Harmonic Potential Research Articles

On the two-dimensional harmonic oscillator with an electric field confined to a circular box

Abstract We revisit the quantum-mechanical two-dimensional harmonic oscillator with an electric field confined to a circular box of impenetrable walls. In order to obtain the energy spectrum we resort to the Rayleigh-Ritz method with polynomial and Gaussian basis sets. We compare present results with those derived recently by other authors. We discuss the limits of large and small box radius and also do some calculations with perturbation theory.

Generalized Path Integral Energy and Heat Capacity Estimators of Quantum Oscillators and Crystals Using Harmonic Mapping.

Imaginary-time path integral (PI) is a rigorous tool to treat nuclear quantum effects in static properties. However, with its high computational demand, it is crucial to devise precise estimators. We introduce generalized PI estimators for the energy and heat capacity that utilize coordinate mapping. While it can reduce to the standard thermodynamic and centroid virial (CVir) estimators, the formulation can also take advantage of harmonic character of quantum oscillators and crystals to construct a coordinate mapping. The method is not applicable to fluids or systems with imaginary modes such as double-well potentials. This yields harmonically mapped averaging (HMA) estimators, with mappings that decouple (HMAc) or couple (HMAq) the centroid and internal modes. The HMAq is constructed with normal mode coordinates (HMAq-NM) with quadratic scaling of cost or harmonic oscillator staging (HMAq-SG) coordinates with linear scaling. The estimator performance is examined for a 1D anharmonic oscillator and a 3D Lennard-Jones crystal using path integral molecular dynamics (PIMD) simulation. The HMA estimators consistently provide more precise estimates compared to CVir, with the best performance obtained by HMAq-NM, followed by HMAq-SG, and then HMAc. We also examine the effect of anharmonicity (for AO), intrinsic quantumness, and Trotter number. The HMA formulation introduced assumes the availability of forces and Hessian matrix; however, an equally efficient finite difference alternative is possible when these derivatives are inaccessible. The remarkable improvement in precision offered by HMAq estimators provides a framework for efficient PI simulation of more challenging systems, such as those based on ab initio calculations.

Integral Representations of Solution in the Problem on Skew Incidence of a Surface Wave on the Straight Shoreline Water Wedge

In the linear approximation of the surface gravitational waves of small amplitude a classical model problem about the incursion of a surface wave under some angle on the shoreline is solved. The problem is formulated for the harmonic potential of velocity of the fluid in the 3D water wedge with the Robin-Steklov boundary condition on the free surface and with the no-flow condition along the normal on the bed of the water domain. Some critical comments about a known in the literature solution having a “non-physical” singularity of the logarithmic type on the coastal line are given. The asymptotics with respect to distance from the shoreline of the obtained solution, bounded on the edge, is constructed. The reflection coefficient of the wave reflected from the shoreline is calculated.

Superstatistical functions of modified shifted Morse potential for diatomic molecules

The application of the molecular potential model to the study of different thermodynamic systems has been a focus of interest in recent time due to its importance in molecular and chemical physics. Hence, the asymptotic iteration method and the Laguerre polynomials are employed to obtain the eigensolutions of the Schrödinger equation with the modified shifted Morse potential model. Vibrational energy results for different diatomic molecules have been presented numerically, and these results are compared with experimental data and other results from available literature. The variation of normalised wave function and probability density of modified shifted Morse potential for some selected diatomic molecules at the ground- and first-excited states have been considered. Results obtained show that the equally spaced sinusoidal wave curves and varying nodes depend on the level of electron concentrations in the diatomic molecules considered. Partition function and other superstatistical function expressions of modified shifted Morse potential for some diatomic molecules within the modified Dirac delta distribution are obtained, using the Poisson summation formula. The superstatistical plots show a high level of dependence on the temperature parameter, maximum vibrational quantum number and deformation parameter. The superstatistical functions for a deformation parameter of zero correspond to the conventional thermodynamic functions.

