Application Of Quantum Mechanics Research Articles

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.

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 that is known as the origin of evolution in the organisms. An approach to clarify the main reason of mutation is quantum mechanics. A brief history of genetics, new-Darwinism and Mendelian inheritance from ancient time to the twentieth century has been presented. Meaning and location of chromosomes and genes in cells, DNA structure, the replication mechanism, genetic code, tautomeric form and application of quantum mechanics have been described. Finally, we present the importance of quantum tunnelling to play 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.

Correction: Applying quantum mechanics to deconvolute benchtop 1H NMR reaction data

Correction for ‘Applying quantum mechanics to deconvolute benchtop 1H NMR reaction data’ by Jiayu Zhang et al., React. Chem. Eng., 2024, https://doi.org/10.1039/D3RE00583F.

Applying quantum mechanics to deconvolute benchtop 1H NMR reaction data

Quantum mechanical model files developed from pure components enable deconvolution of complex 1H NMR reaction dataset at different field strength.

A Study of Traveling Wave Structures and Numerical Investigations into the Coupled Nonlinear Schrödinger Equation Using Advanced Mathematical Techniques

This article explores adapted mathematical methods to solve the coupled nonlinear Schrödinger (C-NLS) equation through analytical and numerical methods. To obtain exact solutions for the (C-NLS) equation, we utilize the improved modified, extended tanh-function method. By separating the Schrödinger equation into real and imaginary parts, we can obtain four coupled equations, which we then analyze using the generalized tanh method to extract exact solutions. This system of equations is essential for understanding the behavior of quantum systems and has various applications in quantum mechanics. We obtain an analytical solution and demonstrate numerical solutions using implicit finite difference. Studies have shown that this scheme is second-order in space and time, and the von Neumann stability analysis confirms its unconditional stability. We introduce the comparison between numerical and exact solutions.

Weak measurements and quantum-to-classical transitions in free electron–photon interactions

How does the quantum-to-classical transition of measurement occur? This question is vital for both foundations and applications of quantum mechanics. Here, we develop a new measurement-based framework for characterizing the classical and quantum free electron–photon interactions and then experimentally test it. We first analyze the transition from projective to weak measurement in generic light–matter interactions and show that any classical electron-laser-beam interaction can be represented as an outcome of weak measurement. In particular, the appearance of classical point-particle acceleration is an example of an amplified weak value resulting from weak measurement. A universal factor, exp left(-{Gamma }^{2}/2right), quantifies the measurement regimes and their transition from quantum to classical, where Gamma corresponds to the ratio between the electron wavepacket size and the optical wavelength. This measurement-based formulation is experimentally verified in both limits of photon-induced near-field electron microscopy and the classical acceleration regime using a DLA. Our results shed new light on the transition from quantum to classical electrodynamics, enabling us to employ the essence of the wave-particle duality of both light and electrons in quantum measurement for exploring and applying many quantum and classical light–matter interactions.

Semiclassical calculation of the pendulum period

Abstract In this paper, we calculate the swing period of the classical pendulum via semiclassical path-integration. We point out the significance of the classical periodic orbits and the equivalence of pendulum’s classical isochronism to the equidistance of the quantum energy levels. We derive the swing period in terms of the semiclassical tunneling time and the fractional revival period. A possible definition of a critical value for the quantum ‘bounce time’ is proposed. This paper intends for graduate students as an illustrating example of applying quantum mechanics to a classical system. It offers valuable insight into some characteristics that the classical and quantum pendulum possess in common. It also intends for a specialist in quantum chemistry where the quantum pendulum dynamics appears in what is known as hindered rotation about some chemical bonds.

An Extended Zeta Function with Applications in Model Building and Bayesian Analysis

In certain problems in model building and Bayesian analysis, the results end up in forms connected with generalized zeta functions. This necessitates the introduction of an extended form of the generalized zeta function. Such an extended form of the zeta function is introduced in this paper. In model building situations and in various types of applications in physical, biological and social sciences and engineering, a basic model taken is the Gaussian model in the univariate, multivariate and matrix-variate situations. A real scalar variable logistic model behaves like a Gaussian model but with a thicker tail. Hence, for many of industrial applications, a logistic model is preferred to a Gaussian model. When we study the properties of a logistic model in the multivariate and matrix-variate cases, in the real and complex domains, invariably the problem ends up in the extended zeta function defined in this paper. Several such extended logistic models are considered. It is also found that certain Bayesian considerations also end up in the extended zeta function introduced in this paper. Several such Bayesian models in the multivariate and matrix-variate cases in the real and complex domains are discussed. It is stated in a recent paper that “Quantum Mechanics is just the Bayesian theory generalized to the complex Hilbert space”. Hence, the models developed in this paper are expected to have applications in quantum mechanics, communication theory, physics, statistics and related areas.

