Simple Harmonic Oscillator Research Articles

A higher-dimensional model of the nucleon-nucleon central potential

Based on a theory of extra dimensional confinement of quantum particles [E. R. Hedin, Physics Essays, 2012, 25(2): 177], a simple model of a nucleon-nucleon (NN) central potential is derived which quantitatively reproduces the radial profile of other models, without adjusting any free parameters. It is postulated that a higher-dimensional simple harmonic oscillator confining potential localizes particles into three-dimensional (3D) space, but allows for an evanescent penetration of the particles into two higher spatial dimensions. Producing an effect identical with the relativistic quantum phenomenon of zitterbewegung, the higher-dimensional oscillations of amplitude ħ/(mc) can be alternatively viewed as a localized curvature of 3D space back and forth into the higher dimensions. The overall spatial curvature is proportional to the particle’s extra-dimensional ground state wave function in the higher-dimensional harmonic confining potential well. Minimizing the overlapping curvature (proportional to the energy) of two particles in proximity to each other, subject to the constraint that for the two particles to occupy the same spatial location one of them must be excited into the 1st excited state of the harmonic potential well, gives the desired NN potential. Specifying only the nucleon masses, the resulting potential well and repulsive core reproduces the radial profile of several published NN central potential models. In addition, the predicted height of the repulsive core, when used to estimate the maximum neutron star mass, matches well with the best estimates from relativistic theory incorporating standard nuclear matter equations of state. Nucleon spin, Coulomb interactions, and internal nucleon structure are not considered in the theory as presented in this article.

$\mathcal{PT}$ -symmetric nonlinear metamaterials and zero-dimensional systems

A one dimensional, parity-time (${\cal PT}$)-symmetric magnetic metamaterial comprising split-ring resonators having both gain and loss is investigated. In the linear regime, the transition from the exact to the broken ${\cal PT}$-phase is determined through the calculation of the eigenfrequency spectrum for two different configurations; the one with equidistant split-rings and the other with the split-rings forming a binary pattern (${\cal PT}$ dimer chain). The latter system features a two-band, gapped spectrum with its shape determined by the gain/loss coefficient as well as the inter-element coupling. In the presense of nonlinearity, the ${\cal PT}$ dimer chain with balanced gain and loss supports nonlinear localized modes in the form of novel discrete breathers below the lower branch of the linear spectrum. These breathers, that can be excited from a weak applied magnetic field by frequency chirping, can be subsequently driven solely by the gain for very long times. The effect of a small imbalance between gain and loss is also considered. Fundamendal gain-driven breathers occupy both sites of a dimer, while their energy is almost equally partitioned between the two split-rings, the one with gain and the other with loss. We also introduce a model equation for the investigation of classical ${\cal PT}$ symmetry in zero dimensions, realized by a simple harmonic oscillator with matched time-dependent gain and loss that exhibits a transition from oscillatory to diverging motion. This behaviour is similar to a transition from the exact to the broken ${\cal PT}$ phase in higher-dimensional ${\cal PT}-$ symmetric systems. A stability condition relating the parameters of the problem is obtained in the case of piecewise constant gain/loss function that allows for the construction of a phase diagram with alternating stable and unstable regions.

The stochastic dynamics of a nanobeam near an optomechanical resonator in a viscous fluid

