Harmonic Oscillator Problem Research Articles

Morris–Thorne-type wormhole with a cosmic string effects on harmonic oscillator problem

Morris–Thorne-type wormhole with a cosmic string effects on harmonic oscillator problem

The uncertainty principle and quantum wave functions that are their own Fourier transforms

We present several variations of a proof of the position-momentum uncertainty principle that are based on the calculus of variations and does not rely on the Cauchy–Schwartz inequality. We show that the stationary uncertainty wave functions are the Hermite–Gaussian solutions to the quantum harmonic oscillator problem, that the minimum uncertainty wave function is the Gaussian, and that stationary uncertainty wave functions must be their own Fourier transforms. We also provide a calculus of variations proof of the Cauchy–Schwartz inequality. Finally, we discuss the properties of wave functions that are their own Fourier transforms and provide examples of such functions that may be of interest to teachers of undergraduate and graduate level quantum mechanics courses.

On the path integral formulation of Wigner–Dunkl quantum mechanics

Feynman’s path integral approach is studied in the framework of the Wigner–Dunkl deformation of quantum mechanics. We start with reviewing some basics from Dunkl theory and investigate the time evolution of a Gaussian wave packet, which exhibits the same dispersion relation as observed in standard quantum mechanics. Feynman’s path integral approach is then extended to Wigner–Dunkl quantum mechanics. The harmonic oscillator problem is solved explicitly. We then look at the Euclidean time evolution and the related Dunkl process. This process, which exhibit jumps, can be represented by two continuous Bessel processes, one with reflection and one with absorption at the origin. The Feynman–Kac path integral for the harmonic oscillator problem is explicitly calculated.

Comment on “Harmonic oscillator problem in the background of a topologically charged Ellis-Bronnikov–type wormhole” by Ahmed Faizuddin

Abstract We analyse some results appeared in a paper published recently in this journal and show that the author obtained wrong solutions for the two eigenvalue equations studied. In particular, he obtained bound states for a problem that does not support them. The mistake can be traced back to a failed truncation of the power-series ansatz proposed as a possible solution. We also show that the problem is conditionally solvable and one can obtain analytical solutions only for particular values of the model parameters.

Rotational and inverse-square potential effects on harmonic oscillator confined by flux field in a space–time with screw dislocation

This research paper delves into the study of a nonrelativistic quantum system, considering the interplay of noninertial effects induced by a rotating frame and confinement by the Aharonov–Bohm (AB) flux field with potential in the backdrop of topological defects, specifically a screw dislocation. We first focus on the harmonic oscillator problem, incorporating an inverse-square repulsive potential. Notably, it becomes evident that the energy eigenvalues and wave functions are intricately influenced by multiple factors: the topological defect parameter [Formula: see text] (representing the screw dislocation), the presence of a rotating frame engaged in constant angular motion with speed [Formula: see text] and the external potential. Then we study the quantum behavior of nonrelativistic particles, engaging in interactions governed by an inverse-square potential, all while taking into account the effects of the rotating frame. In both scenarios, a significant observation is made: the quantum flux field’s existence brings about a shift in the energy spectrum. This phenomenon bears a resemblance to the electromagnetic Aharonov–Bohm effect.

Stability of Finite Difference Solution of Time-Dependent Schrodinger Equations

In this paper, the stability of finite difference methods for time-dependent Schrodinger equation with Dirichlet boundary conditions on a staggered mesh was considered with explicit and implicit discretization. Using the matrix representation for the numerical algorithm, it is shown that for the explicit finite difference method, the solution is conditionally stable while it becomes unconditionally stable for implicit finite difference methods. A 1D Harmonic Oscillator problem shall be used to illustrate this behaviour.