Detect and Engage: A New American Way of War

The threat that paces American force planning is not from Russia, but rather from China. This is leading to a new way of warfare. The authors call it ‘detect and engage’. The confluence of artificial intelligence, applied quantum mechanics and satellite networking permits dispersed and diverse units and platforms to operate as a unified joint force across sea, air, land and space. Cyber operations, practically indifferent to location, further enable the United States to reduce reliance on geographically concentrated and exposed forces. Advanced technology is just one step in operationalising new forms of warfare. Others include anticipating adversary responses; building robust inventories of extended-range weapons; knowing when and how to conduct close-in operations; integrating and empowering defence allies; and maintaining US forward presence. Harnessing leading-edge technology to improve global US military effectiveness reflects neither an isolationist nor an insular policy, but rather one of ongoing engagement.

About the notion of eigenstates for C*-algebras and some application in quantum mechanics

This work is concerned with the notion of eigenstates for C*-algebras. After reviewing some basic and structural results, we explore the possibility of reinterpreting certain typical concepts of quantum mechanics (e.g., dynamical equilibrium states, ground states, gapped states, Fermi surfaces) in terms of (algebraic) eigenstates.

Some Properties of Hessenberg Matrices with Conditional Elements

Hessenberg matrices are a class of structured matrices that play a fundamental role in various areas of mathematics and scientific computing. These matrices have diverse applications in numerical analysis, optimization, graph theory, and quantum mechanics, among others. This paper provides the determinants of the conditional Hessenberg matrices whose elements are the generalized bi-periodic Fibonacci numbers. Moreover, we find the inverse of some special type matrices which help us to find the determinant of the conditional Hessenberg matrices.

A mild-slope formulation based on Weyl rule of association with application to coastal wave modelling

Weyl rule of association, proposed by Hermann Weyl for quantum mechanics applications (Weyl, 1931), can be used to associate between the dispersion relation of water waves and a non-local pseudo-differential operator. The central result of this study is that this operator correctly approximates the Dirichlet-to-Neumann operator derived for linear waves over a slowly varying bathymetry. This opens the door to a formal use of Weyl’s operational calculus, and consequently, allowing straightforward derivations and generalizations of water waves’ models over mild slopes. Specifically, within the framework of linear wave theory, the formulation based on Weyl rule of association provides a generalized mild-slope model which does not impose a limit on the spectral width. Most significantly, the mild-slope formulation based on Weyl rule of association allows to derive a general linear kinetic equation for which the widely used energy balance equation (the central equation of forecasting models such as SWAN and WAVEWATCH) serves as a special case. This result not only provides a formal link between the deterministic description (i.e., Euler equations) and the stochastic description (i.e., the energy balance equation), but also establishes the theoretical foundations for the statistical description of bathymetry-induced wave interferences. Such a statistical description is especially important over coastal waters, where through the interaction with the bathymetry, waves are rapidly scattered and tend to form focal zones and associated interference patterns.

Different forms for exact traveling wave solutions of unstable and hyperbolic nonlinear Schrödinger equations

In general, Schrödinger’s equations have lots of applications in quantum mechanics. Accordingly, in this paper, some new forms of the exact traveling wave solutions have been obtained for two different types of Schrödinger’s equations, namely the unstable generalized nonlinear Schrödinger’s equation and the hyperbolic nonlinear Schrödinger’s equation. In the literature one can see different forms of the solution by various existing methods. However, the applied methodology or obtained results are sometimes complicated in nature. So here to obtain the new and simple forms of the solutions for both of the equations we have used a very simple and important method known as sine-cosine method. In addition, the probability density functions (PDFs) for both of the considered problems have also been computed. To visualize the impact of the solutions, graphical representations have been made with respect to various parameters. From the results one can see, they are in terms of complex-valued functions. In special cases comparison has also been given. Furthermore, the results are computed and validated using the Maple software.

