Simple Harmonic Oscillator Research Articles

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.

Insights from the Gaussian Process Method for the Fast Radio Burst–associated X-Ray Burst of SGR 1935+2154

The Gaussian process method is employed to analyze the light curves of bursts detected by Insight-HXMT, Neutron Star Interior Composition Explorer (NICER), and Gravitational Wave High-energy Electromagnetic Counterpart All-sky Monitor from SGR 1935+2154 between 2020 and 2022. It is found that a stochastically driven damped simple harmonic oscillator (SHO) is necessary to capture the characteristics of the X-ray bursts (XRBs). A variability timescale of the XRBs, corresponding to the broken frequencies in the SHO power spectral densities (PSDs), is extracted. In particular, a high broken frequency of 35 Hz where the index of the SHO PSD changes from −4 to −2 is constrained by the HXMT-HE burst associated with fast radio burst (FRB) 200428. It is suggested that the corresponding timescale of 0.03 s could be the retarding timescale of the system driven by some energy release, and the production of the HE photon should be quasi-simultaneous with the response. The other special event is a NICER burst with a retarding timescale of 1/(39 Hz) ≈ 0.02 s. In the normal XRBs, no retarding timescale is constrained; a long relax/equilibrium timescale (corresponding to a broken frequency of 1–10 Hz, where the index of the SHO PSD changes from −4/−2 to 0 in the SHO PSD) is obtained. The results indicate that the FRB-associated HXMT-HE XRB could be produced immediately when the system is responding to the energy disturbance, far before the equilibrium state.

Fermi model of a quantum black hole

We propose a quantum model of the Schwarzschild black hole as a quantum mechanics of a system of fermionic degrees of freedom. The system has a constant density of states and a Fermi energy that is inversely proportional to the size of the system. Assuming the equivalence principle, we show that the degeneracy pressure of the Fermi degrees of freedom is able to withstand the collapse of gravity if the radius of the system is given precisely by the horizon radius of the Schwarzschild black hole. In our model, the fermionic degrees of freedom at each energy level can be entangled in certain different ways, giving rise to a multitude of degenerate ground states of the system. The counting of these microstates reproduces precisely the Bekenstein-Hawking entropy. This simple Fermi model is universal and works also for the Reissner-Nordström charged black hole as well as black holes with a cosmological constant. From the properties of the Fermi variables, we propose that quantum gravity is characterized by a principle of where there can be no more than V/lP3 quantum states in any volume V. It implies a loss of spatial locality below the Planck length and suggests that any singularity predicted by general relativity is resolved and replaced by a quantum space in quantum gravity. In our model, a black hole spacetime is equipped with a uniform distribution of energy levels. This is another reason why black holes can be considered a simple harmonic oscillator of quantum gravity. Published by the American Physical Society 2024

Quantized Microcavity Polariton Lasing Based on InGaN Localized Excitons.

Exciton-polaritons, which are bosonic quasiparticles with an extremely low mass, play a key role in understanding macroscopic quantum effects related to Bose-Einstein condensation (BEC) in solid-state systems. The study of trapped polaritons in a potential well provides an ideal platform for manipulating polariton condensates, enabling polariton lasing with specific formation in k-space. Here, we realize quantized microcavity polariton lasing in simple harmonic oscillator (SHO) states based on spatial localized excitons in InGaN/GaN quantum wells (QWs). Benefiting from the high exciton binding energy (90 meV) and large oscillator strength of the localized exciton, room-temperature (RT) polaritons with large Rabi splitting (61 meV) are obtained in a strongly coupled microcavity. The manipulation of polariton condensates is performed through a parabolic potential well created by optical pump control. Under the confinement situation, trapped polaritons are controlled to be distributed in the selected quantized energy sublevels of the SHO state. The maximum energy spacing of 11.3 meV is observed in the SHO sublevels, indicating the robust polariton trapping of the parabolic potential well. Coherent quantized polariton lasing is achieved in the ground state of the SHO state and the coherence property of the lasing is analyzed through the measurements of spatial interference patterns and g(2)(τ). Our results offer a feasible route to explore the manipulation of macroscopic quantum coherent states and to fabricate novel polariton devices towards room-temperature operations.

