2D Harmonic Oscillator Research Articles

Algebraic DVR Approaches Applied to Describe the Stark Effect

Two algebraic approaches based on a discrete variable representation are introduced and applied to describe the Stark effect in the non-relativistic Hydrogen atom. One approach consists of considering an algebraic representation of a cutoff 3D harmonic oscillator where the matrix representation of the operators r2 and p2 are diagonalized to define useful bases to obtain the matrix representation of the Hamiltonian in a simple form in terms of diagonal matrices. The second approach is based on the U(4) dynamical algebra which consists of the addition of a scalar boson to the 3D harmonic oscillator space keeping constant the total number of bosons. This allows the kets associated with the different subgroup chains to be linked to energy, coordinate and momentum representations, whose involved branching rules define the discrete variable representation. Both methods, although originating from the harmonic oscillator basis, provide different convergence tests due to the fact that the associated discrete bases turn out to be different. These approaches provide powerful tools to obtain the matrix representation of 3D general Hamiltonians in a simple form. In particular, the Hydrogen atom interacting with a static electric field is described. To accomplish this task, the diagonalization of the exact matrix representation of the Hamiltonian is carried out. Particular attention is paid to the subspaces associated with the quantum numbers n=2,3 with m=0.

A numerical damped oscillator approach to constrained Schrödinger equations

This article explains and illustrates the use of a set of coupled dynamical equations, second order in a fictitious time, which converges to solutions of stationary Schrödinger equations with additional constraints. In fact, the method is general and can solve constrained minimization problems in many fields. We present the method for introductory applications in quantum mechanics including three qualitative different numerical examples: the radial Schrödinger equation for the hydrogen atom; the 2D harmonic oscillator with degenerate excited states; and a nonlinear Schrödinger equation for rotating states. The presented method is intuitive, with analogies in classical mechanics for damped oscillators, and easy to implement, either with coding or with software for dynamical systems. Hence, we find it suitable to introduce it in a continuation course in quantum mechanics or generally in applied mathematics courses which contain computational parts. The undergraduate student can, for example, use our derived results and the code (supplemental material (https://stacks.iop.org/EJP/41/065406/mmedia)) to study the Schrödinger equation in 1D for any potential. The graduate student and the general physicist can work from our three examples to derive their own results for other models including other global constraints.

Hybrid derivation of the 1D harmonic oscillator propagator

We provide an elementary derivation of the one-dimensional quantum harmonic oscillator propagator, using a mix of approaches, such as path integrals, canonical operators, and ladder operators. This way we take the best of each world, and find the propagator with as few tears as possible.

Expansion of deformed harmonic oscillator wavefunctions in terms of spherical wavefunction bases: deformation effects

The interbasis expansion of deformed cylindrical 3D harmonic oscillator (HO) wavefunctions in terms of spherical wavefunction bases is introduced. The spatial overlap integral is rewritten in a simplified form and presented in terms of the axis ratio of the deformed nucleus. Deformed nuclear spheroidal shapes are considered, including both quadrupole and hexadecapole deformations. The effect of deformation on the oscillator lengths of deformed 3D-HO, and in turn on the spatial overlap coefficients, is discussed. For the description of the geometrical shape of deformed nuclear systems, the collective and extended Nilsson models for surface parametrization are used.

Controlling Energy Spectra and Whispering Gallery Modes of Electrons in a Few Electrons Lateral QD

In this work, we study energy spectra and Whispering Gallery Modes (WGMs) of electrons in Quantum Dot (QD). It is shown that the WGMs can exist between two lowest energy levels of a QD, s- and p-states by taking into account the Coulomb interaction between the electrons in the QD, since in a few electron QD, Coulomb Interaction plays a crucial role in its energy spectrum. In literature, lateral QDs are usually modeled by a 2D harmonic oscillator. However, our calculations reveal that some degenerate energy levels would be created due to the height of the QD. The effect of the dimension of the QD, and magnetic field are also investigated on the energy spectra.

Numerical solution of the Schrödinger equation in polar coordinates using the finite-difference time-domain method

In quantum mechanics, many concepts, equations, and interactions are expressed as functions of the radius and angles and are therefore best understood and handled directly in polar or spherical coordinates. A finite-difference time-domain (FDTD) method for solving the two-dimensional Schrodinger equation in polar coordinates is proposed herein. In this method and through a subgridding approach, new nodes are added on rings far from the origin to retain the precision of the mesh grids; then a trigonometric interpolation is used to calculate the derivatives at these nodes. A comparison with analytic solutions for a two-dimensional (2D) harmonic oscillator is carried out to verify the performance of the code. A simple method based on the spatial Fourier transform is presented for the separation of degenerate eigenstates. A 2D quantum dot is also simulated and analyzed. When using this polar FDTD method along with proposed subgridding approach, the resolution of the solutions and Hamiltonian terms are conserved in the whole space of a plane of polar coordinates, and all operators or results expressed in polar coordinates can be easily implemented or obtained.