Enhanced nonlinear optomechanics in a coupled-mode photonic crystal device

The nonlinear component of the optomechanical interaction between light and mechanical vibration promises many exciting classical and quantum mechanical applications, but is generally weak. Here we demonstrate enhancement of nonlinear optomechanical measurement of mechanical motion by using pairs of coupled optical and mechanical modes in a photonic crystal device. In the same device we show linear optomechanical measurement with a strongly reduced input power and reveal how both enhancements are related. Our design exploits anisotropic mechanical elasticity to create strong coupling between mechanical modes while not changing optical properties. Additional thermo-optic tuning of the optical modes is performed with an auxiliary laser and a thermally-optimised device design. We envision broad use of this enhancement scheme in multimode phonon lasing, two-phonon heralding and eventually nonlinear quantum optomechanics.

Quantum chemistry simulations in an undergraduate project: tellurophenes as narrow bandgap semiconductor materials

The convenient graphical user-interfaces now available with advanced simulation software offer a powerful didactic tool for research-led teaching of methods in quantum chemistry and wider applications of quantum mechanics. In the student project work reported here, a homologous series of semiconducting chalcogenophenes (encompassing poly-thiophenes, poly-selenophenes and poly-tellurophenes) with varying polymer chain lengths were simulated in detail using density functional theory (DFT). Following geometry optimization, energy calculations reveal that increasing the length of the polymer chain (N) from a monomer to a hexamer leads to a narrowing and large-N convergence of the bandgap. It is found that hexa-tellurophene has significantly favourable electronic properties as compared to the other analogues, with a greatly enhanced electron affinity (−2.74 eV), and a corresponding bandgap energy of 2.18 eV, giving a superior matching to the solar spectrum.

Oscillatory path integrals for radio astronomy

We introduce a new method for evaluating the oscillatory integrals which describe natural interference patterns. As an illustrative example of contemporary interest, we consider astrophysical plasma lensing of coherent sources like pulsars and fast radio bursts in radio astronomy. Plasma lenses are known to occur near the source, in the interstellar medium, as well as in the solar wind and the earth’s ionosphere. Such lensing is strongest at long wavelengths, hence it is generally important to go beyond geometric optics and into the full wave optics regime. Our computational method is a spinoff of new techniques two of us, and our collaborators, have developed for defining and performing Lorentzian path integrals with applications in quantum mechanics, condensed matter physics, and quantum gravity. Cauchy’s theorem allows one to transform a computationally fragile and expensive, highly oscillatory integral into an exactly equivalent sum of absolutely and rapidly convergent integrals which can be evaluated in polynomial time. We require only that it is possible to analytically continue the lensing phase, expressed in the integrated coordinates, into the complex domain. We give a first-principles derivation of the Fresnel–Kirchhoff integral, starting from Feynman’s path integral for a massless particle in a refractive medium. We then demonstrate the effectiveness of our method by computing the detailed diffraction patterns of Thom’s caustic catastrophes, both in their “normal forms” and within a variety of more realistic, local lens models, for all wavelengths. Our numerical method, implemented in a freely downloadable code, provides a fast, accurate tool for modeling interference patterns in radioastronomy and other fields of physics.

Characterization of modulation spaces by symplectic representations and applications to Schrödinger equations

In the last twenty years modulation spaces, introduced by H. G. Feichtinger in 1983, have been successfully addressed to the study of signal analysis, PDE's, pseudodifferential operators, quantum mechanics, by hundreds of contributions. In 2011 M. de Gosson showed that the time-frequency representation Short-time Fourier Transform (STFT), which is the tool to define modulation spaces, can be replaced by the Wigner distribution. This idea was further generalized to τ-Wigner representations in [11].In this paper time-frequency representations are viewed as images of symplectic matrices via metaplectic operators. This new perspective highlights that the protagonists of time-frequency analysis are metaplectic operators and symplectic matrices A∈Sp(2d,R). We find conditions on A for which the related symplectic time-frequency representation WA can replace the STFT and give equivalent norms for weighted modulation spaces. In particular, we study the case of covariant matrices A, i.e., their corresponding WA are members of the Cohen class.Finally, we show that symplectic time-frequency representations WA can be efficiently employed in the study of Schrödinger equations. In fact, modulation spaces and WA representations are the frame for a new definition of wave front set, providing a sharp result for propagation of micro-singularities in the case of the quadratic Hamiltonians. This new approach may have further applications in quantum mechanics and PDE's.