential due to their applications in quantum mechanics, numerical analysis, probability, and signal processing. Their orthogonality, recurrence relations, and spectral properties make them a valuable tool for both theoretical research and practical applications. From a different perspective, we introduced a variant of the Hermite matrix that incorporates triple factorials and demonstrated that this matrix satisfies various properties. By utilizing effective matrix algebra techniques, various algebraic properties of this matrix have been determined, including the product formula, inverse matrix and eigenvalues. Additionally, we extended this matrix to a more generalized form and derived several identities.</p>

Solutions of Inhomogeneous Multiplicatively Advanced ODEs and PDEs with a q‐Fredholm Theory and Applications to a q‐Advanced Schrödinger Equation

For q > 1, a new Green’s function provides solutions of inhomogeneous multiplicatively advanced ordinary differential equations (iMADEs) of form y(N)(t) − Ay(qt) = f(t) for t ∈ [0, ∞). Such solutions are extended to global solutions on ℝ. Applications to inhomogeneous separable multiplicatively advanced partial differential equations are presented. Solutions to a linear free forced q‐advanced Schrödinger equation are obtained, opening an avenue to applications in quantum mechanics. New q‐Mittag‐Leffler functions qEα,β and ΥN,p govern the allowable decay rate of the inhomogeneities f(t) in the above iMADE. This provides a refinement to standard distribution theory, as we show is necessary for this study of iMADEs. A q‐Fredholm theory is developed and related to the above approach. For f(t) whose antiderivatives provide eigenfuntions of the noncompact integral operator K below, we exhibit solutions of the iMADE. Examples are provided, including a certain class of Dirichlet series.

МОЛЕКУЛЯРНЫЙ ИОН ВОДОРОДА𝑯𝟐+. МАГНИТНЫЕ M1 ПЕРЕХОДЫ

The magnetic dipole transitions in the homonuclear molecular ion 𝐻2+are obtained for a range ofand L, vibrational and total orbital momentum quantum numbers, respectively. Calculations are performed in the nonrelativistic approximation. The effects of the ion spin structure on M1 transitions are also considered. Numerical calculations were carried out on the basis of “exponential” variational expansion. One of the simplest and most fully developed areas of application of quantum mechanics is the theory of atoms with one or two electrons. For hydrogen and hydrogen-like ions, calculations can be performed strictly in both Schrödinger's non-relativistic wave mechanics and Dirac's relativistic electron theory. The exact calculations are rigorous for an electron at a fixed Coulomb potential. Therefore, the hydrogen-like atom provides an excellent material for testing the validity of quantum mechanics. For such an atom, the correction terms that take into account the motion and structure of the atomic nucleus, as well as quantum electrodynamic effects, are small and can be calculated with great accuracy. Since the energy levels of hydrogen and hydrogen-like atoms can be experimentally studied with anamazing degree of accuracy, it turns out that it is possible to test the correctness of quantum electrodynamics to some extent accurately.