Harmonic Oscillator Research Articles

A quadrupole oscillator as an ingtegrable model

A quadrupole oscillator is presented as an integrable model in the Born-Oppenheimer formalism with an electronic Hamiltonian being the quadrupole tensor. The electronic states of present concern are associated with a doubly degenerate positive eigenvalue of the electronic Hamiltonian, and accordingly the nuclear Hamiltonian takes a 2×2 matrix form. While the potential function for nuclear motion is proportional to r2, the kinetic energy operator is rather complicated, containing coupling terms with a Berry connection through adiabatic approximation. The energy eigenvalues, which receive a modification by a Chern number, get closer to those for the 3D isotropic harmonic oscillator if the angular momentum quantum number becomes sufficiently large.

Design of a lightweight broadband vibration reduction structure with embedded acoustic black holes in viscoelastic damping materials

Viscoelastic damping materials (VDMs) are valued for their high damping characteristics in vibration and noise control. However, they typically underperform at lower frequencies and add substantial mass to structures. This study introduces an innovative approach by embedding an acoustic black hole (ABH) structure within VDMs (ABH-VDM) to achieve lightweight and broadband vibration damping. Firstly, a finite element method-based vibration model is developed to analyse the propagation and attenuation characteristics of vibrations in a plate strip embedded with ABH-VDM. This analysis provides insights into the dynamic behaviour and damping effectiveness of the proposed structure. Secondly, the study investigates the vibration reduction capabilities and mass implications of ABH-VDM on large-scale plate structures. The influence of ABH structural parameters, including the power exponent, cutoff thickness, and array configuration, is systematically investigated to optimize damping performance. Finally, experimental validation confirms that ABH-VDM achieves an additional 1.4 dB reduction in vibration across the entire frequency spectrum, with a bandwidth extension of 900 Hz. Moreover, ABH-VDM reduces mass by 8.6 %, demonstrating its potential for lightweight vibration control in structural applications. This research contributes valuable insights into advancing lightweight and broadband damping solutions for enhanced vibration management in engineering systems.

Effect of microstructure on dynamic response of ferrite matrixed steel at high strain rates

Dynamic response of steels at high strain rates are of great practical significance for understanding mechanical structures under extreme load conditions. To explore the relationship between structure and stress during dynamic deformation at high strain rates, steel specimens were prepared with varied microstructures of ferrite, ferrite and pearlite, and ferrite and spherical carbide. It is revealed that different microstructured steels exhibit drastically different characteristic of waves excited by the instantaneous impact of a servo-hydraulic tensile testing. It is also observed that a beat frequency response exists under dynamic deformation at certain high strain rates. It is therefore proposed that dynamic plastic deformation starts with stress fluctuation with a dislocation damped vibration model, and two successively excited waves are superimposed to form the beat frequency response.

Electromechanical characteristics analysis of L-shaped pipelines with enhanced active constrained damping treatment

Abstract Fluid-conveying pipelines are widely employed in various engineering fields, such as aerospace, nuclear, and marine fields. These pipelines work in serious vibration environments, which can quickly damage the pipeline system. The vibration control of pipelines is a prominent challenge in the engineering field. This paper is aimed to investigate the electromechanical analysis characteristics of L-shaped pipelines with the Enhanced Active Constrained Layer Damping (EACLD) structure. A finite element model of the L-shaped pipeline with EACLD is established. The dynamic behaviour of an L-shaped pipeline with an EACLD structure was analysed in both the time and frequency domains. The influence of the voltage and the position, the length and the elastic modulus, the thickness and the edge element parameters are all considered. Additionally, the influence of the EACLD patch orientation on static displacement and stress are considered. Simulation results indicate that reasonable selection of the parameters for the EACLD patch and edge element can enhance vibration damping effectiveness, which can provide effective design guidance for active vibration control of the pipeline system. 

A Polarization‐Insensitive and Adaptively‐Blazed Meta‐Grating Based on Dispersive Metasurfaces

AbstractThe diffraction efficiency of blaze gratings is optimized only at a specific frequency due to a fixed blaze angle, resulting in reduced and variable diffraction efficiencies over the working frequency band. Additionally, blazed gratings demonstrate polarization dependence due to their groove structures and the interaction of light with their surfaces. Consequently, designing gratings with constant diffraction efficiencies across a wide frequency bandwidth while maintaining polarization independence remains a challenge. Here, a design paradigm of dispersion engineerable meta‐grating inspired by orthogonal harmonic oscillations (OHO) is presented. Utilizing the OHO model, the phase dispersion of a metasurface can be precisely controlled, which applies to any unit cell featuring two orthogonal electromagnetic resonances. As a proof of concept, a polarization‐insensitive meta‐grating is showcased, where the blazed angle adapts with the incident frequency, ensuring broadband performance. In the experiment, the adaptively‐blazed grating measured an optimized and constant diffraction efficiency of ≈80% over the working wavelength range, i.e., 8.7–12.2 µm. The difference in diffraction efficiency between the two perpendicular linear polarization states remains within 4.6%. The proposed paradigm paves the way for meta‐device design based on precise dispersion engineering, which has potential applications in spectrometers, broadband beam forming and steering, hyperspectral imaging, etc.