Harmonic oscillator in the context of the extended uncertainty principle

At large-scale distances where the space-time is curved due to gravity, a nonzero minimal uncertainty in the momentum, [Formula: see text], is being estimated to emerge. The presence of minimal uncertainty in momentum allows a modification to the quantum uncertainty principle, which is known as the extended uncertainty principle (EUP). In this work, we handle the harmonic oscillator problem in the EUP scenario and obtain analytical exact solutions in classical and semi-classical domains. In the classical context, we establish the equations of motion of the oscillator and show that the EUP-corrected frequency is depending on the energy and deformation parameter. In the semi-classical domain, we derive the energy eigenvalue levels and demonstrate that the energy spectrum depends on [Formula: see text], as the feature of hard confinement. Finally, we investigate the impact of the EUP on the harmonic oscillator’s thermodynamic properties by using the EUP-corrected partition functions in the classical limit in the (A)dS backgrounds.

Underwater oscillations of rigid plates with H-shaped cross sections: An experimental study to explore their flow physics

In this work, we present a comprehensive experimental study on the problem of harmonic oscillations of rigid plates with H-shaped cross sections submerged in a quiescent, Newtonian, incompressible, viscous fluid environment. Motivated by recent results on the minimization of hydrodynamic damping for transversely oscillating flat plates, we conduct a detailed qualitative and quantitative experimental investigation of the flow physics created by the presence of the flanges, that is, the vertical segments in the plate cross section. Specifically, the main goal is to elucidate the effect of flange size on various aspects of fluid–structure interaction, by primarily investigating the dynamics of vortex shedding and convection. We perform particle image velocimetry experiments over a broad range of oscillation amplitudes, frequencies, and flange size-to-width ratios by leveraging the identification of pathlines, vortex shedding and dynamics, distinctive hydrodynamic regimes, and steady streaming. The fundamental contributions of this work include novel hydrodynamic regime phase diagrams demonstrating the effect of flange ratio on regime transitions, and in the investigation of their relation to qualitatively distinct patterns of vortex–vortex and vortex–structure interactions. Finally, we discuss steady streaming, identifying primary, and secondary structures as a function of the governing parameters.

Harmonic oscillator problem in the background of a topologically charged Ellis-Bronnikov–type wormhole(a)

In this paper, we study the quantum dynamics of non-relativistic particles in the background of a topological chagred Ellis-Bronnikov–type wormhole space-time. We derived the radial wave equation and obtain the eigenvalue solution through the confluent Heun equation. Afterwards, we study the harmonic oscillator problem in the same wormhole background and solve the wave equation using the same technique. In both cases, the ground state energy level E 1,l and the wave function are presented. Finally, we analyze the results and show that the eigenvalue solutions are influenced by the topological defect parameter α and the wormhole throat radius and get them modified.

Effect of electric and magnetic field on thermal property of two dimensional harmonic oscillator

The energy spectrum of a 2D quantum harmonic oscillator confined by a hard wall circular cavity of size r0 in the presence of static electric and the magnetic field is computed numerically using the Bessel function basis set and the linear variational process. The effect of the size of the circular cavity, magnetic field, and electric field strength is studied on the energy levels, the radial transition matrix elements, and the thermal properties of the system like specific heat, Helmholtz free energy, and entropy using canonical partition function. Blueshift is observed for an increase in magnetic and electric field strength. The harmonic oscillator problem is critical for understanding any natural system that undergoes deformation and performs simple harmonic motion.

ANALYTICAL SOLUTION OF THE DIFFERENTIAL EQUATION OF FORCED OSCILLATIONS OF A VIBRATION PROTECTION SYSTEM WITH A PIECEWISE LINEAR STATIC FORCE CHARACTERISTIC UNDER HARMONIC KINEMATIC EXCITATION OF BASE DISPLACEMENTS

The need to protect operators of mobile machines from vibrations determines the relevance of the development of mathematical models for studying the dynamic processes of forced vibrations of the vibration-protected mass of the seat with the operator. Analytical solutions have the advantage of being the most accurate. For the design scheme of a vibration protection system with one degree of freedom in the form of a linear oscillator with kinematic excitation, an analytical solution to the problem of forced harmonic oscillations was obtained for the given values of the parameters: the initial absolute displacements and speed of the vibration-protected mass, the initial phase of the given harmonic oscillations of the seat base. At the same time, the expression of the static power characteristic of the vibration protection system was a straight line with a vertical displacement, which made it possible to break the transient process into separate time intervals, within which the local coordinate, showing the deformation of the vibration protection mechanism, is within a separate rectilinear segment of the piecewise linear static characteristic. The results of the comparative analysis showed a significant discrepancy between the piecewise analytical solution and the solution obtained by the numerical integration method.

