Harmonic Oscillator Potential Research Articles

Infrared Spectra of Hydrogen-Bonded Molecular Complexes Under Spatial Confinement.

Infrared (IR) spectroscopy is commonly used in chemical laboratories to study the geometrical structure of molecules and molecular complexes. The analysis of experimental IR spectra can nowadays be reliably supported by the results of quantum-chemical computations as vibrational frequencies and corresponding vibrational transition intensities are routinely calculated using harmonic approximation by virtually all quantum chemistry packages. In the present study we combine the methodology of computing vibrational spectra using high-level electron correlation treatments with an analytical potential-based approach to take into account spatial confinement effects. Using this approach, we perform a pioneering analysis of the impact of the spatial confinement caused by a cylindrical harmonic oscillator potential on the harmonic vibrational transition intensities and frequencies of two hydrogen-bonded complexes: HCN…HCN and HCN…HNC. The emphasis is put on the largest-intensity bands, which correspond to the stretching vibrations. The obtained results demonstrate that embedding the molecular complexes in an external confining potential causes significant changes of transition intensities and vibrational frequencies.

Explicit Solution of the Focus Locus Problem for the Harmonic Oscillator Orbits in the Plane

Dynamical orbits of the harmonic oscillator potential in the plane are ellipses which depend on a real parameter. Some time ago in this journal it has been proven by purely geometrical methods that the locus of the focuses of these ellipses are Cassinian ovals. Here we present several explicit analytic parameterizations of these remarkable curves. Nominally, their forms depend on the magnitude of the initial distance from the center of attraction and the magnitude of the initial velocity. We have found a few parameterizations in which the roles of the size and shapes can be clearly distinguished.

Disguised electromagnetic connections in classical electron theory

In the first quarter of the 20th century, physicists were not aware of the existence of classical electromagnetic zero-point radiation nor of the importance of special relativity. Inclusion of these aspects allows classical electron theory to be extended beyond its 19th century successes. Here we review spherical electromagnetic radiation modes in a conducting-walled spherical cavity and connect these modes to classical electromagnetic zero-point radiation and to electromagnetic scale invariance. Then we turn to the scattering of radiation in classical electron theory within a simple approximation. We emphasize that, in steady-state, the interaction between matter and radiation is disguised so that the mechanical motion appears to occur without the emission of radiation, even though the particle motion is actually driven by classical electromagnetic radiation. It is pointed out that, for nonrelativistic particles, only the harmonic oscillator potential taken in the low-velocity limit allows a consistent equilibrium with classical electromagnetic zero-point radiation. For relativistic particles, only the Coulomb potential is consistent with electrodynamics. The classical analysis places restrictions on the value of e 2/ℏc.

Position-dependent mass Dirac equation and local Fermi velocity

We present a new approach to study the one-dimensional Dirac equation in the background of a position-dependent mass m. Taking the Fermi velocity v f to be a local variable, we explore the resulting structure of the coupled equations and arrive at an interesting constraint of m turning out to be the inverse square of v f. We address several solvable systems that include the free particle, shifted harmonic oscillator, Coulomb and nonpolynomial potentials. In particular, in the supersymmetric quantum mechanics context, the upper partner of the effective potential yields a new form for an inverse quadratic functional choice of the Fermi velocity.

Nonlinear optical responses of molecular systems with vibronic coupling in fluctuating environments

The topological profiles of the nonlinear optical properties of a two-level molecular system involving simplified versions of intramolecular coupling immersed in a thermal reservoir have been studied in four-wave mixing (FWM) spectroscopy. Solvent effects have been explicitly considered by modeling the nonradiative interaction with the solute as a random variable. To formulate a solution, it is necessary to introduce stochastic considerations. We assume that system–solvent interactions induce random shifts in the Bohr frequency of the adiabatic states, and their manifestation corresponds to the broadening of the upper level. The Born–Oppenheimer (BO) electronic energy curves for this molecular model consist of two intertwined harmonic oscillator potentials shifted in both their position and their minimum equilibrium energy. The critical quantities for this analysis are the transition and permanent dipole moments, which depend strongly on intramolecular coupling. Our results show how absorption and refractive index are affected by vibronic coupling and solvent stochasticity.

Fundamental energy scale of the thick brane in mimetic gravity

In this paper, thick branes generated by the mimetic scalar field with Lagrange multiplier formulation are investigated. We give three typical thick brane background solutions with different asymptotic behaviors and show that all the solutions are stable under tensor perturbations. The effective potentials of the tensor perturbations exhibit as volcano potential, Poöschl–Teller potential, and harmonic oscillator potential for the three background solutions, respectively. All the tensor zero modes (massless gravitons) of the three cases can be localized on the brane. We also calculate the corrections to the four-dimensional Newtonian potential. On a large scale, the corrections to the four-dimensional Newtonian potential can be ignored. While on a small scale, the correction from the volcano-like potential is more pronounced than the other two cases. Combining the specific corrections to the four-dimensional Newtonian potential of these three cases and the latest results of short-range gravity experiments, we get the constraint on the scale parameter as k > rsim 10^{-4}eV, and constraint on the corresponding five-dimensional fundamental scale as M_* > rsim 10^5TeV.