On a Repulsive Short-Range Potential Influence on the Harmonic Oscillator

On a Repulsive Short-Range Potential Influence on the Harmonic Oscillator

Spectroscopic properties under vibrational strong coupling in disordered matter from path-integral Monte Carlo simulations.

Vibrational strong coupling (VSC), the strong coupling between a Fabry-Perrot cavity and molecular vibrations at mid-infrared frequencies, has received important attention in the last years due to its capacity of modifying both vibrational spectra and chemical reactivity. VSC is a collective effect, and in this work, we introduce Path Integral Monte Carlo (PIMC) simulations that not only take into account the quantum character of the molecular vibrations and of the optical resonance of the cavity but also reproduce this collective behavior by considering multiple replicas of the molecular system. Moreover, we show that it is possible to extract from the PIMC simulations the decomposition of the hybrid optical and molecular states in terms of the bare molecular modes. On a model system of an ensemble of disordered Morse oscillators coupled to a single cavity through the Pauli-Fierz Hamiltonian, PIMC can retrieve known features obtained from analytical modes such as the Tavis-Cummings model and obtain a very close agreement with exact diagonalization for a small number of Morse oscillators. We also find that notwithstanding the anhamonic character of the Morse oscillators, the collective mode coupled to the cavity behaves as a harmonic oscillator, following the quantum central limit theorem.

The hydrogen atom perturbed by a 1-dimensional Simple Harmonic Oscillator (1d-SHO) potential

Abstract The hydrogen atom perturbed by a constant 1-dimensional weak quadratic potential λz2 is solved at first-order perturbation theory using the eigenstates of the total angular momentum operator - the coupled basis. Physical applications of this result could be found, for example, in the study of a quadratic Zeeman effect weaker than fine-structure effects, or in a perturbation caused by instantaneous generalised van der Waals interactions.

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.

Effects of conical singularity and Wu-Yang magnetic monopole on thermodynamic properties of harmonic oscillator interacting with potential

Abstract This paper investigates the thermodynamic properties of the harmonic oscillator system in a conical singularities background. Moreover, we incorporate a Wu-Yang magnetic monopole (WYMM) and inverse square potential into the quantum system and analyze the resulting effects. Using analytical methods, we derive the energy eigenvalues and study the thermodynamic properties, such as the partition function, Helmholtz free energy, Gibbs's free energy and specific heat capacity at finite temperature $T \neq 0$. Our findings demonstrate that these physical parameters are influenced by the topological parameter, strength of WYMM and potential parameters. These results provide valuable insights into the interplay between thermodynamic properties of a quantum mechanical problem and the gravitational effects.

Similarity among quantum-mechanical states: Analysis and applications for central potentials

Abstract The similarity of quantum-mechanical solutions for central potentials is analytically determined and numerically explored for arbitrary dimensionalities. The study here provided focuses on hydrogenic systems and the harmonic oscillator, in respective non-relativistic frameworks. A diversity of analytical expressions for the Quantum Similarity Measure (QSM) and Index (QSI) are provided. Relevant conclusions are derived from the analyses grounded on state quantum numbers, space dimensionality and on the role played by the main characteristic parameters of these systems, namely the nuclear charge in the hydrogenic case, and the angular frequency for the oscillator. For this purpose, a statistical analysis of the QSI values has been performed for a large number both of states and combinations of them in each system. Considering the factorization of QSI into a radial and an angular part, particular attention is paid to the individual contribution of each part in both systems.

Inverted oscillator: quantum discrete spectrum

Abstract We use the invariant operator method to investigate the quantum characteristics of the inverted oscillator. We introduce a unitary transformation that maps a time-dependent general Hermitian linear invariant operator to a time-independent one, which describes a harmonic oscillator with a unit mass and a constant frequency. Our investigation includes three distinguished cases: negative frequency, zero frequency, and positive frequency. Our most interesting result concerns the last case. For the case of negative frequency, we propose a new inner product by the introduction of the new metric operator η. Coherent state for this case will be constructed.

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.

Effects of nanodiamonds on the mechanical properties, acoustic properties, vibration damping, and flame retardancy of ramie-fiber epoxy composite panels used for indoor applications