Solution for the simple harmonic oscillator problem via unilateral Fourier transform

Solution for the simple harmonic oscillator problem via unilateral Fourier transform

Solving non-Hermitian Dirac equation in the presence of PDM and local Fermi velocity

We present a new approach to study a class of non-Hermitian ([Formula: see text])-dimensional Dirac Hamiltonian in the presence of local Fermi velocity. We apply the well-known Nikiforov–Uvarov method to solve such a system. We discuss applications and explore the solvability of both [Formula: see text]-symmetric and non[Formula: see text]-symmetric classes of potentials. In the former case, we obtain the solution of a harmonic oscillator in the presence of a linear vector potential while in the latter case we solve the shifted harmonic oscillator problem.

On the Quantization of Length in Noncommutative Spaces

We consider canonical/Weyl-Moyal type noncommutative (NC) spaces with rectilinear coordinates. Motivated by the analogy of the formalism of the quantum mechanical harmonic oscillator problem in quantum phase-space with that of the canonical-type NC 2-D space, and noting that the square of length in the latter case is analogous to the Hamiltonian in the former case, we arrive at the conclusion that the length and area are quantized in such an NC space, if the area is expressed entirely in terms of length. We extend our analysis to the 3-D case and formulate a ladder operator approach to the quantization of length in 3-D space. However, our method does not lend itself to the quantization of spacetime length in 1 + 1 and 2 + 1 Minkowski spacetimes if the noncommutativity between time and space is considered. If time is taken to commute with spatial coordinates and the noncommutativity is maintained only among the spatial coordinates in 2 + 1 and 3 + 1 dimensional spacetime, then the quantization of spatial length is possible in our approach.

Application of matrices for transfer function values calculation in mechanical systems

The paper describes the formalism and methodology that can be employed for calculation of transfer function in distributed parameter elastic mechanical systems. Generalized formalism for derivation of transfer function matrices is presented as developed in accordance with the Hilbert-Schmidt theorem. Matrix-based method to determine elastic mechanical system transfer function is offered. The solution of the problem of forced harmonic oscillations of an elastic body of a mechanical system with the presence of friction is considered. The described technique can be applied to find the transfer functions of technological equipment in order to determine the dynamic characteristics of the system.

Studying the interaction of a reservoir with an ideal compressible fluid with an elastic half-space

We solve in a flat formulation the problem of harmonic oscillations for a basin with an ideal compressible fluid on an elastic half-space exposed to a localized surface vibration load. The problem reduces to an integral equation (IE) of the first kind for the amplitude of the contact hydrodynamic pressure with a kernel that depends on the difference and the sum of arguments. The IE was solved by the factorization method. A semi-analytical method is presented for determining the main parameters of the contact interaction in hydroelastic systems «liquid-soil» taking into account the effect of natural and man-made vibration loads on them. This makes it possible to identify the conditions for the occurrence of dynamic modes that are dangerous for the construction integrity and to estimate their frequencies range depending on defining characteristics of the system.

A new series solution associated with the quantum harmonic oscillator

In the non-relativistic case, the problem of quantum harmonic oscillator is solved by the aid of the Schrödinger equation. An additional term of potential energy that is related with the square of the spatial coordinate is also included in the equation. A second order linear differential equation is obtained in the stationary state. The direct solution gives the wave function in terms of the parabolic cylinder functions. This solution leads to the Hermite polynomials when the quantization of energy is considered. However, the derived expression of the wave function does not yield the correct functions for the limiting case, in which the potential energy is zero. We put forth an alternative solution for the second order linear differential equation. First of all, a new series expression is suggested as an ansatz for the power series solution of differential equations. The coefficients of the series are obtained in terms of recurrence relations. Some computational simulations are given.

