Two-dimensional Quantum Harmonic Oscillator Research Articles

Control of the oscillation frequency of a vortex cluster in the trapped polariton condensate

An optically trapped exciton–polariton condensate forms states corresponding to excited energy levels of the confining potential. Recently, it was shown that non-uniformity of the ring-shaped trapping potential leads to the simultaneous occupation of two split energy levels. This results in the formation of an oscillating vortex cluster with periodically changing signs of topological charges. Here, we experimentally demonstrate the control of the topological charge oscillation frequency by tuning the ellipticity of the excitation profile. Our observations are corroborated using the linear Schrödinger equation for a two-dimensional quantum harmonic oscillator. Our findings open a promising avenue for the investigation of optical vorticity properties and their applications in controllable settings.

The B<sub>2</sub> Harmonic Oscillator with Reflections and Superintegrability

The two-dimensional quantum harmonic oscillator is modified with reflection terms associated with the action of the Coxeter group B<sub>2</sub>, which is the symmetry group of the square. The angular momentum operator is also modified with reflections. The wavefunctions are known to be built up from Jacobi and Laguerre polynomials. This paper introduces a fourth-order differential-difference operator commuting with the Hamiltonian but not with the angular momentum operator; a specific instance of superintegrability. The action of the operator on the usual orthogonal basis of wavefunctions is explicitly described. The wavefunctions are classified according to the representations of the group: four of degree one and one of degree two. The identity representation encompasses the wavefunctions invariant under the group. The paper begins with a short discussion of the modified Hamiltonians associated to finite reflection groups, and related raising and lowering operators. In particular, the Hamiltonian for the symmetric groups describes the Calogero-Sutherland model of identical particles on the line with harmonic confinement.

A colloquium on the variational method applied to excitons in 2D materials

In this colloquium, we review the research on excitons in van der Waals heterostructures from the point of view of variational calculations. We first make a presentation of the current and past literature, followed by a discussion on the connections between experimental and theoretical results. In particular, we focus our review of the literature on the absorption spectrum and polarizability, as well as the Stark shift and the dissociation rate. Afterwards, we begin the discussion of the use of variational methods in the study of excitons. We initially model the electron-hole interaction as a soft-Coulomb potential, which can be used to describe interlayer excitons. Using an \emph{ansatz}, based on the solution for the two-dimensional quantum harmonic oscillator, we study the Rytova-Keldysh potential, which is appropriate to describe intralayer excitons in two-dimensional (2D) materials. These variational energies are then recalculated with a different \emph{ansatz}, based on the exact wavefunction of the 2D hydrogen atom, and the obtained energy curves are compared. Afterwards, we discuss the Wannier-Mott exciton model, reviewing it briefly before focusing on an application of this model to obtain both the exciton absorption spectrum and the binding energies for certain values of the physical parameters of the materials. Finally, we briefly discuss an approximation of the electron-hole interaction in interlayer excitons as an harmonic potential and the comparison of the obtained results with the existing values from both first--principles calculations and experimental measurements.

The classical Lenz vector and the two-dimensional quantum harmonic oscillator

Both the two-dimensional harmonic oscillator and the Newton potential allow particular solutions for the orbits which are ellipses with center of attraction in the center, in the first case, and in one focus, in the second. The same complex map which allows to go from Kepler’s to Hooke’s orbits, and back, is used to transform the Lenz vector, defined for the Kepler orbit, into two conserved quantities for the harmonic motion. Upon quantization, the resulting operators, together with the angular momentum Lz, are found to correspond to the generators of the SU(2) internal symmetry of the two-dimensional quantum oscillator and the connection to the Schwinger model of angular momentum is made apparent. We give a self-contained new look on this topic.

Symmetry algebra of the planar anisotropic quantum harmonic oscillator with rational ratio of frequencies

The symmetry algebra of the two-dimensional quantum harmonic oscillator with rational ratio of frequencies is identified as a non-linear extension of the u(2) algebra. The finite dimensional representation modules of this algebra are studied and the energy eigenvalues are de- termined using algebraic methods of general applicability to quantum superintegrable systems.