Gravity train of variable mass

The gravity train is a virtual vehicle that travels in a tunnel across Earth, and its motion due to only gravity is a simple harmonic oscillator. It can run at a constant velocity by appropriately exhausting fuel. In this case, the train obeys the equation of motion for a variable mass system. We discuss the mass and energy reduction of the gravity train during travel and find the optimal conditions for economical travel. This study is suitable for university students in physics classes.

An accelerated lyapunov function for Polyak’s Heavy-ball on convex quadratics

AbstractIn 1964, Polyak showed that the Heavy-ball method, the simplest momentum technique, accelerates convergence of strongly-convex problems in the vicinity of the solution. While Nesterov later developed a globally accelerated version, Polyak’s original algorithm remains simpler and more widely used in applications such as deep learning. Despite this popularity, the question of whether Heavy-ball is also globally accelerated or not has not been fully answered yet, and no convincing counterexample has been provided. This is largely due to the difficulty in finding an effective Lyapunov function: indeed, most proofs of Heavy-ball acceleration in the strongly-convex quadratic setting rely on eigenvalue arguments. Our work adopts a different approach: studying momentum through the lens of quadratic invariants of simple harmonic oscillators. By utilizing the modified Hamiltonian of Störmer-Verlet integrators, we are able to construct a Lyapunov function that demonstrates an $$O(1/k^2)$$ O ( 1 / k 2 ) rate for Heavy-ball in the case of convex quadratic problems. Our novel proof technique, though restricted to linear regression, is found to work well empirically also on non-quadratic convex problems, and thus provides insights on the structure of Lyapunov functions to be used in the general convex case. As such, our paper makes a promising first step towards potentially proving the acceleration of Polyak’s momentum method and we hope it inspires further research around this question.

Hyperelasticity of blood clots: Bridging the gap between microscopic and continuum scales

The biomechanical properties of blood clots, which are dictated by their compositions and micro-structures, play a critical role in determining their fates, i.e., occlusion, persistency, or embolization in the human circulatory system. While numerous constitutive models have emerged to describe the biomechanics of blood clots, most of these models have primarily focused on the macroscopic deformation of the clots and the resultant strain-stress correlations without depicting the microscopic contributions from their structural components, such as fibrin fibers, fibrin network and red blood cells. This work addresses the gap in current scientific understanding by quantifying how changes in the microstructure of blood clots affect its mechanical responses under different external stresses. We leverage our previous published work to develop a hyperelastic potential model for blood clots, which incorporates six distinct strain-energy components to describe the alignment of fibers, the entropic and enthalpic stretching of fibrin fibers, the buckling of these fibers, clot densification, and clot jamming. These strain-energy components are represented by a combination of simple harmonic oscillators, one-sided harmonic potentials, and a Gaussian potential. The proposed model, which is C0, C1, and C2 continuous with a total of 13 parameters, has been validated against three datasets: 1) fibrin clot in tension, 2) blood clot in compression, and 3) blood clots in shear, demonstrating its robustness. Subsequent simulations of a microscopic blood clot model are performed to uncover mechanistic correlations for a majority of the hyperelastic potential's stiffness/strain parameters. Our results show that only one proposed term concerning fiber buckling needs further refinement, while the remaining five strain-energy terms appear to describe precisely what they were intended to. In summary, the proposed model provides insight into the behavior of thromboembolisms and assistance in computer-aided design of surgical tools and interventions such as thrombectomy.