An MHD spectral theory approach to Jeans’ magnetized gravitational instability

Abstract In this paper, we revisit the governing equations for linear magnetohydrodynamic (MHD) waves and instabilities existing within a magnetized, plane-parallel, self-gravitating slab. Our approach allows for fully non-uniformly magnetized slabs, which deviate from isothermal conditions, such that the well-known Alfvén and slow continuous spectra enter the description. We generalize modern MHD textbook treatments, by showing how self-gravity enters the MHD wave equation, beyond the frequently adopted Cowling approximation. This clarifies how Jeans’ instability generalizes from hydro to magnetohydrodynamic conditions without assuming the usual Jeans’ swindle approach. Our main contribution lies in reformulating the completely general governing wave equations in a number of mathematically equivalent forms, ranging from a coupled Sturm-Liouville formulation, to a Hamiltonian formulation linked to coupled harmonic oscillators, up to a convenient matrix differential form. The latter allows us to derive analytically the eigenfunctions of a magnetized, self-gravitating thin slab. In addition, as an example we give the exact closed form dispersion relations for the hydrodynamical p- and Jeans-unstable modes, with the latter demonstrating how the Cowling approximation modifies due to a proper treatment of self-gravity. The various reformulations of the MHD wave equation open up new avenues for future MHD spectral studies of instabilities as relevant for cosmic filament formation, which can e.g. use modern formal solution strategies tailored to solve coupled Sturm-Liouville or harmonic oscillator problems.

Euler's rigid rotators, Jacobi elliptic functions, and the Dzhanibekov or tennis racket effect

In this paper, the torque-free rotational motion of a general rigid body is developed analytically and is applied to the flipping motion of a T-handle spinning in zero gravity that can be seen in videos on the internet. This flipping motion is known both as the Dzhanibekov effect (after the cosmonaut who reported it) and more recently the tennis racket effect. The presentation is self-contained, accessible to students, and is complementary to the treatment found in most texts in that it involves a time-dependent analytical solution in terms of elliptic functions as opposed to a development based on conservation laws. These two complementary approaches are interesting and useful in different ways. In the present approach, the Euler rigid-body equations are derived and then solved as differential equations that are satisfied by Jacobi elliptic functions. This is analogous to solving the spring–mass harmonic oscillator problem by turning Newton's laws into differential equations that are satisfied by sine and cosine functions. The Jacobi functions are closely related to these trigonometric functions and are only slightly more complicated. They are defined as geometrical ratios on a reference ellipse and developed geometrically without reference to power series or complex variables. However, because these functions are less familiar, they are introduced in a short Appendix where their main properties are derived. Also, a link is provided to a Mathematica script for animating the analytical solution to the present problem.

Dark–dark soliton breathing patterns in multi-component Bose–Einstein condensates

In this work, we explore systematically various SO(2)-rotation-induced multiple dark–dark (DD) soliton breathing patterns obtained from stationary and spectrally stable multiple dark–bright (DB) and DD waveforms in trapped one-dimensional, two-component atomic Bose–Einstein condensates. The stationary states stemming from the associated linear limits (as the eigenfunctions of the quantum harmonic oscillator problem) are parametrically continued to the nonlinear regimes by varying the respective chemical potentials, i.e. from the low-density linear limits to the high-density Thomas–Fermi (TF) regimes. We perform a Bogolyubov–de Gennes spectral stability analysis to identify stable parametric regimes of these states, finding a wide range of stability intervals in the TF regimes for all of the states considered herein. Upon applying an SO(2)-rotation to stable steady states, one-, two-, three-, four-, and many DD soliton breathing patterns are observed in the numerical simulations. Furthermore, analytic solutions up to three DB solitons in the homogeneous setting, and three-component systems are also investigated.