We quantify the Brownian driven, stochastic dynamics of an elastic nanobeam immersed in a viscous fluid that is partially wrapped around a microdisk optical resonator. This configuration has been proposed as an optomechanical and nanoscale analog of the atomic force microscope [Srinivasan et al., Nano Lett. 11, 791 (2011)]. A small gap between the nanobeam and microdisk is necessary for the optomechanical transduction of the mechanical motion of the nanobeam. We compute the stochastic dynamics of the nanobeam in fluid for the precise conditions of the laboratory using deterministic finite element simulations and the fluctuation dissipation theorem. We investigate the dynamics of a nanobeam in water and in air and quantify the significance of the fluid-solid interaction between the nanobeam and the optical resonator. Our results in air show that, despite the complex geometry of the nanobeam, it can still be represented approximately as a damped simple harmonic oscillator. On the other hand, when the nanobeam is immersed in water there are significant deviations from the dynamics of a simple harmonic oscillator. The small gap between the nanobeam and the microdisk is found to be a significant source of additional dissipation. In air, the quality factor of the mechanical oscillation of the nanobeam is reduced by an order of magnitude due to the presence of the microdisk, however, the dynamics remain underdamped even in the presence of the microdisk. On the other hand, when placed in water, the dynamics without the microdisk is underdamped and with the microdisk the dynamics become strongly over damped.

Symplectic semiclassical wave packet dynamics

The paper gives a symplectic-geometric account of semiclassical Gaussian wave packet dynamics. We employ geometric techniques to ‘strip away’ the symplectic structure behind the time-dependent Schrödinger equation and incorporate it into semiclassical wave packet dynamics. We show that the Gaussian wave packet dynamics is a Hamiltonian system with respect to the symplectic structure, apply the theory of symplectic reduction and reconstruction to the dynamics, and discuss dynamic and geometric phases in semiclassical mechanics. A simple harmonic oscillator example is worked out to illustrate the results. We show that the reduced semiclassical harmonic oscillator dynamics is completely integrable by finding the action–angle coordinates for the system, and calculate the associated dynamic and geometric phases explicitly. We also propose an asymptotic approximation of the potential term that provides a practical semiclassical correction term to the approximation by Heller. Numerical results for a simple one-dimensional example show that the semiclassical correction term realizes a semiclassical tunneling.

Novel frequency masking curves for noise-robust automatic speech recognition

We investigate the incorporation of frequency masking curves in the feature extraction module of automatic speech recognition systems to improve noise robustness. Frequency-masking curves are mathematically derived based on an auditory model, in which a basilar membrane is modeled as a cascade system of damped simple harmonic oscillators. Based on the analysis of the motion under speech signals, we derive the relationship between the amplitudes of neighboring oscillators and convert it into frequency-masking curves, which are used in the computation of spectral-masking thresholds to modify the speech spectrum. Evaluated on the Aurora 2.0 noisy-digit speech database, the proposed methodology achieves a significant improvement in noise-robustness.

Decomposition of the total momentum in a linear dielectric into field and matter components

The long-standing resolution of the Abraham–Minkowski electromagnetic momentum controversy is predicated on a decomposition of the total momentum of a closed continuum electrodynamic system into separate field and matter components. Using a microscopic model of a simple linear dielectric, we derive Lagrangian equations of motion for the electric dipoles and show that the dielectric can be treated as a collection of stationary simple harmonic oscillators that are driven by the electric field and produce a polarization field in response. The macroscopic energy and momentum are defined in terms of the electric, magnetic, and polarization fields that travel through the dielectric together as a pulse of electromagnetic radiation. We conclude that both the macroscopic total energy and the macroscopic total momentum are entirely electromagnetic in nature for a simple linear dielectric in the absence of significant reflections.

About a class of nonlinear oscillators with amplitude-independent frequency

This work is concerned with nonlinear oscillators that have a fixed, amplitude-independent frequency. This characteristic, known as isochronicity/isochrony, is achieved by establishing the equivalence between the Lagrangian of the simple harmonic oscillator and the Lagrangian of conservative oscillators with a position-dependent coefficient of the kinetic energy, which can stem from their mass that changes with the displacement or the geometry of motion. Conditions under which such systems have an isochronous center in the origin are discussed. General expressions for the potential energy, equation of motion as well as solutions for a phase trajectory and time response are provided. A few illustrative examples accompanied with numerical verifications are also presented.

