Harmonic Oscillator Potential Research Articles

One dimensional Bose–Einstein condensate under the effect of the extended uncertainty principle

In this study, an analytical investigation was conducted to assess the effects of the extended uncertainty principle (EUP) on a Bose–Einstein condensate (BEC) described by the deformed one-dimensional Gross–Pitaevskii equation (GPE). Analytical solutions were derived for null potential while we used variational and numerical methods for a harmonic oscillator potential. The effects of EUP on stability, probability density, position, and momentum uncertainties of BEC are analyzed. The EUP is found to be applicable for the free dark soliton solution and in the presence of a harmonic potential within specific ranges of the deformation parameter α, while it is not valid for the free bright soliton solution.

Spectrum conversion and pattern preservation of Airy beams in fractional systems with a dynamical harmonic-oscillator potential

We investigate the dynamics of optical Airy beams in the one-dimensional fractional Schrödinger equation with a harmonic-oscillator (HO) potential subjected to modulation along the propagation distance. Deriving general solutions for propagating beams and particular solutions for Airy waves with/without chirp, we study analytically the spectrum conversion and pattern preservation for the chirp-free and chirped Airy beams in the fractional system including the HO potential with moiré-lattice, hyperbolic-secant, and delta-functional modulation formats. For the HO-moiré-lattice potential, it is found that the chirp-free Airy beam experiences multiple spectrum conversions between the Airy and Gaussian patterns in the momentum space, preserving the Airy pattern in the coordinate space. The chirp magnitude of the chirped Airy beam determines whether the spectrum conversion occurs in the momentum space, and the splitting and evolution direction of the beam in the coordinate space. For the HO-hyperbolic-secant potential, the chirp-free Airy beam undergoes spectrum conversion and tunneling, with the positions of the spectrum conversion and tunneling significantly depending on parameters of the hyperbolic-secant potential; however, the spectrum conversion and pattern preservation of the chirped Airy beam occurs only under a certain relation of the chirp and parameters of the potential. In the case of the HO-delta-functional potential, the chirp-free Airy beam experiences abrupt spectrum conversion and a two-step spectrum shift; however, for the chirped Airy beam, the spectrum conversion is affected by the relation between the chirp and height of the potential. Effects of the fractional Lévy index on the spectrum conversion and pattern preservation of the Airy beams under the action of the three modulation patterns considered here are also explored in detail.

Bound states of relativistic spinless particles in a mix of circularly symmetric vector and scalar harmonic oscillators

We study the dynamics of relativistic spinless particles moving in a plane when there is circular symmetry. The general formalism for solving the Klein–Gordon equation in cylindrical coordinates for such systems is presented, as well as the conserved observables and the corresponding quantum numbers. We look for bound solutions of the corresponding Klein–Gordon equation when one has vector and scalar circularly symmetric harmonic oscillator potentials. Both positive and negative bound solutions are considered when there is either equal vector and scalar potentials or symmetric vector and scalar potentials, and it is shown how both cases are related through charge conjugation. We compute the non-relativistic limit for those cases, and show that for symmetric scalar and vector potentials the limit does not exist in the first order of an harmonic oscillator frequency, recovering a known result from the Dirac equation with the same kind of potentials.

The turbulent jet in the scale-relativity framework

The turbulent jet plays an important role in fluid mechanics since it is a prototype of free turbulent flow which presents many unsolved problems, among which the absence of closure of the governing equations. Despite the chaotic nature of turbulence, it is characterized by the existence of many universal structures, in particular dimensionless invariants such as its opening angle or the correlation coefficient of velocities. After having given new exact solutions of the Reynolds Averaged Navier–Stokes equations (RANS), we suggest a solution to this puzzle in the scale-relativity framework. In this theory, the time derivative of the Navier–Stokes equations is integrated in terms of a macroscopic Schrödinger equation acting in velocity-space. This equation involves a constant ℏv which can be identified with the energy dissipation rate, while the pressure gradient manifests itself as a quantum harmonic oscillator (QHO) potential. The squared modulus of its solutions yields the probability density function (PDF) of velocities. The Reynolds stresses can then be derived from this PDF, so that the closure problem is solved in this case. This allows us to obtain a theoretical prediction for the turbulent intensity radial profile (and therefore for the pressure) which agrees with the experimental data. The ratio of axial over radial velocity fluctuations is found to be R=1.3–1.4 from QHO properties, in good agreement with its experimental values; we theoretically predict a jet opening angle α=1/2R3 accounting for its universal value ≈1/5; the mean ratio of turbulent intensity amplitudes over jet centerline axial velocity is predicted to be X=19/80, in good agreement with its universally measured value ≈1/4; from these parameters we derive possible values of the radial turbulent intensity amplitude μ=0.19–0.22 consistent with experiments; finally, we find a correlation coefficient of velocities ρ=1/R3=2 α, suggesting an explanation for the universal value ≈0.4 observed for the turbulent jet and for all free shear flows.

