Simple Harmonic Oscillator Research Articles

Anharmonic oscillator: a playground to get insight into renormalization

In the presence of interactions the frequency of a simple harmonic oscillator deviates from the noninteracting one. Various methods can be used to compute the changes to the frequency perturbatively. Some of them resemble the methods used in quantum field theory (QFT) where the bare values of the parameters of the theory run when an interaction is added. In this paper, we review some of these techniques and introduce some new ones in line with QFT methods. Moreover, we investigate the case of more than one oscillator to see how the frequencies of small oscillations change when non-linear terms are added to a linear system and observe an interesting beat phenomenon in degenerate coupled oscillatory systems.

Inferring time-dependent distribution functions from kinematic snapshots

ABSTRACT We propose a method for constructing the time-dependent phase space distribution function (DF) of a collisionless system from an isolated kinematic snapshot. In general, the problem of mapping a single snapshot to a time-dependent function is intractable. Here, we assume a finite series representation of the DF, constructed from the spectrum of the system’s Koopman operator. This reduces the original problem to one of mapping a kinematic snapshot to a discrete spectrum rather than to a time-dependent function. We implement this mapping with a convolutional neural network. The method is demonstrated on two example models: the quantum simple harmonic oscillator and a self-gravitating isothermal plane. The latter system exhibits phase space spiral structure similar to that observed in Gaia Data Release 2.

Dynamical Invariant and Exact Mechanical Analyses for the Caldirola-Kanai Model of Dissipative Three Coupled Oscillators.

We study the dynamical invariant for dissipative three coupled oscillators mainly from the quantum mechanical point of view. It is known that there are many advantages of the invariant quantity in elucidating mechanical properties of the system. We use such a property of the invariant operator in quantizing the system in this work. To this end, we first transform the invariant operator to a simple one by using a unitary operator in order that we can easily manage it. The invariant operator is further simplified through its diagonalization via three-dimensional rotations parameterized by three Euler angles. The coupling terms in the quantum invariant are eventually eliminated thanks to such a diagonalization. As a consequence, transformed quantum invariant is represented in terms of three independent simple harmonic oscillators which have unit masses. Starting from the wave functions in the transformed system, we have derived the full wave functions in the original system with the help of the unitary operators.

A new look at quantal time evolution

Time evolution of quantum systems is normally taught using the Schrödinger equation. Here, we use the Heisenberg approach to consider three problems that are exactly solvable in the Schrödinger picture when the initial wavepacket is Gaussian: the free particle, the simple harmonic oscillator and a free particle with a spin-orbit interaction. We show that complete solutions can be obtained for the free particle (with or without spin-orbit interaction). For the simple harmonic oscillator, however, the fact that the wavepacket remains a Gaussian as time progresses has to be inferred on the basis of the skewness and kurtosis as opposed to a study of all the relevant moments.

On the Poincar’e-Lindstedt perturbation method for a Non-Linear Rayleigh Oscillator with periodic damping coefficient

In this paper the Poincar’e-Lindstedt perturbation method will be applied to analyse an oscillator with Rayleigh type with Periodic Damping Coefficient. The mathematical model of the oscillator describes flow-induced vibrations in a uniform wind field. The horizontal cylinder supported by springs as a model can be designed vibrate in vertical direction. It will be studied a solution approximation of the oscillator by using Poincar’e-Lindstedt perturbation method. The basic idea of the Poincar’e-Lindstedt perturbation is that from the simple harmonic oscillator, the period of oscillation depends on the amplitude of the motion. The Lindstedt perturbation expansion allows the frequency to adapt to the nonlinearity by defining the “stretched time variable” The periodic solution of Limit cycle will be studied in this paper.

Position-dependent finite symmetric mass harmonic like oscillator: Classical and quantum mechanical study

Abstract We formulated the oscillators with position-dependent finite symmetric decreasing and increasing mass. The classical phase portraits of the systems were studied by analytical approach (He’s frequency formalism). We also study the quantum mechanical behaviour of the system and plot the quantum mechanical phase space for necessary comparison with the same obtained classically. The phase portrait in all the cases exhibited closed loop reflecting the stable system but the quantum phase portrait exhibited the inherent signature (cusp or kink) near origin associated with the mass. Although the systems possess periodic motion, the discrete eigenvalues do not possess any similarity with that of the simple harmonic oscillator having m = 1.

State-space Representation of Matérn and Damped Simple Harmonic Oscillator Gaussian Processes

Abstract Gaussian processes (GPs) are used widely in the analysis of astronomical time series. GPs with rational spectral densities have state-space representations which allow  ( n ) evaluation of the likelihood. We calculate analytic state space representations for the damped simple harmonic oscillator and the Matérn 1/2, 3/2 and 5/2 processes.