Simulating quantum transport for a quasi-one-dimensional Bose gas in an optical lattice: the choice of fluctuation modes in the truncated Wigner approximation

We study the effect of quantum fluctuations on the dynamics of a quasi-one-dimensional Bose gas in an optical lattice at zero temperature using the truncated Wigner approximation with a variety of basis sets for the initial fluctuation modes. The initial spatial distributions of the quantum fluctuations are very different when using a limited number of plane-wave (PW), simple-harmonic-oscillator (SHO) and self-consistently determined Bogoliubov (SCB) modes. The short-time transport properties of the Bose gas, characterized by the phase coherence in the PW basis, are distinct from those gained using the SHO and SCB basis. The calculations using the SCB modes predict greater phase decoherence and stronger number fluctuations than the other choices. Furthermore, we observe that the use of PW modes overestimates the extent to which atoms are expelled from the core of the cloud, while the use of the other modes only breaks the cloud structure slightly, which is in agreement with the experimental observations by Fertig et al (2005 Phys. Rev. Lett. 94 120403).

Investigation into the Gaussian time-dependence of the rate coefficient in dispersive kinetic models applied to simple gas-phase chemical reactions

Dispersive kinetic models (DKMs) based on the concept of a distribution of activation energies have shown much promise in describing complex reaction transients. In particular, DKMs exhibiting a time (t)-dependent rate coefficient, k, of approximately Gaussian functional form provide the most insight into the kinetics because they do not rely on unit-less, empirical parameters [P.J. Skrdla, J. Phys. Chem. A 113, 9329 (2009)]. The physical basis of the Gaussian behaviour is investigated using a dispersive simple harmonic oscillator (DSHO) description of Marcus-type reactions studied on the ultrafast timescale. The DSHO description is found to support the Gaussian t-dependence of k over relatively short time periods. Moreover, in cases where the transition state survives more than one vibration of the predominant reactive mode, a periodicity in k is observed that is also consistent with empirical observations.

A maximum entropy thermodynamics of small systems

We present a maximum entropy approach to analyze the state space of a small system in contact with a large bath, e.g., a solvated macromolecular system. For the solute, the fluctuations around the mean values of observables are not negligible and the probability distribution P(r) of the state space depends on the intricate details of the interaction of the solute with the solvent. Here, we employ a superstatistical approach: P(r) is expressed as a marginal distribution summed over the variation in β, the inverse temperature of the solute. The joint distribution P(β, r) is estimated by maximizing its entropy. We also calculate the first order system-size corrections to the canonical ensemble description of the state space. We test the development on a simple harmonic oscillator interacting with two baths with very different chemical identities, viz., (a) Lennard-Jones particles and (b) water molecules. In both cases, our method captures the state space of the oscillator sufficiently well. Future directions and connections with traditional statistical mechanics are discussed.

Bias Sources and Corrections in TEOM Technology

Researchers at the National Institute for Occupational Safety and Health have used tapered element oscillating microbalances (TEOM) in laboratory settings. The TEOMs have assessed the mass concentration of laboratory-generated particulates in experimental dust chambers and provided a reference method for comparison with other particulate-measuring instruments. Current NIOSH research is focused on further adapting TEOM technology as a wearable personal dust monitor (PDM) for coal mine workers. The history of TEOM technology describes the oscillating tapered tube mathematically as a simple harmonic oscillator. However, analysis of the new PDM test data showed a bias dependency on the starting frequency f o. This result prompted a rigorous investigation to uncover the source of the bias and if the bias source is applicable to the 1400 TEOM. Based on the above results, a significantly improved theoretical description of TEOM performance has been developed. Average bias for each group of PDMs is calculated and compared to the results of the accuracy tests performed. Accompanying these biases are estimates of the average bias spans of the new PDMs in comparison to the pre-commercial PDMs. The theory was also applied to the Model 1400 TEOM data to evaluate whether there is agreement. The new theory of TEOM operation provides a good account for both the bias and bias span. Given that TEOM technology has been used for decades around the world to monitor atmospheric particulate contaminants as well as many other aerosols, quantification and correction of this source of bias should result in more accurate assessments.