Solving the 3D Schrödinger Equation on a GPU

Abstract Solving the 3D Schrodinger equation is of great interests in many branches of physical sciences and is usually a computing intensive task. Modern graphics processing units (GPUs) are extensively employed in the general purpose computing domain due to their high performance parallel processing capability. In this paper, an algorithm for solving the time-independent 3D Schrodinger equation with a finite difference time domain (FDTD) method is presented, together with an implementation of the algorithm by using CUDA (Compute Unified Device architecture) C. The GPU-based solver is validated in four cases: 3D Coulomb potential, 3D harmonic oscillator, three coupled anharmonic oscillators, and H 2+. Additionally, a CPU-based solver that is a serial program is developed according to the same FDTD method for the purpose of testing the effectiveness of the GPU-based solver. Relative to the CPU-based solver that employs one core of a multi-core CPU, the GPU-based solver can achieve a speed-up of more than 90X, 60X, 40X, and 90X in the case of 3D Coulomb potential, 3D harmonic oscillator, three coupled anharmonic oscillators, and H 2+, respectively. The GPU-based solver can accelerate the solution of the 3D Schrodinger equation.

Coherent States for the Isotropic and Anisotropic 2D Harmonic Oscillators

In this paper we introduce a new method for constructing coherent states for 2D harmonic oscillators. In particular, we focus on both the isotropic and commensurate anisotropic instances of the 2D harmonic oscillator. We define a new set of ladder operators for the 2D system as a linear combination of the x and y ladder operators and construct the S U ( 2 ) coherent states, where these are then used as the basis of expansion for Schrödinger-type coherent states of the 2D oscillators. We discuss the uncertainty relations for the new states and study the behaviour of their probability density functions in configuration space.

Unitary group approach to describe interatomic potentials in 3D systems

ABSTRACTAn algebraic approach based on the unitary algebra is proposed to describe 3D systems for effective potentials. Our approach is based on the 3D vibron model where the addition of a scalar boson is introduced into the space of a 3D harmonic oscillator. Our approach is based on the mapping between the dynamical symmetry and the harmonic oscillator states. A minimisation procedure is used in order to determine the coefficients involved in the algebraic expansions. This allows the kets associated with the different subgroup chains to be linked to energy, coordinate and momentum representations. As an application of this approach the eigensystem of the 3D-Morse potential is analysed in detail. The analyses of inertia moments as well as the dipole moment strengths are also included. To show the convenience of this approach, a comparison with one ab-initio calculation for vibrational and rotational levels for O is presented in the framework of an analytical potential. In addition, a notch test is done to show how the formalism allows to check the sensitivity of the ro-vibrational results as a function of the precise structure of the potential energy surface as a function of r.

A quantum system with a non-self-adjoint 2D-harmonic oscillator

Non-self-adjoint operators with a real discrete spectrum exhibit, in general, a quite different behavior from the one of self-adjoint operators, with the same spectrum. This fact has serious mathematical and physical implications, namely in the rigorous formulation of quantum mechanics (QM), where self-adjointness plays a central role, in particular in the spectral analysis of the involved operators. In this framework, a non-self-adjoint 2D-harmonic oscillator is considered. Its eigenvalues and eigenfunctions, as well as those of its adjoint, are explicitly determined. Although the eigensystems are complete they may not form bases, a drawback for the mathematical formulation of the QM of the system. The existence of a metric operator defining an inner product which renders the Hamiltonian self-adjoint, is discussed.

A simple approach to solve the time independent Schröedinger equation for 1D systems

A simple algebraic approach based on the well known angular momentum SU(2) algebra is presented to describe 1D systems for arbitrary potentials. The approach is based on the dimension increase of the 1D harmonic oscillator space through the addition of a scalar boson, keeping constant the total number of bosons. In this new space the realization of the coordinate and momentum correspond to components of the angular momentum algebra, which in turn define the coordinate and momentum representation bases. This remarkable result allows the Hamiltonian matrix representationto be expressed in terms of a diagonal matrix characterizing the potential we are considering. The solutions are obtained as an expansion of harmonic oscillator functions by purely algebraic means. As an example of our approach the Morse and asymmetric double Morse potentials are considered.

Unitary group approach for effective potentials in 2D systems: Application to carbon suboxide C3O2

A U(3) algebraic approach is proposed to describe 2D systems for effective potentials. Our approach is based on the 2D vibron model where the addition of a scalar boson is introduced into the 2D harmonic oscillator space. As a crucial ingredient of our approach an algebraic realization of the coordinates and momenta is obtained. This feature provides the tools to obtain the algebraic representation of a 2D Hamiltonian in terms of similitude transformation of a diagonal matrix. As an application of our approach the rotation-bending energy levels of carbon suboxide C3O2 are described in good agreement with experimental data.