Probability Calculations Within Stochastic Electrodynamics

Several stochastic situations in stochastic electrodynamics (SED) are analytically calculated from first principles. These situations include probability density functions, as well as correlation functions at multiple points of time and space, for the zero-point (ZP) electromagnetic fields, as well as for ZP plus Planckian (ZPP) electromagnetic fields. More lengthy analytical calculations are indicated, using similar methods, for the simple harmonic electric dipole oscillator bathed in ZP as well as ZPP electromagnetic fields. The method presented here makes an interesting contrast to Feynman’s path integral approach in quantum electrodynamics (QED). The present SED approach directly entails probabilities, while the QED approach involves summing weighted paths for the wave function.

MEMS displacement generator for atomic force microscopy metrology

Atomic force microscopy enables three-dimensional high-resolution imaging of surfaces with nanoscale features. In order to obtain the quantitative information about surface geometry, the atomic force microscope’s scanning system must be calibrated. This is usually done by using calibration samples of known and/or defined shape based on either lithographic or crystal structures. In this work we report on a microelectromechanical device, referred to as a displacement generator, whose vertical deflection is controlled electronically. The designed, fabricated and applied device is formed out of a silicon nitride doubly clamped lever, referred to as a microbridge, with a deposited pair of platinum strips. When the MEMS displacement generator is immersed in a magnetic field and when it is electrically biased, the associated Lorentz force induces a structural displacement. In the presented design, the silicon nitride microbridges were fabricated on a (110) silicon wafer in a Wheatstone bridge configuration. A second reference cantilever was mechanically supported by the silicon substrate. In this way, a highly symmetrical structure was fabricated, making it possible to control precisely deflection in Z direction with sub-nanometre precision. The cantilever’s high resonance frequency, of ca. 500 kHz, makes the constructed device insensitive to external vibration sources which are typically at much lower frequencies. As the stage function can be described using the simple harmonic oscillator model, it is clear that the system can operate with sub-nanometre resolution, which, for the purpose of microscope calibration, is extremely beneficial. By placing of the atomic force microscope tip on the actuated reference device it is possible to determine the response of the system over a wide frequency bandwidth. In this work we will describe the fabrication process of the MEMS displacement generator, interferometric and traceable investigations of thermomechanical and electromagnetic actuation schemes. Moreover, we will present the results of the calibration of an atomic force microscope operating in contact and intermittent contact modes.

Leggett-Garg tests for macrorealism: Interference experiments and the simple harmonic oscillator

Leggett-Garg (LG) tests for macrorealism were originally designed to explore quantum coherence on the macroscopic scale. Interference experiments and systems modeled by harmonic oscillators provide useful examples of situations in which macroscopicity has been approached experimentally and are readily turned into LG tests for a single dichotomic variable $Q$. Applying this approach to the double-slit experiment in which a noninvasive measurement at the slits is included, we exhibit LG violations. We find that these violations are always accompanied by destructive interference. The converse is not true in general and we find that there are nontrivial regimes in which there is destructive interference but the two-time LG inequalities are satisfied, which implies that it is in fact often possible to assign (indirectly determined) probabilities for the interferometer paths. Similar features have been observed in recent work involving a LG analysis of a Mach-Zehnder interferometer and we compare with those results. We extend the analysis to the triple-slit experiment again finding LG violations, and we also exhibit examples of some surprising relationships between LG inequalities and NSIT conditions that do not exist for dichotomic variables. For the simple harmonic oscillator, we find an analytically tractable example showing a two-time LG violation with a gaussian initial state, echoing in simpler form recent results of Bose et al. [Phys. Rev. Lett. 120, 210402 (2018)].

The forming mechanism of spontaneous emission noise flux radiated from hydrogen-like atoms by means of vibrational Hamiltonian

The main goal of this article is to investigate the origin of spontaneous emission radiation by applying a new method. In this method, the vibrational Hamiltonian of hydrogen-like atoms (HLAs) is first constructed in accordance with the Hamiltonian of a simple harmonic oscillator. The Hamiltonian is then used to derive the vibrational motion equations of HLAs in a Heisenberg picture. The Langevin equation will ultimately be formed by adding the dissipative term and fluctuating (Langevin) force according to the fluctuation–dissipation theorem. The solution of the Langevin equation provides the exact data on the fluctuations that occur for the electron position during its rather fast vibrational oscillations. On the other hand, the fluctuations in the position of the oscillating electron, in turn, lead to the fluctuations in its potential and kinetic energies so that the appearance of the potential and kinetic noise fluxes is vulnerable. It is finally demonstrated that the potential and kinetic noise fluxes commonly act as an internal source of producing the external noise flux emitted from HLAs in the form of spontaneous emission with a Lorentzian profile. In contrast with all previous procedures, no ambient effect has been involved to describe the forming mechanism of spontaneous emission for the first time.