Study of thermodynamic properties of N particles in one-dimensional harmonic oscillator potential

We have calculated the partition function and hence the energy and the specific heat of a two-particle system in n numbers of finite nondegenerate energy levels of equal spacing distributed according to classical and quantum statistics. We have then extended our study for N identical particles placed in 1D harmonic oscillator potential. It is observed that the specific heat of bosons and fermions are exactly the same in this potential at any temperature, and at high temperature the results are similar to that for the classical particles. The partition function of a system of two particles placed in 2D harmonic oscillator potential has also been determined as comparison.

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.

Demonstrating Shrodenger equation involving harmonic oscillator potential with a position dependent mass in an external electric field

An external electric field is provided to the familiar one-dimensional quantum harmonic oscillator model with a various position mass m(x)=(a m_0)/(a+x). The Nikiforov-Uvarov approach is involved to investigate a particular solution to the exact Schrödinger equation. The exactly solvable confined model of the quantum harmonic oscillator in an external electric field was proposed. Starting with the BenDanial-Duke kinetic energy operator approximations, the construction of the position-dependent mass Schrödinger equation is studied. The analytic representation of the wave functions of the stationary states is expressed analytically and graphically using modified Laguerre polynomials as well as the energy spectrum. In contrast with the absence of an external electric field, when the energy spectrum totally overlaps with that of the harmonic oscillator potential in an external electric field: the energy spectrum becomes non-equidistant and varies depending on some factors. The Nikiforov-Uvarov approach succeeds broadly in demonstrating the wave function and energy spectrum and showing good sense.

Rationally extended harmonic oscillator potential, isospectral family and the uncertainty relations

We consider the rationally extended harmonic oscillator potential which is isospectral to the conventional one and whose solutions are associated with the exceptional, Xm- Hermite polynomials and discuss its various important properties for different even codimension of m. The uncertainty relations are obtained for different m and it is shown that for the ground state, the uncertainty increases as m increases. A one parameter (λ) family of exactly solvable isospectral potential corresponding to this extended harmonic oscillator potential is obtained. Special cases corresponding to the λ=0 and λ=−1, which give the Pursey and the Abraham-Moses potentials respectively, are discussed. The uncertainty relations for the entire isospectral family of potentials for different m and λ are also calculated.

Behavior of spin-0 scalar particles in Eddington-inspired Born–Infeld gravity global monopole with harmonic oscillator potential

In this paper, we investigate the behavior of a scalar bosonic field within the backdrop of Eddington-inspired Born–Infeld gravity with a global monopole configuration. Additionally, we consider the presence of both scalar and vector potentials. To introduce the scalar potential, we perform a transformation on the mass term, replacing [Formula: see text] with [Formula: see text]. For the vector potential, we apply a minimal coupling scheme, transforming [Formula: see text] within the relativistic Klein–Gordon equation. We opt for a harmonic oscillator scenario where the scalar and vector potentials are equal, i.e. [Formula: see text]. Through this setup, we analytically solve the wave equation using the confluent Heun equation form. Subsequently, we delve into the quantum system in the absence of any potential, determining the permissible values for bound-state relativistic energy levels and the scalar field’s wave function. Our analysis reveals that the energy levels and wave function are influenced by various parameters present in the eigenvalue expression.

Describing geophysical turbulence with a Schrödinger–Coriolis equation in velocity space

In this paper, we examine the predictions of the scale-relativity approach for a turbulent fluid in rotation. We first show that the time derivative of the governing Navier–Stokes equation in the usual x-space can be transformed into a Schrödinger-like equation in velocity space with an external vectorial field to account for the rotation, together with a local velocity harmonic oscillator (VHO) potential in the v-space. The coefficients of this VHO are given by the second order x-derivatives of the pressure. We can then give formulas for the velocity and acceleration probability distribution functions (PDF). Using a simple model of anisotropic harmonic oscillator, we compare our predictions with relevant data from both direct numerical simulations (DNS) and oceanic drifter velocity measurements. We find a good agreement of the predicted acceleration PDF with that observed from drifters and some possible support in DNS for the existence of gaps in the local velocity PDF, expected in the presence of a Coriolis force.