Exact prime density reproduced through resonant tunneling across a double barrier system

A solid-state experiment based on quantum tunneling is proposed to reproduce the natural numbers and prime numbers as resonant tunneling energies in a double barrier system (DBS). For getting the prime numbers as eigenvalues the well potential is considered as the superposition of a smooth potential which is estimated using a semi-classical approach and a weak local fluctuating potential. We use the transfer matrix approach and finite element method by taking only the smooth part of the potential to obtain resonant energies which reproduces the local average prime density and the local average prime gap exactly. The methodology when applied to a quadratic potential of the well, produces whole numbers as eigenvalues except for a constant zero-point energy —the energy levels of a simple harmonic oscillator.

Ultra‐Narrow Linewidth Polariton Lasing in Optically Trapped Bose‐Einstein Condensates at Room Temperature

AbstractPolariton lasing via Bose‐Einstein condensation (BEC) provides a peculiar method to achieve low threshold coherent light sources. Until now, the cryogenic operating temperature required for polariton lasers has hampered the development of polaritonics. Here, a novel approach is first reported to realize the ultra‐narrow linewidth polariton lasing under quasi‐3D quantum confinement at room‐temperature (RT). The potential trap landscape is constructed by ring‐shaped optical excitation, wherein the continuum polariton dispersion is modulated into discrete simple harmonic oscillator (SHO) states. The coherent condensation lasing of trapped polaritons occurs beyond the threshold power. Benefiting from the horizontal quantum confinement, the trapped polariton lasing exhibits an ultra‐narrow linewidth (0.7 meV) and excellent quality factor (Q = 3510). Moreover, the second‐order quantum coherence properties of trapped polariton condensation are determined to reveal the quantum phase transition of Bosonic system. The results offer a feasible route for realizing a low‐threshold ultra‐narrow linewidth polariton laser at RT, which significantly facilitates the fabrication of polariton laser and switches on opportunities for future applications.

Non‐Markovian Cost Function for Quantum Error Mitigation

AbstractIn near‐term quantum computers like noisy intermediate‐scale quantum (NISQ) devices, reducing the impact of errors and decoherence is critical for practical implementation. Existing studies have primarily focused on Markovian noise sources; however, understanding the relationship between quantum error mitigation (QEM) and non‐Markovian noise sources is essential, as these effects are practically unavoidable in most solid‐state devices used for quantum computing. Here, a non‐Markovian model of quantum state evolution and a QEM cost function of controlled‐NOT (CNOT) gate operation are presented for NISQ devices interacting with an environment characterized by simple harmonic oscillators as a noise source. Using the projection operator formalism and both advanced and retarded propagators in time, the reduced‐density operator is derived for output quantum states in a time‐convolutionless form by solving the quantum Liouville equation. Output quantum state fluctuations are analyzed for identity and CNOT gate operations in two‐qubit operations across various input states and compare these results with experimental data from ion‐trap and superconducting quantum computing systems to estimate the key parameters of the QEM cost functions. These findings demonstrate that the QEM cost function increases as the coupling strength between the quantum system and its environment intensifies. This study highlights the significance of non‐Markovian models for understanding quantum state evolution and the practical implications of the QEM cost function in assessing experimental results from NISQ devices.

Quantum ballet by gravitational waves: Generating entanglement's dance of revival-collapse and memory within the quantum system

Recent proposals are emerging for the experimental detection of entanglement mediated by classical gravity, carrying significant theoretical and observational implications. In fact, the detection of gravitational waves (GWs) in LIGO provides an alternative laboratory for testing various gravity-related properties. By employing LIGO's arms as oscillators interacting with gravitational waves (GWs), our study demonstrates the potential for generating quantum entanglement between two mutually orthogonal modes of simple harmonic oscillators. Our findings reveal unique entanglement dynamics, including periodic “collapse and revival” influenced by GW oscillations, alongside a distinct “quantum memory effect.” Effectively, each harmonic oscillator feels a temperature. We believe that these forecasts may hold significance towards both theoretically probing and experimentally verifying various properties of classical gravitational waves.

