Simple Harmonic Oscillator Research Articles

Newton’s Second Law on Positive Real Line R_+

In this work, a classical particle that moves on will be investigated by using the affine commutator relation which corresponds to the Poisson bracket in classical mechanics where the equation of motions and the Newton’s second law will be modified. Two example of physical systems will be studied further using the modified Newton’s second law namely free particle and harmonic oscillator. The result concludes that for a free particle, the force on is no longer zero and a linear force due positive real line and fixed momentum. While, classical dynamics on for simple harmonic oscillator system is a qubic force.

Slamming forces during water entry of a simple harmonic oscillator

When a blunt body impacts an air–water interface, large hydrodynamic forces often arise, a phenomenon many of us have unfortunately experienced in a failed dive or ‘belly flop’. Beyond assessing risk to biological divers, an understanding and methods for remediation of such slamming forces are critical to the design of numerous engineered naval and aerospace structures. Herein we systematically investigate the role of impactor elasticity on the resultant structural loads in perhaps the simplest possible scenario: the water entry of a simple harmonic oscillator. Contrary to conventional intuition, we find that ‘softening’ the impactor does not always reduce the peak impact force, but may also increase the force as compared with a fully rigid counterpart. Through our combined experimental and theoretical investigation, we demonstrate that the transition from force reduction to force amplification is delineated by a critical ‘hydroelastic’ factor that relates the hydrodynamic and elastic time scales of the problem.

Dynamics of the modified emden and pseudo-modified emden equations: position-dependent mass, invariance and exact solvability

We consider the modified Emden equation (MEE) and introduce its most general solution, using the most general solution for the simple harmonic oscillator’s linear dynamical equation (i.e. the initial conditions shall be identified by the PDM-MEE problem at hand). We use a general nonlocal point transformation and show that modified Emden dynamical equation is transformed to describe position-dependent mass (PDM) classical particles. Two PDM-MEE-type classical particles are used as illustrative examples, and their exact solutions are reported. Under specific parametric considerations, the phase-space trajectories are reported for the MEE-type and for PDM-MEE-type classical particles.

Leading quantum correction to the classical free energy

The quantum free energy of a system governed by a standard Hamiltonian is larger than its classical counterpart. The lowest-order correction, first calculated by Wigner, is proportional to ℏ2 and involves the sum of the mean squared forces. We present an elementary derivation of this result by drawing upon the Zassenhaus formula, an operator-generalization for the main functional relation of the exponential map. Our approach highlights the central role of non-commutativity between kinetic and potential energy and is more direct than Wigner's original calculation, or even streamlined variations thereof found in modern textbooks. We illustrate the quality of the correction for the simple harmonic oscillator (analytically) and the purely quartic oscillator (numerically) in the limit of high temperature. We also demonstrate that the Wigner correction fails in situations with sufficiently rapidly changing potentials, for instance, the particle in a box.

Quantum particles in a suddenly accelerating potential

We study the behavior of a quantum particle trapped in a confining potential in one dimension under multiple sudden changes of velocity and/or acceleration. We develop the appropriate formalism to deal with such situation and we use it to calculate the probability of transition for simple problems such as the particle in an infinite box and the simple harmonic oscillator. For the infinite box of length L under two and three sudden changes of velocity, where the initial and final velocity vanish, we find that the system undergoes quantum revivals for Δt=τ0≡4mL2πℏ, regardless of other parameters (Δt is the time elapsed between the first and last change of velocity). For the simple harmonic oscillator we find that the states obtained by suddenly changing (one change) the velocity and/or the acceleration of the potential, for a particle initially in an eigenstate of the static potential, are coherent states. For multiple changes of acceleration or velocity we find that the quantum expectation value of the Hamiltonian is remarkably close (possibly identical) to the corresponding classical expectation values. Finally, the probability of transition for a particle in an accelerating harmonic oscillator (no sudden changes) calculated with our formalism agrees with the formula derived long time ago by Ludwig [Z. Phys. 130(4), 468–475 (1951)], and recently modified by Dodonov [J. Russ. Laser Res. 42(3), 243–249 (2021)], but with a different expression for the dimensionless parameter γ. Our probability agrees with the one of Dodonov for γ ≪ 1 but is not periodic in time (it decays monotonously), contrary to the result derived by Dodonov.

Simulations and theory of power spectral density functions for time-dependent and anharmonic Langevin oscillators.