Squeezing and evolution of single particle by frequency jumping in two-dimensional rotating harmonic

The control of microscopic particle behavior based on a specific external field has always been a research hotspot in the field of physics. Many studies have been exploring various methods to manipulate and control the behavior of particles at a microscopic level. In this work, we investigate the phenomenon of single-particle squeezing induced by frequency jumping in a two-dimensional rotating harmonic oscillator potential. Squeezing, as a quantum mechanical phenomenon, has attracted significant attention due to its potential applications in various fields. It refers to the reduction of fluctuations in certain physical quantities, allowing for more precise measurement results. Squeezing phenomena have been extensively studied in different physical systems, including optics, atomic physics, and solid-state physics. However, there have been few reports on the quantum state squeezing phenomenon induced by frequency jumping in a rotating harmonic oscillator potential. Therefore, our study aims to fill this gap and shed light on this intriguing phenomenon. To explore the squeezing phenomenon induced by frequency jumping, we focus on the fluctuations and squeezing of the single particle’s cyclotron radius coordinate and center-guided coordinate in the two-dimensional rotating harmonic oscillator potential. Through numerical simulations and theoretical analysis, we can understand the influence of frequency jumping on the degree of squeezing and reveal the underlying physical mechanism of squeezing evolution. In this work, we first investigate the influence of frequency jumping on the squeezing evolution of the cyclotron radius mode. By carefully selecting appropriate jumping moments, we analyze the influence of frequency jumping on the degree of squeezing. Our research results show that the degree of squeezing in the cyclotron radius coordinate remains unchanged at the jumping moment. However, we observe a stronger squeezing phenomenon in the subsequent evolution process. This indicates that frequency jumping plays a crucial role in squeezing evolution of the cyclotron radius mode. Furthermore, we focus on the squeezing evolution of the center-guided mode during frequency jumping. By selecting suitable parameters, we analyze the squeezing and evolution of two squeezing modes: the divergent mode and the oscillatory mode. Interestingly, we discover the existence of a critical potential trap aspect ratio, which is determined by the rotation angular velocity of the external potential. When the aspect ratio approaches this critical value, the squeezing mode undergoes a transition, and a significant squeezing phenomenon appears in the oscillatory mode. This finding provides valuable insights into the origin and control of squeezing phenomena. Finally, we discuss the potential applications of these squeezing phenomena. Squeezing has significant implications in the fields of quantum sensing and quantum information processing. Through a deeper understanding of the squeezing evolution process caused by frequency jump, we can better control the microscopic particle behavior through external field. This knowledge opens up new possibilities for future physical research and technical applications.

Does canonical quantization lead to GKSL dynamics?

We introduce a generalized classical model of Brownian motion for describing thermal relaxation processes which is thermodynamically consistent. Applying the canonical quantization to this model, a quantum equation for the density operator is obtained. This equation has a thermal equilibrium state as its stationary solution, but the time evolution is not necessarily a Completely Positive and Trace-Preserving (CPTP) map. In the application to the harmonic oscillator potential, however, the requirement of the CPTP map is shown to be satisfied by choosing parameters appropriately and then our equation reproduces a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation satisfying the detailed balance condition. This result suggests a quantum-classical correspondence in thermal relaxation processes and will provide a new insight to the study of decoherence.

Exact solution of three dimensional schrödinger equation with power function superposition potential.

Finding an analytical solution to the Schrödinger equation with power function superposition potential is essential for the development of quantum theory. For example, the harmonic oscillator potential, Coulomb potential, and Klazer potential are all classed as power function superposition potentials. In this study, the general form of the power function superposition potential was used to decompose the second-order radial Schrödinger equation with this potential into the first-order Ricatti equation. Furthermore, two forms of the power function superposition potential are constructed with an exact analytical solution, and the exact bound-state energy level formula is obtained for these two potentials. Finally, the energy levels of some of the diatomic molecules were determined through calculation. And our results are actually consistent with those obtained by other methods.

Three-dimensional eigenvalues of harmonic oscillator and Coulomb-type potentials from one-dimensional generalized Morse potential: perturbative analysis based on generalized Laguerre polynomials

Three-dimensional eigenvalues of harmonic oscillator and Coulomb-type potentials from one-dimensional generalized Morse potential: perturbative analysis based on generalized Laguerre polynomials

The Parametric Generalized Fractional Nikiforov-Uvarov Method and Its Applications

   By using generalized fractional derivative, the parametric generalized fractional Nikiforov-Uvarov (NU) method is introduced. The second-order parametric generalized differential equation is exactly solved in the fractional form. The obtained results are applied on the extended Cornell potential, the pesudoharmonic potential, the Mie potential, the Kratzer-Fues potential, the harmonic oscillator potential, the Morse potential, the Woods-Saxon potential, the Hulthen potential, the deformed Rosen-Morse potential and the P schl-Teller potential which play an important role in the fields of molecular and atomic physics. The special of classical cases are obtained from the fractional cases at  which are agreement with recent works.