Back-decay of the muonic molecular ion (αμt)012+ resonantly formed in (tμ)1s+4HeH+ collision

Abstract Resonant formation of the muonic molecular ion ( α μ t ) 01 2 + in the collision of the muonic tritium atom with the helium hydride ion, ( t μ ) 1 s + 4 HeH + → [ ( α μ t ) 01 2 + , p , e e ] + , is proposed. The ( α μ t ) 01 2 + ion is formed in the rotational-vibrational state (0,1) as one of the two nuclei of the final complex. Back-decay process [ ( α μ t ) 01 2 + , p , e e ] K n + → ( t μ ) 1 s + 4 HeH K ′ n ′ + , where K and n ( K ′ and n ′ ) is the rotational and vibrational quantum number, respectively, is considered in detail and the corresponding decay widths for all possible rotational-vibrational ( K , n ) → ( K ′ , n ′ ) transitions are calculated. Numerous inelastic decay channels of ( α μ t ) 01 2 + are also presented and discussed. The possible fast resonant formation of the ( α μ t ) 01 2 + ion may provide an interesting tool for experimental studies of t ( α , γ ) 7 Li nuclear fusion, which is one of the most important reactions in Big-Bang nucleosynthesis models. Studying this reaction may prove important for solving the lithium problem.

Entropy production of active Brownian particles going from liquid to hexatic and solid phases

Due to its inherent intertwinement with irreversibility, entropy production is a prime observable to monitor in systems of active particles. In this numerical study, entropy production in the liquid, hexatic and solid phases of a two-dimensional system of active Brownian particles is examined at both average and fluctuation level. The trends of averages as functions of density show no singularity and marked changes in their derivatives at the hexatic-solid transition. Distributions show instead peculiar tail structures interpreted by looking at microscopic configurations. Particles in regions of low local order generate tail values according to different dynamical mechanisms: they move towards empty regions or bounce back and forth into close neighbours. The tail structures are reproduced by a simple single-particle model including an intermittent harmonic potential.

Accurate and Fast Estimation of the Continuum Limit in Path Integral Simulations of Quantum Oscillators and Crystals.

Convergence of imaginary-time path integral results requires a substantial number of beads (n) when quantum effects are significant. Traditional Trotter scaling approaches estimate the continuum limit (n → ∞) through extrapolation; however, their validity is restricted to the asymptotic domain of large n. We introduce an efficient extrapolation approach for quantum oscillators and crystals with harmonic character. The fitting function is inspired by the analytic solution of the harmonic oscillator (HO), or the Einstein crystal for solids. While the formulation is designed for first derivative properties (energy and pressure), its extension to second derivative properties (e.g., heat capacity) is straightforward. We apply the method to a one-dimensional HO and anharmonic oscillator (AO), as well as a three-dimensional Lennard-Jones crystal. Configurations are sampled using path integral molecular dynamics simulations in the canonical ensemble, at a low temperature of T = 0.1 (simulation units). Compared to Trotter extrapolation approaches, the new method demonstrates a substantial accuracy in estimating the continuum limit, using only a few simulations of relatively small system sizes. This efficiency significantly reduces computational cost, providing a powerful tool to facilitate computations for more complex and challenging systems, such as molecules and crystals modeled using ab initio methods.

Calculation of α-decay half-lives of 186−218Po isotopes and actinide nuclei using a new phenomenological potential

This paper is devoted to the calculations of alpha decay half-lives of [Formula: see text]Po [Formula: see text] isotopes and actinide nuclei between the atomic numbers Z = 90 and Z = 102, using an energy-dependent potential model. The [Formula: see text]-nucleus effective potential employed in this study includes a Coulomb potential, a centrifugal barrier potential, and a nuclear potential. The nuclear potential is chosen to be the energy-dependent singularized harmonic potential, typical of nuclear interactions. The role of the energy-dependent potential is very significant in this study. Indeed, when compared to the experimental data, our results are found to be improved when energy-dependent interaction potential is employed compared to when energy-independent interactions are used, especially for small values of the parameter [Formula: see text] as evidenced by the values of the root mean square deviation.