Simulations and theory are presented for the power spectral density functions (PSDs) of particles in time-dependent and anharmonic potentials, including the effects of a thermal environment leading to damping and fluctuating forces. We investigate three one-dimensional perturbations to the harmonic oscillator of which two are time-dependent changes in the natural frequency of the oscillator, while the other is a time-independent extension of the quadratic potential to include a quartic term. We investigate the effect of these perturbations on two PSDs of the motion that are used in experiments on trapped nano-oscillators. We also derive and numerically test the PSDs for the motion of a spherical nanoparticle in a Paul trap. We found that the simple harmonic Langevin oscillator's PSDs are good approximations for the x and y coordinates' PSDs for small values of the parameter q of the Mathieu equation, but the difference can be more than a factor of two as "q" increases. We also numerically showed that the presence of a permanent electric dipole on the nanosphere can significantly affect the PSDs in the x and y coordinates.

Bekenstein-Smarr formula and quantization of Kerr black holes

By solving the zero surface equation and studying the black holes law, we derive the equivalent Bekenstein-Smarr formula for Kerr black holes. With the reduced phase space method and canonical transformation, we establish the quantum mechanical model for Kerr black hole which is a standard two-mode simple harmonic oscillator model. With the help of this model, we further derive the spectra of the event horizon area and corresponding eigen wave functions for the Kerr black holes, which are proved that the area of a black hole event horizon is quantized.

Touchdown dynamics of a liquid droplet on a patterned substrate: A generalized lumped parameter-based model

Droplet dynamics post-impingement on a solid surface is of immense technological importance due to its applicability in self-cleaning, anti-frosting, anti-biofouling, anti-icing, thermal management, and inkjet printing. While sustained droplet oscillation is desirable in some applications, a fast seizure of oscillations after the surface touchdown of the droplets is necessary for others. The post-impingement dynamics is a complex non-linear interaction of the fluid and solid properties and a common framework to characterize the short-time response of a droplet is provided here. We describe these post-impact droplet dynamics on a patterned surface in terms of lumped simple harmonic oscillator parameters. We delineate the effect of the substrate and fluid properties and attribute the same to one or more terms in the dynamical model of the oscillator, describing an analogy between the terms. Our analogy gives a universal mapping between the fluid–solid parameters and droplet dynamics, making it feasible to predict with accuracy the post-impact behavior. Here for the first time, we developed a simplified universal approach to predict the droplet oscillation dynamics impacting any solid surface for a broad range of fluid and solid properties and impingement conditions.

Quantifying population dynamics via a geometric mean predator-prey model.

An integrable Hamiltonian variant of the two species Lotka-Volterra (LV) predator-prey model, shortly referred to as geometric mean (GM) predator-prey model, has been recently introduced. Here, we perform a systematic comparison of the dynamics underlying the GM and LV models. Though the two models share several common features, the geometric mean dynamics exhibits a few peculiarities of interest. The structure of the scaled-population variables reduces to the simple harmonic oscillator with dimensionless natural time TGM varying as ωGMt with ωGM=c12c21. We found that the natural timescales of the evolution dynamics are amplified in the GM model compared to the LV one. Since the GM dynamics is ruled by the inter-species rather than the intra-species coefficients, the proposed model might be of interest when the interactions among the species, rather than the individual demography, rule the evolution of the ecosystems.

Canonical quantization of modified non-gauge invariant Einstein-Maxwell gravity and stability of spherically symmetric electrostatic stars

We consider a non-minimally coupled Einstein-Maxwell gravity with no U(1) symmetry property to study stability of an electrostatic star via canonical quantization approach and obtain that the stability is free of gauge field effects. By calculating the Hamiltonian density of the stellar system we show that the corresponding Wheeler-DeWitt wave functional is similar to a simple harmonic quantum Oscillator for which a non zero ADM mass of the system causes a quantization condition on the metric fields. Probability wave packets are described by the Hermit polynomials. Our mathematical calculations show that in this approach of quantum gravity the metric fields are regular for all values of the electric potential and so the quantized spacetime has not both of event and apparent horizons. The most probability of the quantized line element is for ground state of the system. To check validation of the model we use Bohr's correspondence principal and generate directly semi classical approach of the quantized metric states at large quantum numbers where they reach to Schwarzschild like metric according to the Birkhoff’s theorem. Also we check that the generated semi classical solutions are satisfied exact classical metric solutions which are obtained from Euler–Lagrange equations. We show that ‘charge to mass ratio’ of the electrostatic star is a constant defined by the coupling constant of the model and it is in accord to other alternative approaches.