Harmonically confined n -electron systems coupled to light in a cavity: Time-dependent case

An analytically solvable time-dependent coupled light-matter problem is presented. An $n$-electron system is confined by a harmonic-oscillator potential and interacts with photons in a cavity. Both the electrons and the photons can interact with a time-dependent external field. The light-matter coupling is described by the Pauli-Fierz Hamiltonian. By separating the relative and center-of-mass motion, the Hamiltonian of the system can be simplified to a sum of a Hamiltonian of the relative and the center-of-mass motion. The Hamiltonian of the relative motion is time-independent, not coupled to light, and it can be solved by conventional approaches. The Hamiltonian of the center-of-mass motion reduces to that of a time-dependent harmonic oscillator and can be solved analytically. The analytical solution will be used to study excitations, the high-harmonic-generation spectrum, and nonlinear optical properties in a cavity.

Shell Model Investigation of Some p and sd-Shell Nuclei with Harmonic Oscillator and Skyrme Interactions

In this study, the longitudinal charge and form factors for the nuclei 9Be and 28Si lying in the p and sd shells are studied by employing the Harmonic Oscillator potential (HO) and Skyrme effective interaction (Sk35−Skzs∗). The C0 and C2 from factors calculated for the ground state 3/2-, the 5/2- (2.429 MeV) and 7/2- (6.380 MeV) for 9Be, while the ground state 0+ and 2+ (1.779 MeV) state for 28Si nucleus. Calculations of microscopic perturbations that involve intermediate one-particle, one-hole excitation from the core and MS orbits into all upper orbits with excitations are utilized to generate the effective charges necessary to account for the “core polarization effect”. The shell model calculations are utilized on the extended model space to include all 1s, 1p, 2s–1d, 2p‑1f orbits with  truncation. Bohr-Mottelson collective model and Tassie model with properly estimated effective neutron and proton charges are taken into account to consider the effect of the core contribution. The estimated form factors were compared with the measured available data and they were in good agreement for most of the studied states. A conclusion can be drawn that  truncation is very good choice to study the longitudinal form factors.
 
 The choice of Harmonic Oscillator potential (HO) and Skyrme effective interaction (Sk35−Skzs∗) is adequate for form estimation of longitudinal form factors.
 The estimation of the effective charges based on microscopic perturbations that involve intermediate one-particle, one-hole excitation from the core and MS orbits into all upper orbits with excitations is adequate.
 The truncation proves to be very successful to perform the study.

The Magnetic Form Factors for Some Nuclei 51V, 59Co, 93Nb, 115In by using Valence with and Without Core Polarization Effects Models

The magnetic electron scattering form factor with glekpn, d3f7, ho models space for 51V , 59Co, 93Nb, and 115In nuclei are discussed with and without core polarization effects(CP). The calculations are done with the help of NuShellX@MUS code. The radial wave function for the single-particle matrix elements have been calculated with the SKyrme-Hartree Fock (SKX), Wood-Saxon(WS), and harmonic oscillator (HO) potentials. valence model(Vm) used in these calculation to calculate form factors with core-polarization effects. The results give a good agreement with available experimental data.

Three-dimensional approach applied to quasistationary states of deformed α emitters

We apply a three-dimensional (3D) approach to investigate the quasistationary states of well-deformed $\ensuremath{\alpha}$ emitters. With a splitting of the anisotropic 3D potential into internal and external parts at a separation surface, the 3D $\ensuremath{\alpha}$-cluster decay width is determined by the initial wave function of a true bound state of an anisotropic harmonic oscillator potential and a nonresonance scattering wave function of Coulomb potential. Substantial difference between the one-dimensional (1D) and 3D decay width is found for typical $\ensuremath{\alpha}$ emitters with large quadrupole and hexadecapole deformations.

Effects of the Lorentz Symmetry Violation Caused by the Coupling Between the Vector Field $$f^{\mu }\gamma ^{5}$$ and the Derivative of the Fermionic Field on the Modified Pöschl-Teller, Woods-Saxon and Harmonic Oscillator Potentials

Effects of the Lorentz Symmetry Violation Caused by the Coupling Between the Vector Field $$f^{\mu }\gamma ^{5}$$ and the Derivative of the Fermionic Field on the Modified Pöschl-Teller, Woods-Saxon and Harmonic Oscillator Potentials