An Inductive Approach to Simulating Multispectral MODIS Surface Reflectance Time Series

In this letter, a first-order Moderate Resolution Imaging Spectroradiometer time-series simulator, which uses a colored simple harmonic oscillator, is proposed. The simulated data can be used to augment data sets so that data intensive classification and change detection algorithms can be applied without enlarging the available ground truth data sets. The simulator's validity is tested by simulating data sets of natural vegetation and human settlement areas and comparing it with the ground truth data in Gauteng province located in South Africa. The difference found between the real and simulated data sets, which is reported in the experiments, is negligent. The simulated and real-world data sets are compared by using a wide selection of class and pixel metrics. In particular, the average temporal Hellinger distance between the real and simulated data sets is 0.2364 and 0.2269 for the vegetation and settlement classes, respectively, whereas the average parameter Hellinger distance is 0.1835 and 0.2554, respectively.

Lie groups and quantum mechanics

Mathematical modelling should present a consistent description of physical phenomena. We illustrate an inconsistency with two Hamiltonians—the standard Hamiltonian and an example found in Goldstein—for the simple harmonic oscillator and its quantisation. Both descriptions are rich in Lie point symmetries and so one can calculate many Jacobi Last Multipliers and therefore Lagrangians. The Last Multiplier provides the route to the resolution of this problem and indicates that the great debate about the quantisation of dissipative systems should never have occurred.

Periodic relative orbits of two spacecraft subject to differential gravity and electrostatic forcing

Coulomb forces between charged close-flying satellites can be used for formation control, and constant electric potentials enable static equilibria solutions. In this work, open-loop time-varying potential functions, which produce periodic, two-craft, Coulomb formation motions are demonstrated for the first time. This is done in the rotating Hill-Frame, with linearized gravity, and craft position components assumed in the form of simple harmonic oscillators. Substitution of the oscillatory functions into the dynamics, further constrains these functions, and yields necessary potential histories, to produce the periodic flow. The assumed position functions, however, are not arbitrary, since the dynamical model restricts what oscillatory trajectories are allowed. Specifically, a Hill-Frame integral of motion is derived, and this is used to show certain candidate periodic functions to be inadmissible. The system dynamics are then linearized to expose stability properties of the solutions, and it is established that asymptotic stability is impossible for all orbit families. Finally, the degree of instability in the assumed motions, over free parameter ranges, is determined numerically via the Floquet multipliers of the associated full-cycle state-transition matrices.

On a hidden symmetry of quantum harmonic oscillators

We consider a six-parameter family of the square integrable wave functions for the simple harmonic oscillator, which cannot be obtained by the standard separation of variables. They are given by the action of the corresponding maximal kinematical invariance group on the standard solutions. In addition, the phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications are briefly discussed. A visualization of the Heisenberg uncertainty principle is presented.

Plasmon resonances and plasmon‐induced charge transport in linear atomic chains

In linear hydrogen atomic chains, plasmon resonances and plasmon‐induced charge transport are studied by time‐dependent density functional theory. For the large linear chain, it is a general phenomenon that, in the longitudinal excitation, there are high‐energy resonances and a large low‐energy resonance. The energy of the large low‐energy resonance conforms to the results calculated by the classical Drude model. In order to explain the formation mechanism of the high‐energy resonances, we present a simple harmonic oscillator model. This model may reasonably account for the relationship between low‐energy and high‐energy resonances, and has a certain degree of universality. As the interatomic distance decreases, the current shows a gradual transition from insulator to metal. The current enhancement mainly depends on the local field enhancement associated with plasmon excitation, and the enhanced electron delocalization effect as a result of the decrease of the interatomic distance. © 2013 Wiley Periodicals, Inc.