Path integral in position-deformed Heisenberg algebra with maximal length uncertainty

In this work, we study the path integral in a position-deformed Heisenberg algebra with quadratic deformation which implements both minimal momentum and maximal length uncertainties. We construct propagators of path integrals within this deformed algebra using the position space representation on the one hand and the Fourier transform and its inverse representations on the other. The result is remarkably similar to the one obtained by Pramanik (2022) from the Perivolaropoulos’s deformed algebra (Perivolaropoulos, 2017). Then, the propagators and the corresponding actions of a free particle and a simple harmonic oscillator are discussed as examples. We also show that the actions which describe the classical trajectories of both systems are bounded by the ordinary ones of classical mechanics due to the existence of this maximal length. Consequently, particles of these systems travel faster from one point to another with low kinetic and mechanical energies.

Determination of Stretching Frequencies by Isotopic Substitution Using Infrared Spectroscopy: An Upper-Level Undergraduate Experiment for an In-Person or Online Laboratory

A Fourier transform infrared (FTIR) experiment appropriate for an upper-level undergraduate laboratory such as chemical instrumentation is described. Students collect FTIR spectra of four protio-solvents and their deuterated analogues. In addition to qualitatively observing C–H and O–H peaks shift to lower energy upon deuteration, students apply a simple harmonic oscillator model to predict the magnitude of the shifts and gain an understanding of the relationship between reduced mass and vibrational frequencies. Vibrational calculations are then performed to corroborate the experimental results and allow students to visualize the normal modes involved via animations of the vibrations. This experiment is suitable to be used as either an in-person or online laboratory.

Transition form factors and angular distributions of the Λb→Λ(1520)(→NK¯)ℓ+ℓ− decay supported by baryon spectroscopy

We calculate the weak transition form factors of the $\Lambda_b\to\Lambda(1520)$ transition, and further calculate the angular distributions of the rare decays $\Lambda_b\to\Lambda(1520)(\to N\bar{K})\ell^{+}\ell^{-}$ ($N\bar{K}=\{pK^-,n\bar{K}^0\}$) with unpolarized $\Lambda_b$ and massive leptons. The form factors are calculated by the three-body light-front quark model with the support of numerical wave functions of $\Lambda_b$ and $\Lambda(1520)$ from solving the semirelativistic potential model associated with the Gaussian expansion method. By fitting the mass spectrum of the observed single bottom and charmed baryons, the parameters of the potential model are fixed, so this strategy can avoid the uncertainties arising from the choice of a simple harmonic oscillator wave function of the baryons. With more data accumulated in the LHCb experiment, our result can help for exploring the $\Lambda_b\to\Lambda(1520)\ell^+\ell^-$ decay and deepen our understanding on the $b\to s\ell^+\ell^-$ processes.

Dynamics and Control of a Buoyancy-Driven Underwater Vehicle for Estimating and Tracking the Scattering Layer

This paper presents the design of a control strategy for an autonomous buoyancy-driven underwater vehicle, called the Driftcam, tasked with estimating and tracking the pelagic acoustic-scattering layer at depths of up to 700 meters. This reference-tracking control design is an alternative to the onboard control currently implemented on the Driftcam, as the existing control architecture is not designed for time-varying setpoint tracking. Tracking a known reference is presented using state-feedback control for continuous- and discrete-time command inputs. The known reference signal is chosen to represent the pelagic scattering layer vertical migration dynamics, modeled as a simple harmonic oscillator and estimated with a dynamic observer. We define a measurement function for the depth-varying density of organisms observable in the scattering layer. Time-varying reference tracking control is presented using output feedback for continuous- and discrete-time command inputs, where the reference is the online position estimate of the scattering layer. We also present an offboard trajectory motion planning algorithm using the existing onboard controller to emulate the response of the proposed reference tracking strategy. Experimental results from laboratory and field testing of the trajectory motion planner demonstrate that the buoyancy engine can feasibly achieve the desired behavior of the reference-tracking control strategy.

Significance of classically forbidden regions for short baseline neutrino experiments

Classically forbidden regions (CFRs) are common to both non-relativistic quantum mechanics, and to relativistic quantum field theory. It is known since 2001 that CFR contributes roughly sixteen percent of energy to the ground state of a simple harmonic oscillator (Adunas G. Z. et al., Gen. Relativ. Gravit., 33 (2001) 183). Similarly, quantum field theoretic arguments yield a non-zero amplitude for a massive particle to cross the light cone (that is, into the CFR). The signs of these amplitudes are opposite for fermions and antifermions. This has given rise to an erroneous conclusion that amplitude to cross the light cone is identically zero. This is true as long as a measurement does not reveal the considered object to be a particle or antiparticle. However, neutrino oscillation experiments do measure a neutrino ν, or an antineutrino . Here we show that in the context of neutrino oscillations these observations have the potential to resolve various short baseline anomalies for a sufficiently light lowest mass eigenstate. In addition, we make a concrete prediction for the upcoming results to be announced later this year by JSNS2.

