Isotropic Harmonic Oscillator Research Articles

First and second-order relativistic corrections to the two and higher-dimensional isotropic harmonic oscillator obeying the spinless Salpeter equation

We study the relativistic version of the d-dimensional isotropic quantum harmonic oscillator based on the spinless Salpeter equation. This has no exact analytical solutions. We use perturbation theory to obtain compact formulas for the first and second-order relativistic corrections; they are expressed in terms of two quantum numbers and the spatial dimension d. The formula for the first-order correction is obtained using two different methods and we illustrate how this correction splits the original energy into a number of distinct levels each with their own degeneracy. Previous authors obtained results in one and three dimensions and our general formulas reduce to them when d = 1 and d = 3 respectively. Our two-dimensional results are novel and we provide an example that illustrates why two dimensions is of physical interest. We also obtain results for the two-dimensional case using a completely independent method that employs ladder operators in polar coordinates. In total, three methods are used in this work and the results all agree.

Exploration of near the origin and the asymptotic behaviors of the Kohn-Sham kinetic energy density for two-dimensional quantum dot systems with parabolic confinement.

The behaviors of the positive definite Kohn-Sham kinetic energy density near the origin and at the asymptotic region play a major role in designing meta-generalized gradient approximations (meta-GGAs) for exchange in low-dimensional quantum systems. It is shown that near the origin of the parabolic quantum dot, the Kohn-Sham kinetic energy differs from its von Weizsäcker counterpart due to the p orbital contributions, whereas in the asymptotic region, the difference between the above two kinetic energy densities goes as ∼ρ(r)r2. All these behaviors have been explored using the two-dimensional isotropic quantum harmonic oscillator as a test case. Several meta-GGA ingredients are then studied by making use of the above findings. Also, the asymptotic conditions for the exchange energy density and the potential at the meta-GGA level are proposed using the corresponding behaviors of the two kinetic energy densities.

Alternative angle-based approach to the $\mathcal{KS}$-Map. An interpretation through symmetry and reduction

The $\mathcal{KS}$ map is revisited in terms of an $S^1$-action in $T^*\mathbb{H}_0$ with the bilinear function as the associated momentum map. Indeed, the $\mathcal{KS}$ transformation maps the $S^1$-fibers related to the mentioned action to single points. By means of this perspective a second twin-bilinear function is obtained with an analogous $S^1$-action. We also show that the connection between the 4-D isotropic harmonic oscillator and the spatial Kepler systems can be done in a straightforward way after regularization and through the extension to 4 degrees of freedom of the Euler angles, when the bilinear relation is imposed. This connection incorporates both bilinear functions among the variables. We will show that an alternative regularization separates the oscillator expressed in Projective Euler variables. This setting takes advantage of the two bilinear functions and another integral of the system including them among a new set of variables that allows to connect the 4-D isotropic harmonic oscillator and the planar Kepler system. In addition, our approach makes transparent that only when we refer to rectilinear solutions, both bilinear relations defining the $\mathcal{KS}$ transformations are needed.

Экситоны и биэкситоны в сфероидальных квантовых точках А-=SUB=-2-=/SUB=-B-=SUB=-6-=/SUB=-

AbstractIn the limit of strong quantum confinement the lower energy states of excitons and biexcitons in spheroidal quantum dots of semiconductors with a fourfold degenerate vertex of the valence band, which are active in the dipole approximation at one- and two-photon excitation, have been considered. The comparative analysis of the order of energy levels of the hole in the potentials of the infinitely deep quantum well and a three-dimensional harmonic oscillator taking into account the axial anisotropy of the quantum dot (QD) shape is carried out. It is shown that the anisotropy of the QD shape can lead to the opposite sign of splitting with respect to angular momentum projection ±3/2, ±1/2 for spatially odd (1 P _3/2) and even (1 S _3/2) levels of the hole. At the same time, in the case of the potential of an infinitely deep quantum well, an inversion of the order of 1 S _3/2 and 1 P _3/2 levels can be observed at values of the ratio of the effective masses of the light and heavy holes β = m _lh/ m _hh ≈ 0.14. The type of the trial wave functions of the hole for the state 1 P _3/2 in the potential of an isotropic three-dimensional harmonic oscillator depending on β is proposed. The dependence of the binding energy of excitons in the considered potentials on β is presented and the possibility of formation of various biexcitonic states is considered.