Coupled second-quantized oscillators

Second quantization is a powerful technique for describing quantum mechanical processes in which the number of excitations of a single particle is not conserved. A textbook example of second quantization is the presentation of the simple harmonic oscillator in terms of creation and annihilation operators, which, respectively, represent addition or removal of quanta of energy from the oscillator. Our aim in this article is to bolster this textbook example. Accordingly, we explore the physics of coupled second-quantized oscillators. These explorations are phrased as exactly solvable eigenvalue problems, the mathematical structure providing a framework for the physical understanding. The examples we present can be used to enhance the discussion of second-quantized harmonic oscillators in the classroom, to make a connection to the classical physics of coupled oscillators, and to acquaint students with systems employed at the frontiers of contemporary physics research.

INHOMOGENEOUS BAROTROPIC FRW COSMOLOGIES IN CONFORMAL TIME

It is known that the barotropic FRW system of differential equations for zero cosmological constant can be reduced to simple harmonic oscillator (HO) differential equations in the conformal time variable. This is due to the fact that the Hubble rate parameter in conformal time is the solution of a simple Riccati equation of constant coefficients. In previous works, we have used this mathematical result to set the barotropic HO equations in the nonrelativistic supersymmetric approach by factorizing them. If a constant additive parameter, denoted by S, is added to the common Riccati solution of these supersymmetric partner cosmologies one obtains inhomogeneous barotropic cosmologies with periodic singularities in their spatial curvature indices that are counterparts of the non-shifted supersymmetric partners. The zero-mode solutions of these cyclic singular cosmologies are reviewed here as a function of real and imaginary shift parameter. We also notice the modulated zero modes obtained by using the general Riccati solution and comment on their cosmological application.

Relationship between neighbor number and vibrational spectra in disordered colloidal clusters with attractive interactions

We study connections between vibrational spectra and average nearest neighbor number in disordered clusters of colloidal particles with attractive interactions. Measurements of displacement covariances between particles in each cluster permit calculation of the stiffness matrix, which contains effective spring constants linking pairs of particles. From the cluster stiffness matrix, we derive vibrational properties of corresponding "shadow" glassy clusters, with the same geometric configuration and interactions as the "source" cluster but without damping. Here, we investigate the stiffness matrix to elucidate the origin of the correlations between the median frequency of cluster vibrational modes and average number of nearest neighbors in the cluster. We find that the mean confining stiffness of particles in a cluster, i.e., the ensemble-averaged sum of nearest neighbor spring constants, correlates strongly with average nearest neighbor number, and even more strongly with median frequency. Further, we find that the average oscillation frequency of an individual particle is set by the total stiffness of its nearest neighbor bonds; this average frequency increases as the square root of the nearest neighbor bond stiffness, in a manner similar to the simple harmonic oscillator.

Rapid distortion analysis of high Mach number homogeneous shear flows: Characterization of flow-thermodynamics interaction regimes

In high-speed shear flows the nature of flow-thermodynamics interactions, and consequently the character of transition/turbulence, changes markedly with Mach number. We identify and characterize three different regimes of interactions in terms of acoustic frequency-to-shear magnitude ratio employing the linear rapid distortion analysis. We begin with an analysis of the pressure equation and demonstrate that acoustic frequency grows monotonically with time in this initial value problem whereas the shear magnitude is imposed to be constant. Initially when acoustic frequency is smaller than shear magnitude, fluctuations grow rapidly as the velocity field evolves unrestrained by pressure. This corresponds to Regime 1 wherein there is no significant flow-thermodynamics interaction. Flow-thermodynamics interactions commence in Regime 2 as acoustic frequency grows to the level of imposed shear rate. Dilatational velocity and pressure fields in the flow-normal direction are generated. The two fields are coupled as in a simple harmonic oscillator and consequently shear stresses oscillate about a near zero value. Thus production is drastically reduced leading to stabilization of perturbation kinetic energy growth rate. Further increase in acoustic frequency beyond shear magnitude leads to Regime 3. The solenoidal perturbations begin to dominate the flow statistics and the kinetic energy evolution bears much semblance to the incompressible shear flow. Overall, this work leads to improved insight into high-speed shear flow physics.