A New Constructive Method for Solving the Schrödinger Equation

In this paper, a new method for the exact solution of the stationary, one-dimensional Schrödinger equation is proposed. Application of the method leads to a three-parametric family of exact solutions, previously known only in the limiting cases. The method is based on solutions of the Ricatti equation in the form of a quadratic function with three parameters. The logarithmic derivative of the wave function transforms the Schrödinger equation to the Ricatti equation with arbitrary potential. The Ricatti equation is solved by exploiting the particular symmetry, where a family of discrete transformations preserves the original form of the equation. The method is applied to a one-dimensional Schrödinger equation with a bound states spectrum. By extending the results of the Ricatti equation to the Schrödinger equation the three-parametric solutions for wave functions and energy spectrum are obtained. This three-parametric family of exact solutions is defined on compact support, as well as on the whole real axis in the limiting case, and corresponds to a uniquely defined form of potential. Celebrated exactly solvable cases of special potentials like harmonic oscillator potential, Coulomb potential, infinite square well potential with corresponding energy spectrum and wave functions follow from the general form by appropriate selection of parameters values. The first two of these potentials with corresponding solutions, which are defined on the whole axis and half axis respectively, are achieved by taking the limit of general three-parametric solutions, where one of the parameters approaches a certain, well-defined value.

Superintegrability of Calogero–Moser systems associated with the cyclic quiver

We study complex integrable systems on quiver varieties associated with the cyclic Noquiver, and prove their superintegrability by explicitly constructing first integrals. We interpret them as rational Calogero–Moser systems endowed with internal degrees of freedom called spins. They encompass the usual systems in type A n−1 and B n , as well as generalisations introduced by Chalykh and Silantyev in connection with the multicomponent KP hierarchy. We also prove that superintegrability is preserved when a harmonic oscillator potential is added.

Radial excitation of light mesons and the γγ*→π0 transition form factor

We investigate radial excitation of the quark-antiquark pair in the ${\ensuremath{\pi}}^{0}$ meson and its effects on the $\ensuremath{\gamma}{\ensuremath{\gamma}}^{*}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}$ transition form factor in the framework of light cone perturbative QCD. The existing constraints on the light cone wave function of the lowest Fock state $|q\overline{q}⟩$ in the ${\ensuremath{\pi}}^{0}$ meson allow a sizeable radial excitation of the quark-antiquark pair. We construct the light cone wave function for the quark-antiquark pair in the first radially excited state (the 2S state) using a simple harmonic oscillator potential. The distribution amplitude obtained for the 2S state has two nodes in $x$ at low scale of $Q$ and thereby has a much strong scale dependence than the 1S state. Contributions from this radial excitation to the $\ensuremath{\gamma}{\ensuremath{\gamma}}^{*}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}$ transition form factor exhibit different ${Q}^{2}$-dependence behavior from the ground state and thus can modify the prediction for the transition form factor in the medium-large region of ${Q}^{2}$.

Fermi-Pasta-Ulam phenomena and persistent breathers in the harmonic trap.

We consider the long-term weakly nonlinear evolution governed by the two-dimensional nonlinear Schrödinger (NLS) equation with an isotropic harmonic oscillator potential. The dynamics in this regime is dominated by resonant interactions between quartets of linear normal modes, accurately captured by the corresponding resonant approximation. Within this approximation, we identify Fermi-Pasta-Ulam-like recurrence phenomena, whereby the normal-mode spectrum passes in close proximity of the initial configuration, and two-mode states with time-independent mode amplitude spectra that translate into long-lived breathers of the original NLS equation. We comment on possible implications of these findings for nonlinear optics and matter-wave dynamics in Bose-Einstein condensates.

Analytical eigenstate solutions of Schrödinger equation with noncentral generalized oscillator potential by extended Nikiforov-Uvarov method

Exact eigenstate solutions of Schrödinger equation for a generalized oscillator system that includes, as special cases; ring shaped oscillator potential and isotropic harmonic oscillator potential are examined in an analytical treatment by using extended Nikiforov Uvarov method. Radial and angular parts of the Schrödinger equation for the relevant potential can be systematically solved in the same manner. Moreover the relation between the eigenfunction solutions for this potential and Heun polynomials are presented.

Exact solutions to generalized Dunkl oscillator and its thermodynamic properties

In this work, based on the generalized Dunkl derivative in quantum mechanics we study the one-dimensional Schrödinger equation with a harmonic oscillator potential and obtain the energy eigenvalues. The principal thermodynamical properties including the Helmholtz free energy, mean energy and entropy are carried out. The effects of the Dunkl parameters on the thermodynamical quantities for even parity are discussed. The case of the odd parity can be easily obtained by substitution of the and . All results in the limit case are reduced to ordinary statistical mechanics.