1,4-Dihydroquinone and its Derivatives: Vibrational Spectroscopic (FT-IR and Raman), Molecular Structure and Quantum Chemical Investigations

1,4-Dihydroquinone and its Derivatives: Vibrational Spectroscopic (FT-IR and Raman), Molecular Structure and Quantum Chemical Investigations

On the Definition of Velocity in Discrete-Time, Stochastic Langevin Simulations

We systematically develop beneficial and practical velocity measures for accurate and efficient statistical simulations of the Langevin equation with direct applications to computational statistical mechanics and molecular dynamics sampling. Recognizing that the existing velocity measures for the most statistically accurate discrete-time Verlet-type algorithms are inconsistent with the simulated configurational coordinate, we seek to create and analyze new velocity companions that both improve existing methods as well as offer practical options for implementation in existing computer codes. The work is based on the set of GJ methods that, of all methods, for any time step within the stability criteria correctly reproduces the most basic statistical features of a Langevin system; namely correct Boltzmann distribution for harmonic potentials and correct transport in the form of drift and diffusion for linear potentials. Several new and improved velocities exhibiting correct drift are identified, and we expand on an earlier conclusion that, generally, only half-step velocities can exhibit correct, time-step independent Maxwell–Boltzmann distributions. Specific practical and efficient algorithms are given in familiar forms, and these are used to numerically validate the analytically derived expectations. One especially simple algorithm is highlighted, and the ability of one of the new on-site velocities to produce statistically correct averages for a particular damping value is specified.

Out-of-time-ordered-correlators for the pure inverted quartic oscillator: classical chaos meets quantum stability

Out-of-time-ordered-correlators (OTOCs) have been suggested as a means to diagnose chaotic behavior in quantum mechanical systems. Recently, it was found that OTOCs display exponential growth for the inverted quantum harmonic oscillator, mirroring the fact that this system is classically and quantum mechanically unstable. In this work, I study OTOCs for the inverted anharmonic (pure quartic) oscillator in quantum mechanics, finding only oscillatory behavior despite the classically unstable nature of the system. For higher temperature, OTOCs seem to exhibit saturation consistent with a value of –2⟨x2⟩T ⟨p2⟩T at late times. I provide analytic evidence from the spectral zeta-function and the WKB method as well as direct numerical solutions of the Schrödinger equation that the inverted quartic oscillator possesses a real and positive energy eigenspectrum, and normalizable wave-functions.

Generalized Itô’s lemma and the stochastic thermodynamics of diffusion with resetting

Abstract Methods from the theory of stochastic processes are increasingly being used to extend classical thermodynamics to mesoscopic non-equilibrium systems. One characteristic feature of these systems is that averaging the stochastic entropy with respect to an ensemble of stochastic trajectories leads to a second law of thermodynamics that quantifies the degree of departure from thermodynamic equilibrium. A well known mechanism for maintaining a diffusing particle out of thermodynamic equilibrium is stochastic resetting. In its simplest form, the position of the particle instantaneously resets to a fixed position x 0 at a sequence of times generated from a Poisson process of constant rate r. Within the context of stochastic thermodynamics, instantaneous resetting to a single point is a unidirectional process that has no time-reversed equivalent. Hence, the average rate of entropy production calculated using the Gibbs–Shannon entropy cannot be related to the degree of time-reversal symmetry breaking. The problem of unidirectionality can be avoided by considering resetting to a random position or diffusion in an intermittent confining potential. In this paper we show how stochastic entropy production along sample paths of diffusion processes with resetting can be analyzed in terms of extensions of Itô’s formula for stochastic differential equations (SDEs) that include both continuous and discrete processes. First, we use the stochastic calculus of jump-diffusion processes to calculate the rate of stochastic entropy production for instantaneous resetting, and show how previous results are recovered upon averaging over sample trajectories. Second, we formulate single-particle diffusion in a switching potential as a hybrid SDE and develop a hybrid extension of Itô’s stochastic calculus to derive a general expression for the rate of stochastic entropy production. We illustrate the theory by considering overdamped Brownian motion in an intermittent harmonic potential. Finally, we calculate the average rate of entropy production for a population of non-interacting Brownian particles moving in a common switching potential. In particular, we show that the latter induces statistical correlations between the particles, which means that the total entropy is not given by the sum of the 1-particle entropies.