Extreme quantum nonequilibrium, nodes, vorticity, drift and relaxation retarding states

Consideration is given to the behaviour of de Broglie trajectories that are separated from the bulk of the Born distribution with a view to describing the quantum relaxation properties of more ‘extreme’ forms of quantum nonequilibrium. For the 2D isotropic harmonic oscillator, through the construction of what is termed the ‘drift field’, a description is given of a general mechanism that causes the relaxation of ‘extreme’ quantum nonequilibrium. Quantum states are found which do not feature this mechanism, so that relaxation may be severely delayed or possibly may not take place at all. A method by which these states may be identified, classified and calculated is given in terms of the properties of the nodes of the state. Properties of the nodes that enable this classification are described for the first time.

Erratum to: Benchmark Calculation of Radial Expectation Value $$\varvec{\langle r^{-2} \rangle }$$ ⟨ r - 2 ⟩ for Confined Hydrogen-Like Atoms and Isotropic Harmonic Oscillators

Spatially confined atoms have been extensively investigated to model atomic systems in extreme pressures. For the simplest hydrogen-like atoms and isotropic harmonic oscillators, numerous physical quantities have been established with very high accuracy. However, the expectation value of $$\langle r^{-2} \rangle $$ which is of practical importance in many applications has significant discrepancies among calculations by different methods. In this work we employed the basis expansion method with cut-off Slater-type orbitals to investigate these two confined systems. Accurate values for several low-lying bound states were obtained by carefully examining the convergence with respect to the size of basis. A scaling law for $$\langle r^{n} \rangle $$ was derived and it is used to verify the accuracy of numerical results. Comparison with other calculations show that the present results establish benchmark values for this quantity, which may be useful in future studies.

Benchmark Calculation of Radial Expectation Value $$\varvec{\langle r^{-2} \rangle }$$ ⟨ r - 2 ⟩ for Confined Hydrogen-Like Atoms and Isotropic Harmonic Oscillators

Spatially confined atoms have been extensively investigated to model atomic systems in extreme pressures. For the simplest hydrogen-like atoms and isotropic harmonic oscillators, numerous physical quantities have been established with very high accuracy. However, the expectation value of $$\langle r^{-2} \rangle $$ which is of practical importance in many applications has significant discrepancies among calculations by different methods. In this work we employed the basis expansion method with cut-off Slater-type orbitals to investigate these two confined systems. Accurate values for several low-lying bound states were obtained by carefully examining the convergence with respect to the size of basis. A scaling law for $$\langle r^{n} \rangle $$ was derived and it is used to verify the accuracy of numerical results. Comparison with other calculations show that the present results establish benchmark values for this quantity, which may be useful in future studies.

A superintegrable discrete harmonic oscillator based on bivariate Charlier polynomials

A simple discrete model of the two-dimensional isotropic harmonic oscillator is presented. It is superintegrable with su(2) as its symmetry algebra. It is constructed with the help of the algebraic properties of the bivariate Charlier polynomials. This adds to the other discrete superintegrable models of the oscillator based on Krawtchouk and Meixner orthogonal polynomials in several variables.

Incorporating nuclear vibrational energies into the "atom in molecules" analysis: An analytical study.