This research aims to systematically analyze the impact of nanodiamonds (NDs) in ramie fiber-reinforced epoxy resin acoustic panels used for indoor applications. The panels (four variants) are fabricated using six layers of ramie fiber with various weight percentages of NDs (0.0, 0.2, 0.4, and 0.6) in epoxy resin by the vacuum resin infusion process (VRIP). Mechanical characterization (tensile and flexural strength) was carried out using Instron 8801 universal testing machine. The impedance tube-based sound absorption coefficient (SAC) test was performed at low frequency and higher frequency, respectively. The void content in the panel is evaluated using a micro-x-ray computed tomography (CT) scanner. The dispersion of nanoparticles is studied using scanning electron microscopy (SEM) images. The vertical flame-retardant test and the horizontal flame-retardant test were conducted according to the UL-94 standard. The 0.4 wt.% NDs sample showed good tensile strength (59.178 MPa) and flexural strength (69.134 MPa) compared with other panels. The panels showed differences in SAC values for different weight percentages of NDs, both at lower and higher frequencies. The 0.6 wt.% NDs sample attained the maximum SAC of 0.906 at 6000 Hz in the high-frequency and 0.34 at 1112 Hz in the low-frequency. The SEM reveals an even distribution of NDs in both the 0.2 and 0.4 wt.% NDs panels, while agglomerates form in the 0.6 wt.% panels. The CT scan result shows void content both internally and externally in the panels. There are no voids in the panels where NDs are uniformly distributed. The synergistic effect of NDs in epoxy resin forms a char over the burnt surfaces of samples, enhancing the flame-retardant properties of the panels. According to this study, adding NDs to ramie fiber epoxy composites has improved the panels’ necessary qualities, making them suitable for use as indoor acoustic conditioning materials.

Supersymmetric Quesne-Dunkl Quantum Mechanics on Radial Lines

Quantum deformations offer valuable perspectives into quantum mechanics, particularly by advancing our understanding of symmetry and algebraic structures.The Dunkl oscillator, which integrates Dunkl operators into the harmonic oscillator framework, advances the system’s algebraic properties and opens new approaches for exploring quantum phenomena. Supersymmetric quantum mechanics (SSQM) unifies bosonic and fermionic aspects and facilitates the construction of solvable models using generalized Dunkl operators. This paper introduces a new approach to the Dunkl oscillator, employing a complex reflection operator to deepen the understanding of its connection to Hermite polynomials on radial lines. The results offer new perspectives on the Dunkl oscillator’s algebraic structure and its relevance to SSQM and quantum deformation theory, expanding the potential for discovering solvable quantum models.

Two-dimensional simulation of generalized thermoelastic damping in vibrations of strain gradient beam resonators

Two-dimensional simulation of generalized thermoelastic damping in vibrations of strain gradient beam resonators

Quantum instability and Ehrenfest time for an inverted harmonic oscillator

We use out-of-time order correlators (OTOCs) to investigate the quantum instability and Ehrenfest time for an inverted harmonic oscillator (IHO). For initial states located in the stable manifolds of the IHO we find that the corresponding OTOC exhibits identical evolutionary characteristics to the saddle point before the Ehrenfest time. For initial states located in the unstable manifolds, the OTOCs still grow exponentially but the time to maintain exponential growth is related to the center position of its wave packet in phase space. Moreover, we use the Husimi Q function to visualize the quantum wave packets during exponential growth of the OTOCs. Our results show that quantum instability exists at arbitrary orbits in the IHO system, and the Ehrenfest time in the IHO system depends not only on the photon number of the initial system but also on the central positions of the initial states in phase space.

Motional-state analysis of a trapped ion by ultranarrowband composite pulses

In this work, we present a method for measuring the motional state of a two-level system coupled to a harmonic oscillator. Our technique uses ultranarrowband composite pulses on the blue sideband transition to scan through the populations of the different motional states. Our approach does not assume any previous knowledge of the motional state distribution and is easily implemented. It is applicable both inside and outside of the Lamb-Dicke regime. For higher phonon numbers especially, the composite pulse sequence can be used as a filter for measuring phonon number ranges. We demonstrate this measurement technique using a single trapped ion and show good detection results with the numerically evaluated pulse sequence. Published by the American Physical Society 2024

Enhancing Eyringpy: Accurate Rate Constants with Canonical Variational Transition State Theory and the Hindered Rotor Model.

The recrossing effect and hindered rotations can lead to significant inaccuracies in rate constant calculations using transition state theory and the harmonic oscillator approximation. To address these issues, we enhanced Eyringpy, a Python-based computational tool, by integrating the canonical variational transition state theory (CVT) and the hindered rotor model, effectively mitigating the limitations of traditional methods. CVT rate constants are calculated using electronic structure data from nonstationary points along the minimum-energy path. Additionally, we developed an algorithm based on reaction force analysis to autonomously select relevant nonstationary points. Torsions are modeled using one-dimensional hindered rotor approaches proposed by Pitzer-Gwinn and Ayala-Schlegel.

An Experimental Study on Vortex-Induced Vibration Suppression of a Long Flexible Catenary Cable by Using Vibration Dampers

In this paper, an experimental study is conducted to investigate the effectiveness of vibration dampers in suppressing vortex-induced vibration in a long, flexible catenary cable with a low mass ratio. The dampers, consisting of two small, symmetric, lightweight pipes clamped to the cable, are sparsely deployed along the cable to shape the vibration characteristics. The experimental results demonstrate that dampers significantly reduce the vibration amplitude by up to 60% and axial tension by up to 61% at high flow velocities, effectively suppressing the cable vibration in perpendicular flow. In addition, it is observed that the in-line and cross-flow vibration frequencies are approximately equal when the dampers are applied. This behavior contrasts with the conventional undamped catenary cable, where the in-line vibration frequencies are double those of the cross-flow frequencies.