Thermodynamics of the two-dimensional quantum harmonic oscillator system subject to a hard-wall confining potential

We study a two-dimensional isotropic harmonic oscillator with a hard-wall confining potential in the form of a circular cavity defined by the radial coordinate ρ0. When one can normalise the wave function by obtaining polynomial solutions and, in this way, the discrete energy spectrum of the system in an analytical closed form. On the other hand, for , the radial coordinate becomes defined in the range 0 < ρ < ρ0 and the energy spectrum can only be numerically obtained. The thermodynamics of the system can nevertheless be computed and the results compared to the well-studied free oscillator. For instance, the specific heat of the confined oscillator presents a peak and the corresponding limiting value at high temperatures deviates from the expected Dulong–Petit law value. In addition, there is a particular limiting case where a discrete spectrum of energy of the confined oscillator can be obtained in an analytical form. In this case, not only the energy spectrum but also its thermodynamics are comparable to the exact results, even for rather large values of the cavity size.

A model of the two-dimensional quantum harmonic oscillator in an $$AdS_3$$ A d S 3 background

In this paper we study a model of the two-dimensional quantum harmonic oscillator in a 3-dimensional anti-de Sitter background. We use a generalized Schr\"odinger picture in which the analogs of the Schr\"odinger operators of the particle are independent of both the time and the space coordinates in different representations. The spacetime independent operators of the particle induce the Lie algebra of Killing vector fields of the $AdS_3$ spacetime. In this picture, we have a metamorphosis of the Heisenberg's uncertainty relations.

Solution of Time-Independent Schrodinger Equation for a Two-Dimensional Quantum Harmonic Oscillator Using He's Homotopy Perturbation Method

In this paper, time-independent Schr\{o}dinger equation for a charged particle, in the presence of electric potential and vector potential, has been solved using He's Homotopy Perturbation Method (HPM). HPM is one of the newest analytical methods to solve linear and nonlinear differential equations. In contrast to the traditional perturbation methods, the Homotopy method does not require a small parameter in the equation. In this method, according to the homotopy technique, a Homotopy with an embedding parameter $\delta \in \lbrack 0,1]$ is constructed, and the embedding parameter is considered as a small parameter. Using cylindrical coordinates, it has been found that the z-equation of the charged particle is a one-dimensional harmonic oscillator and the r equation is actually a two-dimensional harmonic oscillator. The obtained results show the evidence of simplicity, usefulness, and effectiveness of the HPM for obtaining approximate analytical solutions for the time-independent Schr\{o}dinger equation for a charged particle in parallel electric and magnetic fields.

Quantum cosmology with effects of a preferred reference frame

Recently, we presented a gravity model by generalizing the Brans–Dicke theory which is suitable for studying the metric signature transition dynamics without using an imaginary time parameter. Adding a suitable scalar potential described in terms of the Brans–Dicke scalar field ‘’, this alternative theory is used to study the Wheeler–DeWitt approach of quantum cosmology. We assumed that the universe is defined in a flat Robertson–Walker metric with Lorentzian signature. In that case, the Wheeler–DeWitt wavefunctional is obtained as two-dimensional quantum harmonic oscillator convergent polynomials for both of the choices of positive and negative values of the Brans–Dicke parameter. Here we choose a preferred reference frame with a time coordinate of ‘γ’ which relates to time of cosmological free falling observer ‘t’ as ‘’.

Tracking dynamics of two-dimensional continuous attractor neural networks

We introduce an analytically solvable model of two-dimensional continuous attractor neural networks (CANNs). The synaptic input and the neuronal response form Gaussian bumps in the absence of external stimuli, and enable the network to track external stimuli by its translational displacement in the two-dimensional space. Basis functions of the two-dimensional quantum harmonic oscillator in polar coordinates are introduced to describe the distortion modes of the Gaussian bump. The perturbative method is applied to analyze its dynamics. Testing the method by considering the network behavior when the external stimulus abruptly changes its position, we obtain results of the reaction time and the amplitudes of various distortion modes, with excellent agreement with simulation results.

Fractionalization of Hankel Transforms

The fractional Hankel transform is defined in a way similar to that of the fractional Fourier transform by means of an eigenvalue equation involving the generalized Laguerre polynomials. The integral representation is obtained directly from the Hille-Hardy formula. Generalized operational relations are derived and can be used to solve certain classes of ordinary and partial differential equations which arise in quantum mechanics from quadratic hamiltonians in cylindrical polar co-ordinates. A simple illustration of the method is provided by the two-dimensional quantum harmonic oscillator.