The quantum theory of atoms in molecules (QTAIM) is based on the clamped nucleus paradigm and solely working with the electronic wavefunctions, so does not include nuclear vibrations in the AIM analysis. On the other hand, the recently extended version of the QTAIM, called the multi-component QTAIM (MC-QTAIM), incorporates both electrons and quantum nuclei, i.e., those nuclei treated as quantum waves instead of clamped point charges, into the AIM analysis using non-adiabatic wavefunctions. Thus, the MC-QTAIM is the natural framework to incorporate the role of nuclear vibrations into the AIM analysis. In this study, within the context of the MC-QTAIM, the formalism of including nuclear vibrational energy in the atomic basin energy is developed in detail and its contribution is derived analytically using the recently proposed non-adiabatic Hartree product nuclear wavefunction. It is demonstrated that within the context of this wavefunction, the quantum nuclei may be conceived pseudo-adiabatically as quantum oscillators and both isotropic harmonic and anisotropic anharmonic oscillator models are used to compute the zero-point nuclear vibrational energy contribution to the basin energies explicitly. Inspired by the results gained within the context of the MC-QTAIM analysis, a heuristic approach is proposed within the context of the QTAIM to include nuclear vibrational energy in the basin energy from the vibrational wavefunction derived adiabatically. The explicit calculation of the basin contribution of the zero-point vibrational energy using the uncoupled harmonic oscillator model leads to results consistent with those derived from the MC-QTAIM.

Swings and roundabouts: optical Poincaré spheres for polarization and Gaussian beams.

The connection between Poincaré spheres for polarization and Gaussian beams is explored, focusing on the interpretation of elliptic polarization in terms of the isotropic two-dimensional harmonic oscillator in Hamiltonian mechanics, its canonical quantization and semiclassical interpretation. This leads to the interpretation of structured Gaussian modes, the Hermite-Gaussian, Laguerre-Gaussian and generalized Hermite-Laguerre-Gaussian modes as eigenfunctions of operators corresponding to the classical constants of motion of the two-dimensional oscillator, which acquire an extra significance as families of classical ellipses upon semiclassical quantization.This article is part of the themed issue 'Optical orbital angular momentum'.

Group theory, coherent states, and the N-dimensional oscillator

The isotropic harmonic oscillator in N dimensions is shown to have an underlying symmetry group O(2,1)X O(N)which implies a unique result for the energy spectrum of the system. Raising and lowering operators analogous to those of the one-dimensional oscillator are given for each value of the angular momentum parameter. This allows the construction of an infinite number of coherent states to be carried out. In the N=1 case there is a twofold family of coherent states, a particular linear combination of which coincides with the single set already well known for that case. Wave functions are readily derived which require only the solution of a first order differential equation, an attribute generally characteristic of group theoretical approaches.

Scaling of Harmonic Oscillator Eigenfunctions and Their Nodal Sets Around the Caustic

We study the scaling asymptotics of the eigenspace projection kernels $\Pi_{\hbar, E}(x,y)$ of the isotropic Harmonic Oscillator $- \hbar ^2 \Delta + |x|^2$ of eigenvalue $E = \hbar(N + \frac{d}{2})$ in the semi-classical limit $\hbar \to 0$. The principal result is an explicit formula for the scaling asymptotics of $\Pi_{\hbar, E}(x,y)$ for $x,y$ in a $\hbar^{2/3}$ neighborhood of the caustic $\mathcal C_E$ as $\hbar \to 0.$ The scaling asymptotics are applied to the distribution of nodal sets of Gaussian random eigenfunctions around the caustic as $\hbar \to 0$. In previous work we proved that the density of zeros of Gaussian random eigenfunctions of $\hat{H}_{\hbar}$ have different orders in the Planck constant $\hbar$ in the allowed and forbidden regions: In the allowed region the density is of order $\hbar^{-1}$ while it is $\hbar^{-1/2}$ in the forbidden region. Our main result on nodal sets is that the density of zeros is of order $\hbar^{-\frac{2}{3}}$ in an $\hbar^{\frac{2}{3}}$-tube around the caustic. This tube radius is the `critical radius'. For annuli of larger inner and outer radii $\hbar^{\alpha}$ with $0< \alpha < \frac{2}{3}$ we obtain density results which interpolate between this critical radius result and our prior ones in the allowed and forbidden region. We also show that the Hausdorff $(d-2)$-dimensional measure of the intersection of the nodal set with the caustic is of order $\hbar^{- \frac{2}{3}}$.