Electric quadrupole transitions of 98–112Ru atomic nuclei via conformable fractional Bohr Hamiltonian

In this paper, we investigate the energy spectra, wave functions and the [Formula: see text] transition rates for [Formula: see text]Ru atomic nuclei, using the conformable fractional Bohr Hamiltonian model. For the [Formula: see text]-part of the potential, the newly proposed Yukawa plus modified exponential potential is considered and the harmonic oscillator potential in [Formula: see text]-part, with [Formula: see text] fixed around [Formula: see text]. By using the conformable fractional Nikiforov–Uvarov method, energy spectra and wave functions are obtained analytically. The sensitivity of the potential parameters and the spectra with respect to the fractional order parameter is investigated. The normalized fractional energies and fractional electric quadrupole transition rates are compared with the available experimental predictions and those from existing theoretical studies. Comparisons are made at different values of the fractional derivative order. The results are presented across a broader range of values of [Formula: see text], providing a systematic analysis of how variations in this parameter influence the energy spectra, wave functions, and [Formula: see text] transition rates in Ru nuclei. Overall, the comparison of our results with experimental data shows the good accuracy of our model, especially when the fractional parameter goes to lower values. It can be concluded that the order of fractional derivative plays a crucial role in refining theoretical predictions of electric quadrupole transition rates, especially for transitions involving higher multipolarities.

Is the Cyclic Model of the Universe Possible in the Relativistic Theory of Gravitation?

For the flat FLRW model of Universe evolution in RTG a new model of Dark Energy is proposed. It is a global scalar field Φ with the quadratic potential. It ensures cosmological acceleration at the present time and a bounce at the latе times. At the contraction stage Kazner-like growing anisotropy of Riemannian metrics will break a mass-of- the-graviton bounce mechanism near the Big Bang in FLRW case. There is also noncyclic option, when small enough graviton-mass-terms are significant only at the end of expansion. After bounce, during next contraction epoch, an anisotropy grows and the matter density finally reaches the Planck one.

Effects of gravity rainbow on scalar bosonic and oscillator fields in Bonnor–Melvin space-time with a cosmological constant

In this paper, we explore the relativistic quantum motion of spin-zero scalar charge-free particles influenced by rainbow gravity’s (RG’s) in Bonnor–Melvin magnetic space-time, a four-dimensional solution featuring a positive cosmological constant. We solve the Klein–Gordon equation in this scenario by using two sets of rainbow function: (i) f(χ)=1(1-βχ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(\\chi )=\\frac{1}{(1-\\beta \\,\\chi )}$$\\end{document}, h(χ)=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h(\\chi )=1$$\\end{document} and (ii) f(χ)=1(1-βχ)=h(χ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(\\chi )=\\frac{1}{(1-\\beta \\,\\chi )}=h(\\chi )$$\\end{document}, where 0<χ=|E|Ep≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0 < \\chi \\left( =\\frac{|E|}{E_p}\\right) \\le 1$$\\end{document} with Ep\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_p$$\\end{document} being the Planck’s energy, E is the particle’s energy, and β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document} is the rainbow parameter. Furthermore, we study the relativistic quantum oscillator through the Klein–Gordon oscillator equation in the same Bonnor–Melvin magnetic space-time under RG effects. Employing the first pair of rainbow function, we obtain an approximate eigenvalue solution of the quantum oscillator fields. Notably, we demonstrate that the relativistic energy profile of scalar and oscillator particles are influenced by the topology of the geometry and the cosmological constant which is related with the magnetic field strength lies along the symmetry axis. Additionally, we see the impact of rainbow parameter on these approximate relativistic energy profiles in both quantum systems.

Role of Intermolecular Interactions in Deep Eutectic Solvents for CO2 Capture: Vibrational Spectroscopy and Quantum Chemical Studies.