Stability analysis of an ensemble of simple harmonic oscillators

In this paper, we investigate the stability of the configurations of harmonic oscillator potential that are directly proportional to the square of the displacement. We derive expressions for fluctuations in partition function due to variations of the parameters, viz. the mass, temperature and the frequency of oscillators. Here, we introduce the Hessian matrix of the partition function as the model embedding function from the space of parameters to the set of real numbers. In this framework, we classify the regions in the parameter space of the harmonic oscillator fluctuations where they yield a stable statistical configuration. The mechanism of stability follows from the notion of the fluctuation theory. In Secs. 7 and 8, we provide the nature of local and global correlations and stability regions where the system yields a stable or unstable statistical basis, or it undergoes into geometric phase transitions. Finally, in Sec. 9, the comparison of results is provided with reference to other existing research.

Taylor Series and Getting the General Solutions for the Equations of Motion Using Poisson Bracket Relations

In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called canonical transformations, which map canonical coordinate systems into canonical coordinate systems. In this work we study some examples from the classical mechanics of particles and apply mathematical method for building the equation of motion. In the present paper Poisson Brackets and their properties are presented, by using Poisson brackets and their properties we calculate some brackets. We use the Poisson bracket with Hamiltonians to express the time dependence of a function u (t), the main idea Taylor series is taken as the required solution for equation of motion using the properties of the Poisson Brackets, We have examined examples from the classical mechanics to illustrate the idea such as motion with a constant acceleration, simple harmonic oscillator, freely falling particle. The solutions are compatible with what is known in classical mechanics. The work is fundamental and sheds new light onto classical mechanics. Poisson brackets are a powerful and sophisticated tool in the Hamiltonian formalism of Classical Mechanics. They also happen to provide a direct link between classical and quantum mechanics.

Auto-correlation functions of astrophysical processes, and their relation to Gaussian processes

Context. Accounting for the effects of stellar magnetic phenomena is indispensable to fully exploit radial velocities (RVs) obtained using modern exoplanet-hunting spectrometers. Correlated time variations are often mitigated by non-trivial noise models in the framework of Gaussian processes. These models rely on fitting kernel functions that are motivated on mathematical grounds, and whose physical interpretation is often elusive. Aims. We aim to establish a clear connection between stellar magnetic activity affecting RVs and their corresponding correlations with physical parameters, and compare this connection with kernels used in the literature. Methods. We use simple activity models to investigate the relationship between the physical processes generating the signals and the covariances typically found in data, and to demonstrate the qualitative behaviour of this relationship. We use the StarSim code to calculate RVs of an M dwarf with different realistic evolving spot configurations. The auto-correlation function (ACF) of a synthetic data set shows a very specific behaviour and is explicitly related to the kernel. Gaussian process regression is performed using the quasi-periodic (QP) and simple harmonic oscillator kernels of the george and celerite codes, respectively. Comparison of the resulting kernels with the exact ACFs allows us to cross-match the kernel hyper-parameters with the introduced physical values, study the overall capabilities of the kernels, and improve their definition. Results. We find that the QP kernel provides a more straightforward interpretation of the physics. It is able to consistently recover both the introduced rotation period Prot and the spot lifetime. Our study indicates that the performance can be enhanced by fixing the form factor w and adding a physically motivated cosine term with period Prot∕2, where the contribution to the ACF for the different spot configurations differs significantly. The newly proposed quasi-periodic with cosine (QPC) kernel leads to significantly better model likelihoods, can potentially distinguish between different spot configurations, and can thereby improve the sensitivity of RV exoplanet searches.

A completely algebraic solution of the simple harmonic oscillator

We present a full algebraic derivation of the wavefunctions of a simple harmonic oscillator. This derivation illustrates the abstract approach to the simple harmonic oscillator by completing the derivation of the coordinate-space or momentum-space wavefunctions from the energy eigenvectors. It is simple to incorporate into the undergraduate and graduate curricula. We provide a summary of the history of operator-based methods as they are applied to the simple harmonic oscillator. We present the derivation of the energy eigenvectors along the lines of the standard approach that was first presented by Dirac in 1947 (and is modified slightly here in the spirit of the Schrödinger factorization method). We supplement it by employing the appropriate translation operator to determine the coordinate-space and momentum-space wavefunctions algebraically, without any derivatives.