SU(1, 1) and SU(2) Perelomov number coherent states: algebraic approach for general systems

We study some properties of the SU(1, 1) Perelomov number coherent states. The Schrödinger's uncertainty relationship is evaluated for a position and momentum-like operators (constructed from the Lie algebra generators) in these number coherent states. It is shown that this relationship is minimized for the standard coherent states. We obtain the time evolution of the number coherent states by supposing that the Hamiltonian is proportional to the third generator K0 of the su(1, 1) Lie algebra. Analogous results for the SU(2) Perelomov number coherent states are found. As examples, we compute the Perelomov coherent states for the pseudoharmonic oscillator and the two-dimensional isotropic harmonic oscillator.

Photoionization cross-section of isotropic defects or impurity centers in isolators

An analytical expression to calculate the photoionization cross-section of isotropic defects or impurity centers is being proposed by using the time-dependent perturbation theory. The ground-state wave function of the electron captured in the impurity state is described by a three-dimensional isotropic harmonic oscillator and the electron excited state in the continuum conduction band is described by a plane wave. The expression has been obtained considering all multipoles terms in the Hamiltonian, and that the radiation field which interacts with electrons is semi-classical and linearly polarized. This approximation is assumed because the effects of the linear contribution are dominant. The available data of the Al2O3:C and Lu2SiO5:Ce systems are in good agreement with our predictions. Such satisfactory comparison is a strong indication that the present model can be used to provide good predictions of the photoionization cross-section in several areas.

Vortex knots in tangled quantum eigenfunctions

Tangles of string typically become knotted, from macroscopic twine down to long-chain macromolecules such as DNA. Here, we demonstrate that knotting also occurs in quantum wavefunctions, where the tangled filaments are vortices (nodal lines/phase singularities). The probability that a vortex loop is knotted is found to increase with its length, and a wide gamut of knots from standard tabulations occur. The results follow from computer simulations of random superpositions of degenerate eigenstates of three simple quantum systems: a cube with periodic boundaries, the isotropic three-dimensional harmonic oscillator and the 3-sphere. In the latter two cases, vortex knots occur frequently, even in random eigenfunctions at relatively low energy, and are constrained by the spatial symmetries of the modes. The results suggest that knotted vortex structures are generic in complex three-dimensional wave systems, establishing a topological commonality between wave chaos, polymers and turbulent Bose–Einstein condensates.

Discrete Laguerre Gaussian Transforms and Their Applications

Laguerre Gaussian functions serve as a complete and orthonormal basis for a variety of physical problems, such as 2D isotropic quantum harmonic oscillators and circularly symmetric laser modes. In this paper, we propose “discrete Laguerre Gaussian functions,” which are defined such that some elegant physical properties are preserved and a fast computation algorithm of complexity $O(N \log N)$ is available. Discrete Laguerre Gaussian transforms, as introduced in this paper, inherit nice properties from discrete Laguerre Gaussian functions and admit signal analysis over circularly symmetric patterns. It is demonstrated through examples that discrete Laguerre Gaussian transforms find applications in circular pattern keypoints selection, object detection, image compression, rotational invariance feature for pattern recognition, and rotational angle estimation.

Anisotropic multicluster model in light nuclei

Multicluster models consider that the nucleons can be moving around different centers in the nuclei. These models have been widely used to describe light nuclei but always considering that the mean field is composed of isotropic harmonic oscillators with different centers. In this work, we propose an extension of these models by using anisotropic harmonic oscillators. The strengths of these oscillators, the distance among the different centers and the disposition of the nucleons inside every cluster are free parameters which have been fixed using the variational criterion. All the one-body and two-body matrix elements have been analytically calculated. Only a numerical integration on the Euler angles is needed to carry out the projection on the values of the total spin of the state and its third component. We have studied the ground state and the first excited states of 8Be, 12C and 10Be getting good results for the energies. The disposition of the nucleons in the different clusters have also been analyzed by using projection on the different Cartesian planes getting much more information than when the radial one-body density is used.

Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive δ-impurities symmetrically situated around the origin

In this presentation, we wish to provide an overview of the spectral features for the self-adjoint Hamiltonian of the three-dimensional isotropic harmonic oscillator perturbed by either a single attractive delta-interaction centered at the origin or by a pair of identical attractive delta-interactions symmetrically situated with respect to the origin. Given that such Hamiltonians represent the mathematical model for quantum dots with sharply localized impurities, we cannot help having the renowned article by Bruning, Geyler and Lobanov [1] as our key reference. We shall also compare the spectral features of the aforementioned three-dimensional models with those of the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive delta'-interaction in one dimension, fully investigated in [2, 3], given the existence in both models of the remarkable spectral phenomenon called level crossing. The rigorous definition of the self-adjoint Hamiltonian for the singular double well model will be provided through the explicit formula for its resolvent (Green's function). Furthermore, by studying in detail the equation determining the ground state energy for the double well model, it will be shown that the concept of positional disorder, introduced in [1] in the case of a quantum dot with a single delta-impurity, can also be extended to the model with the twin impurities in the sense that the greater the distance between the two impurities is, the less localized the ground state will be. Another noteworthy spectral phenomenon will also be determined; for each value of the distance between the two centers below a certain threshold value, there exists a range of values of the strength of the twin point interactions for which the first excited symmetric bound state is more tightly bound than the lowest antisymmetric bound state. Furthermore, it will be shown that, as the distance between the two impurities shrinks to zero, the 3D-Hamiltonian with the singular double well (requiring renormalization to be defined) does not converge to the one with a single delta-interaction centered at the origin having twice the strength, in contrast to its one-dimensional analog for which no renormalization is required. It is worth stressing that this phenomenon has also been recently observed in the case of another model requiring the renormalization of the coupling constant, namely the one-dimensional Salpeter Hamiltonian perturbed by two twin attractive delta-interactions symmetrically situated at the same distance from the origin.

Examination of the Feynman-Hibbs Approach in the Study of NeN-Coronene Clusters at Low Temperatures.

Feynman-Hibbs (FH) effective potentials constitute an appealing approach for investigations of many-body systems at thermal equilibrium since they allow us to easily include quantum corrections within standard classical simulations. In this work we apply the FH formulation to the study of NeN-coronene clusters (N = 1-4, 14) in the 2-14 K temperature range. Quadratic (FH2) and quartic (FH4) contributions to the effective potentials are built upon Ne-Ne and Ne-coronene analytical potentials. In particular, a new corrected expression for the FH4 effective potential is reported. FH2 and FH4 cluster energies and structures-obtained from energy optimization through a basin-hopping algorithm as well as classical Monte Carlo simulations-are reported and compared with reference path integral Monte Carlo calculations. For temperatures T > 4 K, both FH2 and FH4 potentials are able to correct the purely classical calculations in a consistent way. However, the FH approach fails at lower temperatures, especially the quartic correction. It is thus crucial to assess the range of applicability of this formulation and, in particular, to apply the FH4 potentials with great caution. A simple model of N isotropic harmonic oscillators allows us to propose a means of estimating the cutoff temperature for the validity of the method, which is found to increase with the number of atoms adsorbed on the coronene molecule.

Generalized Hopf Fibration and Geometric SO(3) Reduction of the 4DOF Harmonic Oscillator

It is shown that the generalized Hopf map ℍ×ℍℍ×ℍ → ℍ×ℝ×ℝℍ×ℝ×ℝquaternion formulation can be interpreted as an SO(3) orbit map for a symplectic SO(3) action. As a consequence the generalized Hopf fibration S7S7 → S4S4 appears in the SO(3) geometric symplectic reduction of the 4DOF isotropic harmonic oscillator. Furthermore it is shown how the Hopf fibration and associated twistor fibration play a role in the geometry of the Kepler problem and the rigid body problem.