Quasi-exactly solvable Schrödinger equation for a modified ring-shaped harmonic oscillator potential

Quasi-exactly solvable Schrödinger equation for a modified ring-shaped harmonic oscillator potential

Do equidistant energy levels necessitate a harmonic potential?

Experimental results from literature show equidistant energy levels in thin Bi films on surfaces, suggesting a harmonic oscillator description. Yet this conclusion is by no means imperative, especially considering that any measurement only yields energy levels in a finite range and with a nonzero uncertainty. Within this study we review isospectral potentials from the literature and investigate the applicability of the harmonic oscillator hypothesis to recent measurements. First, we describe experimental results from literature by a harmonic oscillator model, obtaining a realistic size and depth of the resulting quantum well. Second, we use the shift-operator approach to calculate anharmonic non-polynomial potentials producing (partly) equidistant spectra. We discuss different potential types and interpret the possible modeling applications. Finally, by applying nth order perturbation theory we show that exactly equidistant eigenenergies cannot be achieved by polynomial potentials, except by the harmonic oscillator potential. In summary, we aim to give an overview over which conclusions may be drawn from the experimental determination of energy levels and which may not.

Relativistic Treatment of Quantum Mechanical Gravitational-Harmonic Oscillator Potential

The solutions of the Klein- Gordon equation for the quantum mechanical gravitational plus harmonic oscillator potential with equal scalar and vector potential have been presented using the parametric Nikiforov-Uvarov method. The energy eigenvalues were obtained in relativistic and non-relativistic regime and the corresponding un-normalized eigenfunctions in terms of Laguerre polynomials. The numerical values for the S – wave bound state were obtained.

Superstatistics of the Dunkl oscillator

We use the momentum operator with the Dunkl derivative in quantum mechanics and derive its Schrödinger equation in one dimension with a harmonic oscillator potential. With the energy eigenvalues of such systems, we calculate their principal thermodynamical properties, the Helmholtz free energy, mean energy and entropy, and discuss the effects of the Dunkl parameter and modified Dirac delta function parameters on thermodynamical quantities. Then we use the superstatistics and calculate the thermodynamical properties of the system and analyze the effects of all the parameters on the thermodynamical quantities. We reduce all the results to the limit case of ordinary statistical mechanics.

Skyrme ve Gogny Modelleriyle Bakır İzotoplarının Yoğunluk Hesabı

The ground state properties of the nuclei are generally calculated using two different methods, namely Skyrme and Gogny force methods. We have calculated nucleon densities of Copper isotopes by using Hartree-Fock-Skyrme (using Woods-Saxon Potential) (SHF-WS), Hartree-Fock-Skyrme (using Harmonic Oscillator Potential) (SHF-HO), Hartree-Fock-Bogolyubov-Skyrme (HFB-S) and the Hartree-Fock-Bogolyubov-Gogny (HFB-G) methods. In the first two methods, the densities and rms (root mean square) radii for both proton and neutron of copper isotopes were calculated by different Skyrme parameters set. Theoretical calculated charge density was compared with experimental data of Angeli and Marinova to determine the best parameter set for each copper isotope. Then all evaluated nucleon densities via four different methods were compared each other. All methods gave similar results for all copper isotopes.

Dirac particles in the presence of a constant magnetic field and harmonic potential with spin symmetry

In this paper, we study the effect of the constant magnetic field on energy levels of the Dirac particles such as electron, proton and heavy ions. We calculate the energy eigenvalues of the Dirac equation in the presence of the magnetic field and two-dimensional harmonic oscillator potential with spin symmetry by using the supersymmetric quantum mechanics and asymptotic iteration methods.

Large Scale Shell Model Calculations of the Negative-Parity States Structure in 24Mg Nucleus

The negative-parity states of 24Mg nucleus are investigated within the shell model. We are based on the calculations of energy levels, total squared form factors, and transition probability using the p-sd-pf (PSDPF) Hamiltonian in a large model space (0 + 1) hW. The comparison between the experimental and theoretical states showed a good agreement within a truncated model space. The PSDPF-based calculations successfully reproduced the data on the total squared form factors and transition probabilities of the negative-parity states in 24Mg nucleus. These quantities depend on the one-body density matrix elements that are obtained from the PSDPF Hamiltonian. The wave functions of radial one-particle matrix elements calculated with the harmonic-oscillator potential are suitable to predict experimental data by changing the center-of-mass corrections.

Bound states of Dirac equation using the proper quantization rule

Using the proper quantization rule, we investigate the exact solution of Dirac equation for Hartmann and the ring-shaped non-spherical harmonic oscillator potentials under the condition of equal scalar and vector potentials. By considering the proper quantization condition within angular and radial variables, the exact relativistic energy spectra are obtained for each system. Then by the mean of suitable changes of variables, the corresponding spinor wave-functions are constructed where the normalization constants are exactly calculated. We also derived the non-relativistic limit of energy spectra.