Wide temperature operation of piezoelectric sensors for detecting precursor levels in a canister

Liquid level detection using piezoelectric actuators and sensors is superior to other technologies in accuracy, stability, and durability. In the semiconducting industry, the accurate detection of precursor levels in a canister is directly connected to the quality of the atomic layer growth through chemical vapor deposition and atomic layer deposition processes. However, the sensitivity of the level detection using piezoelectric devices often decreases at a specific temperature range, limiting the wide temperature operation of the canister. We demonstrate reduced sensitivity of the piezoelectric sensors due to a change in detuning frequency by temperature. A model with a simple harmonic oscillator exhibits the fundamental behavior of the actuator amplitude after a finite number of driving pulses. The impedance measurement of a sensor assembly demonstrated a significant shift in the primary resonance frequency due to a change in environmental temperature. By analyzing the simulation data, we established a temperature-dependent number of driving pulses that could extend the operating temperature of the piezoelectric actuators, which can easily be applied to a wide temperature operation for a canister.

Observability for Schrödinger equations with quadratic Hamiltonians

We consider time dependent harmonic oscillators and construct a parametrix to the corresponding Schrödinger equation using Gaussian wavepackets. This parametrix of Gaussian wavepackets is precise and tractable. Using this parametrix we prove L^2 and L^2-L^{infty } observability estimates on unbounded domains omega for a restricted class of initial data. This data includes a class of compactly supported piecewise C^1 functions which have been extended from characteristic functions. Initial data of this form which has the bulk of its mass away from omega ^c=Omega , a connected bounded domain, is observable, but data centered over Omega must be very nearly a single Gaussian to be observable. We also give counterexamples to established principles for the simple harmonic oscillator in the case of certain time dependent harmonic oscillators.

Two New Methods in Stochastic Electrodynamics for Analyzing the Simple Harmonic Oscillator and Possible Extension to Hydrogen

The position probability density function is calculated for a classical electric dipole harmonic oscillator bathed in zero-point plus Planckian electromagnetic fields, as considered in the physical theory of stochastic electrodynamics (SED). The calculations are carried out via two new methods. They start from a general probability density expression involving the formal integration over all probabilistic values of the Fourier coefficients describing the stochastic radiation fields. The first approach explicitly carries out all these integrations; the second approach shows that this general probability density expression satisfies a partial differential equation that is readily solved. After carrying out these two fairly long analyses and contrasting them, some examples are provided for extending this approach to quantities other than position, such as the joint probability density distribution for positions at different times, and for position and momentum. This article concludes by discussing the application of this general probability density expression to a system of great interest in SED, namely, the classical model of hydrogen.

On the quantum dynamics of a general time-dependent coupled oscillator

By using the Lewis–Riesenfeld invariants theory, we investigate the quantum dynamics of a two-dimensional (2D) time-dependent coupled oscillator. We introduce a unitary transformation and show the conditions under which the invariant operator is uncoupled to describe two simple harmonic oscillators with time-independent frequencies and unit masses. The decouplement allows us to easily obtain the corresponding eigenstates. The generalization to three-dimensional (3D) time-dependent coupled oscillator is briefly discussed where a diagonalized invariant, which is exactly the sum of three simple harmonic oscillators, is obtained under specific conditions on the parameters.

Thermally activated motion of cold excitons in conjugated polymers

AbstractThermally activated migration of localized cold excitons were investigated in archetypal conjugated polymers; trans‐polyacetylene, polythiophene, and poly(p‐phenylene vinylene). Room temperature thermal energy has been found to provide sufficient energy to avail substantial mobility for excitons. A simple undistorted but displaced harmonic oscillator model has been assumed to predict the activation energy for exciton hopping between identical sites in perfectly planar conjugated chains, promoted by vibrational modes with Bu symmetry. The activation energies of exciton hopping has been found to increase in the order of trans‐polyacetylene (6.4 meV), polythiophene (29.5 meV), and poly(p‐phenylene vinylene) (147.4 meV). A rough Monte‐Carlo simulation based on incoherent hopping mechanism reveals excitons can displace in an average of 2.17 nm in polythiophene whereas those distances are much smaller in poly(p‐phenylene vinylene) (0.24 nm) and trans‐polyacetylene (0.37 nm). The simulation results partially explain why polythiophene based polymers are better suited for organic photovoltaic applications whereas poly(p‐phenylene vinylene) are more appropriate for organic light emitting diode applications.