Recent research and reviews on CO2 capture methods, along with advancements in industry, have highlighted high costs and energy-intensive nature as the primary limitations of conventional direct air capture and storage (DACS) methods. In response to these challenges, deep eutectic solvents (DESs) have emerged as promising absorbents due to their scalability, selectivity, and lower environmental impact compared to other absorbents. However, the molecular origins of their enhanced thermal stability and selectivity for DAC applications have not been explored before. Therefore, the current study focuses on a comprehensive investigation into the molecular interactions within an alkaline DES composed of potassium hydroxide (KOH) and ethylene glycol (EG). Combining Fourier transform infrared (FT-IR) and quantum chemical calculations, the study reports structural changes and intermolecular interactions induced in EG upon addition of KOH and its implications on CO2 capture. Experimental and computational spectroscopic studies confirm the presence of noncovalent interactions (hydrogen bonds) within both EG and the KOH-EG system and point to the aggregation of ions at higher KOH concentrations. Additionally, molecular electrostatic potential (MESP) surface analysis, natural bond orbital (NBO) analysis, quantum theory of atoms-in-molecules (QTAIM) analysis, and reduced density gradient-noncovalent interaction (RDG-NCI) plot analysis elucidate changes in polarizability, charge distribution, hydrogen bond types, noncovalent interactions, and interaction strengths, respectively. Evaluation of explicit and hybrid models assesses their effectiveness in representing intermolecular interactions. This research enhances our understanding of molecular interactions in the KOH-EG system, which are essential for both the absorption and desorption of CO2. The study also aids in predicting and selecting DES components, optimizing their ratios with salts, and fine-tuning the properties of similar solvents and salts for enhanced CO2 capture efficiency.

Rapidly rotating quantum droplets confined in a harmonic potential

We consider a “symmetric” quantum droplet in two spatial dimensions, which rotates in a harmonic potential, focusing mostly on the limit of “rapid” rotation. We examine this problem using a purely numerical approach, as well as a semianalytic Wigner-Seitz approximation (first developed by Baym, Pethick, and their co-workers) for the description of the state with a vortex lattice. Within this approximation we assume that each vortex occupies a cylindrical cell, with the vortex-core size treated as a variational parameter. Working with a fixed angular momentum, as the angular momentum increases and depending on the atom number, the droplet accommodates none, few, or many vortices, before it turns to center-of-mass excitation. For the case of a “large” droplet, working with a fixed rotational frequency of the trap Ω, as Ω approaches the trap frequency ω, a vortex lattice forms, the number of vortices increases, the mean spacing between them decreases, while the “size” of each vortex increases as compared to the size of each cell. In contrast to the well-known problem of contact interactions, where we have melting of the vortex lattice and highly correlated many-body states, here no melting of the vortex lattice is present, even when Ω=ω. This difference is due to the fact that the droplet is self-bound. For Ω=ω, the “smoothed” density distribution becomes a flat top, very much like the static unconfined droplet. When Ω exceeds ω, the droplet maintains its shape and escapes to infinity, via center-of-mass motion. Published by the American Physical Society 2024

Scorer beams in highly nonlocal media with a nonlinearity coefficient and an external potential

The Snyder-Mitchell model of accessible solitons is a simple model that reduces the dynamics of solitons in highly nonlocal nonlinear media to a linear dynamical system with harmonic potential. Utilizing this model in a system with a nonlinearity coefficient and an external potential generated in highly nonlocal media, we explore its solution by the methods of variable separation and self-similar transformation. We discover a special solution of the model that includes Scorer functions, for which reason we call it the Scorer beam. The transmission dynamics of the Scorer beam in strongly nonlocal nonlinear media is analytically and numerically investigated. Under the specific condition of applying an exponential truncation factor, the evolution of the Scorer beam is more stable and converges faster. We also find that the Scorer beam exhibits self-bending and self-healing characteristics. Our results provide theoretical and numerical guidance for generating Scorer beams that might prove useful for future experimental exploration.

Multi-hump solitons under fractional diffraction and inhomogeneous cubic nonlinearity in a quadratic potential

Multi-hump solitons under fractional diffraction and inhomogeneous cubic nonlinearity in a quadratic potential

Theoretical computation of thermodynamic functions of sodium dimer with modified shifted Morse potential

In this article, the asymptotic iteration method and the Laguerre polynomials are employed to obtain the eigensolutions of the Schrödinger equation with modified shifted Morse potential model. Vibrational energy results for sodium dimer have been presented numerically and these results are compared with experimental data from available literatures. With the help of the energy expression, the partition function and other thermodynamic function expressions of this potential model for sodium dimer are obtained, using the Euler MacLaurin’s formula. For various vibrational quantum numbers considered, the thermodynamic functions show high level of dependence on the temperature parameter. These results are justified by the numerical results presented for the thermodynamic functions of sodium dimer. Our results also agree with the available results in literatures and can also serve as a guiding reference in further studies concerning chemical physics and spectroscopy.