Employing an operator form of the Rodrigues formula to calculate wavefunctions without differential equations

The factorization method of Schrödinger shows us how to determine the energy eigenstates without needing to determine the wavefunctions in position or momentum space. A strategy to convert the energy eigenstates to wavefunctions is well known for the one-dimensional simple harmonic oscillator by employing the Rodrigues formula for the Hermite polynomials in position or momentum space. In this work, we illustrate how to generalize this approach in a representation-independent fashion to find the wavefunctions of other problems in quantum mechanics that can be solved by the factorization method. We examine three problems in detail: (i) the one-dimensional simple harmonic oscillator; (ii) the three-dimensional isotropic harmonic oscillator; and (iii) the three-dimensional Coulomb problem. This approach can be used in either undergraduate or graduate classes in quantum mechanics.

Using a hall sensor with GoGo board to determine angular speed and angular frequency

This research focuses on modifying two physics experiments, a rotating wheel and a simple harmonic oscillator, to train pre-service physics teachers with an emphasis on digital and maker skills. The primary aims are twofold: first, to demonstrate the application of a Hall sensor, combined with the GoGo Board, in determining the angular speed in the rotating-wheel experiment and the angular frequency in the oscillation experiment. The second aim involves comparing these results with those obtained from high-speed video analysis using the Tracker software. To measure the period, a neodymium magnet was attached to both the wheel and mass. A block-based code was developed to control the Hall sensor and record the time when the magnet entered the detectable range of the sensor. In both experiments, high-speed videos at 250 fps were recorded and analysed using Tracker software. The study revealed that while the Hall sensor yields accurate measurements at lower speeds, its performance declines at higher speeds owing to limitations in accurately detecting the position of the magnet. In a simple harmonic oscillation experiment, the Hall sensor proved to be more effective at slower speeds. Although the Hall sensor approach resulted in a higher percentage error, it significantly enhanced students’ learning experience in terms of computational thinking and digital/maker skills.

Slamming forces during water entry of a simple harmonic oscillator – CORRIGENDUM

Slamming forces during water entry of a simple harmonic oscillator – CORRIGENDUM

Non-KAM classical chaos topology for electrons in superlattice minibands determines the inter-well quantum transition rates

We investigate the quantum-classical correspondence for a particle tunnelling through a periodic superlattice structure with an applied bias voltage and an additional tilted harmonic oscillator potential. We show that the quantum mechanical tunnelling rate between neighbouring quantum wells of the superlattice is determined by the topology of the phase trajectories of the analogous classical system. This result also enables us to estimate, with high accuracy, the tunnelling rate between two spatially displaced simple harmonic oscillator states using a classical model, and thus gain new insight into this generic quantum phenomenon. This finding opens new directions for exploring and understanding the quantum-classical correspondence principle and quantum jumps between displaced harmonic oscillators, which are important in many branches of natural science.

A simple harmonic quantum oscillator: fractionalization and solution

A quantum mechanical system that mimics the behavior of a classical harmonic oscillator in the quantum domain is called a simple harmonic quantum oscillator. The time-independent Schrödinger equation describes the quantum harmonic oscillator, and its eigenstates are quantized energy values that correspond to various energy levels. In this work, we first fractionalize the time-independent Schrödinger equation, and then we solve the generated problem with the use of the Adomian decomposition approach. It has been shown that fractional quantum harmonic oscillators can be handled effectively using the proposed approach, and their behavior can then be better understood. The effectiveness of the method is validated by a number of numerical comparisons.