Quantum Groverian geodesic paths with gravitational and thermal analogies

We present a unifying variational calculus derivation of Groverian geodesics for both quantum state vectors and quantum probability amplitudes. In the first case, we show that horizontal affinely parametrized geodesic paths on the Hilbert space of normalized vectors emerge from the minimization of the length specified by the Fubini–Study metric on the manifold of Hilbert space rays. In the second case, we demonstrate that geodesic paths for probability amplitudes arise by minimizing the length expressed in terms of the Fisher information. In both derivations, we find that geodesic equations are described by simple harmonic oscillators (SHOs). However, while in the first derivation the frequency of oscillations is proportional to the (constant) energy dispersion $$\Delta E$$ of the Hamiltonian system; in the second derivation the frequency of oscillations is proportional to the square root $$\sqrt{\mathcal {F}}$$ of the (constant) Fisher information. Interestingly, by setting these two frequencies equal to each other, we recover the well-known Anandan–Aharonov relation linking the squared speed of evolution of an Hamiltonian system with its energy dispersion. Finally, upon transitioning away from the quantum setting, we discuss the universality of the emergence of geodesic motion of SHO type in the presence of conserved quantities by analyzing two specific phenomena of gravitational and thermodynamical origin, respectively.

Parametric Approach for Satellite Relative Orbit Designs and Constellations

This paper proposes a dynamics solution that creates an identical constellation pattern of multiple satellites with the parameterization of relative motion dynamics. The parameterization illustrates that the complexity of relative motion dynamics is interpreted by a Ptolemaic model of an epicycle and deferent system. All possible relative orbits can be represented by a hypo-epi curve or a deformed curve on the x-y plane and a simple z-axis harmonic oscillator motion. The designed relative orbits are used to create identical constellation patterns of multiple satellites, called parametric constellations in this study, by closed-form formulas and satellite phasing rules obtained from the geometrical relationship and periodic conditions. In conclusion, the resulting solution contributes to solving technical dynamics and tracking problems by simply designing complex relative motions in free space.

Forced harmonic oscillator interpreted as diffraction of light.

We investigate a simple forced harmonic oscillator with a natural frequency varying with time. It is shown that the time evolution of such a system can be written in a simplified form with Fresnel integrals, as long as the variation of the natural frequency is sufficiently slow compared to the time period of oscillation. Thanks to such a simple formulation, we found that a forced harmonic oscillator with a slowly varying natural frequency is essentially equivalent to diffraction of light.

Towards spectrally selective catastrophic response.

We study the large-amplitude response of classical molecules to electromagnetic radiation, showing the universality of the transition from linear to nonlinear response and breakup at sufficiently large amplitudes. We demonstrate that a range of models, from the simple harmonic oscillator to the successful Peyrard-Bishop-Dauxois type models of DNA, which include realistic effects of the environment (including damping and dephasing due to thermal fluctuations), lead to characteristic universal behavior: formation of domains of dissociation in driving force amplitude-frequency space, characterized by the presence of local boundary minima. We demonstrate that by simply following the progression of the resonance maxima in this space, while gradually increasing intensity of the radiation, one must necessarily arrive at one of these minima, i.e., a point where the ultrahigh spectral selectivity is retained. We show that this universal property, applicable to other oscillatory systems, is a consequence of the fact that these models belong to the fold catastrophe universality class of Thom's catastrophe theory. This in turn implies that for most biostructures, including DNA, high spectral sensitivity near the onset of the denaturation processes can be expected. Such spectrally selective molecular denaturation could find important applications in biology and medicine.

Endpoint of the up-down instability in precessing binary black holes

Binary black holes in which both spins are aligned with the binary's orbital angular momentum do not precess. However, the up-down configuration, in which the spin of the heavier (lighter) black hole is aligned (anti-aligned) with the orbital angular momentum, is unstable to spin precession at small orbital separations. We first cast the spin precession problem in terms of a simple harmonic oscillator and provide a cleaner derivation of the instability onset. Surprisingly, we find that following the instability, up-down binaries do not disperse in the available parameter space but evolve toward precise endpoints. We then present an analytic scheme to locate these final configurations and confirm them with numerical integrations. Namely, unstable up-down binaries approach mergers with the two spins coaligned with each other and equally misaligned with the orbital angular momentum. Merging up-down binaries relevant to LIGO/Virgo and LISA may be detected in these endpoint configurations if the instability onset occurs prior to the sensitivity threshold of the detector. As a by-product, we obtain new generic results on binary black hole spin-orbit resonances at 2nd~post-Newtonian order. We finally apply these findings to a simple astrophysical population of binary black holes where a formation mechanism aligns the spins without preference for co- or counteralignment, as might be the case for stellar-mass black holes embedded in the accretion disk of a supermassive black hole.