Building a mathematical model for a simple harmonic oscillator that uses educational methods found in both physics education research and in the language disciplines that make it accessible to undergraduate students in an introductory musical acoustics course

At the basis of any course in acoustics is the fundamental idea of the simple harmonic oscillator. The term alone is confusing to students with little to no background in physics or math. For courses in musical acoustics at the undergraduate level, this topic is often minimized due to the lack of preparation. This, in turn, results in a more superficial approach to the advanced topics. While deriving the mathematical treatment from first principles is out of reach to these students, approaching the math itself as a language where they are building a description of a simple mass and spring system in their new, yet somewhat familiar, language can be accomplished through pictures, graphs and hands on activities. Students begin to build a vocabulary of “words” that can be strung together in “sentences” that tell the story of how the motion of a mass on a spring is produced. Emphasis is placed on the analogous comparison of physical properties by relating variables such as amplitude and frequency to that of volume and pitch. This model can then be used as a building block to the understanding of how sound is produced, propagated and perceived.

Supramolecular self-sorting predicted by a simple harmonic oscillator model.

Macrocycles that encapsulate two guests can self-sort those into homo- and heterodimers. We report here a family of self-sorting homobimetallic Pt(II) terpyridyl acetylide dimers secured together with a pair of Cucurbit[8]uril macrocycles (CB[8]). The rigid bridging unit between both Pt centers introduces varying "hinge" angles, resulting in disparities in Pt-Pt distances in the heterodimers, and leads to recognition motif mismatch. We found that the self-sorting process can be quantified using a simple model, in which each complex behaves as a simple harmonic oscillator, whose heteroassembly tends to minimize geometry distorsions through C(aryl)-CC-Pt axis deformation.

Algebraic Solution of the Quantum Harmonic Oscillator

The Wave-particle duality and the Schrodinger cat paradox pose important notions on quantum mechanical systems, in which a quantum system can exist in two physically opposite or distinct states. This motivates scientists from the days of Louis de Broglie to confidently look at matter as having a dual nature, which is a physical particle that can also be visualized as a wave entity. This paper seeks to provide an elegant method of solving the time-independent Schrodinger Equation (TISE) for a quantum simple harmonic oscillator (QSHO). A fully algebraic method is presented, the paper explains in detail how simple algebra, coupled with basic quantum mechanical operator commutation algebra can be employed to solve the TISE without resorting to complicated mathematical methods of solving second-order differential equations. The construction of the Hamiltonian is explained and an analogy between the classical simple harmonic oscillator (CSHO) and the QSHO is drawn to develop the solution method. The action of creation and annihilation operators on wave functions is exploited to generate states with higher and lower energies respectively and to generate a Hamiltonian that can act on a system and generate required quantum states. The aim is to generate the ground state of the QSHO and its associated energy value. From this state, other higher energy states can be generated by the action of the creation operator. The use of algebraic methods has been seen to be versatile as it does not explicitly rely on any particular basis or coordinate system. The method allows operators to be manipulated in different bases for example the momentum operator can be given as P ̂ x=-iℏ (∂/∂x) in a spatial coordinate basis or as a pure number p ̂ =p in the momentum basis.

Non-Markovian cost function for quantum error mitigation with Dirac Gamma matrices representation

This paper investigates the non-Markovian cost function in quantum error mitigation (QEM) and employs Dirac Gamma matrices to illustrate two-qubit operators, significant in relativistic quantum mechanics. Amid the focus on error reduction in noisy intermediate-scale quantum (NISQ) devices, understanding non-Markovian noise, commonly found in solid-state quantum computers, is crucial. We propose a non-Markovian model for quantum state evolution and a corresponding QEM cost function, using simple harmonic oscillators as a proxy for environmental noise. Owing to their shared algebraic structure with two-qubit gate operators, Gamma matrices allow for enhanced analysis and manipulation of these operators. We evaluate the fluctuations of the output quantum state across various input states for identity and SWAP gate operations, and by comparing our findings with ion-trap and superconducting quantum computing systems' experimental data, we derive essential QEM cost function parameters. Our findings indicate a direct relationship between the quantum system's coupling strength with its environment and the QEM cost function. The research highlights non-Markovian models' importance in understanding quantum state evolution and assessing experimental outcomes from NISQ devices.