Damped harmonic oscillator and bosonic string theory through noncommutativity

Some features of noncommutativity are investigated in this work. A noncommutative (NC) versions for the 2D harmonic oscillator and scalar field theory are obtained via the Noncommutative Mapping and fourfold discussion are presented: first, the Lagrangian of the damped harmonic oscillator (DHO), which was an original explanation for the dissipative process presenting in the atomic nuclei collision, is reobtained through noncommutativity. Here, we show that the NC parameter plays the role of the viscous damping coefficient; second, the NC harmonic oscillator is mapped into the bosonic string theory in the light cone frame, which appears as a bosonic string theory attached to a D3-brane; third, we discuss the connection between DHO and bosonic string attached to a D3-brane; fourth, the NC scalar field is indicating into 2D-NC harmonic oscillator.

Unitary group approach for effective molecular potentials: 1D systems

ABSTRACTAn algebraic approach based on the algebra is proposed to describe 1D systems. Our approach starts adding a scalar boson to the boson associated with the 1D harmonic oscillator, whose bilinear products constitute elements of the generators of the algebra. As a next step the identification of coordinates and momenta in the algebraic space is established using the approach recently proposed [Mol. Phys. (2017) doi:10.1080/00268976.2017.1358829]. As a result, the three bases provided by the dynamical group are identified with energy, coordinate and momentum representations. This result provides power tools to obtain the algebraic representation of a 1D Hamiltonian for any potential through the transformation brackets connecting the different bases. As examples of our approach, several potentials are considered: Morse potential as a benchmark case where analytical solutions are known, double Morse potential where exact solutions are known for only restricted subspaces (semi-integral system), and the Harmonic-Gaussian potential where no analytical solutions are known.

An oscillator-representation of elementary particles

Two dynamical systems with same symmetry should have features in common, and as far as their shared symmetry is concerned, one may represent the other. The three light quark constituent of the hadrons, (a) have an approximate flavor SU(3)f symmetry, (b) have an exact color SU(3)c symmetry, and (c) as spin particles, have a Lorentz SO(3, 1) symmetry. So does a 3D harmonic oscillator. (a) Its Hamiltonian has the SU(3) symmetry, breakable if the 3 oscillators are not identical. (b) The 3 directions of oscillation have the permutation symmetry. This enables one to create three copies of unbreakable SU(3) symmetry for each oscillator, and mimic the color of the elementary particles. (c) The Lagrangian of the 3D oscillator has the SO(3,1) symmetry. This can be employed to accommodate the spin of the particles. In this paper we propose a one-to-one correspondence (a) between the eigen modes of the Poisson bracket operator of the 3D oscillator and the flavor multiplets of the particles, and (b) between the permuted modes of the oscillator and the color and anticolor multiplets of the particles. The bi-colored gluons are represented by the generators of the color SU(3)c symmetry of the oscillator. Harmonic oscillators are common place objects and, wherever encountered, are analytically solvable. Elementary particles, on the other hand, are abstract entities far from one’s reach. Understanding of one may help a better appreciation of the other.

Recovery of the Center-of-Mass Galilean invariance of a NR-QCD motivated Hamiltonian

We evaluate the use of 3D-harmonic oscillator eigenfunctions in the expansion of the fermionic fields of an effective QCD Hamiltonian with the 1/r plus a linearly confining potential. Since both the potential and the functions involved in such expansion are non-relativistic, a modified ter-Vehn method is applied to recover the galilean invariance of the center of mass of the mesonic states built with the use of the Random Phase Approximation. Finally, a preliminary calculation for the mesons below 1 GeV is presented.

1-D harmonic oscillator in MONDified inertia

In this paper we study the dynamics of a harmonic oscillator with laws of motion prescribed by MOND (Modified Newtonian Dynamics) in its modified inertia formulation. A differential equation for a 1D harmonic oscillator is obtained and several features of its solution are analyzed. Particular attention is given to the deep MOND limit regime, where the equations of motion are significantly different from the Newtonian one.

Gibbs measures based on 1d (an)harmonic oscillators as mean-field limits

We prove that Gibbs measures based on 1D defocusing nonlinear Schrödinger functionals with sub-harmonic trapping can be obtained as the mean-field/large temperature limit of the corresponding grand-canonical ensemble for many bosons. The limit measure is supported on Sobolev spaces of negative regularity, and the corresponding density matrices are not trace-class. The general proof strategy is that of a previous paper of ours, but we have to complement it with Hilbert-Schmidt estimates on reduced density matrices.

SYMPLECTIC PSEUDOSPECTRAL TIME-DOMAIN SCHEME FOR SOLVING TIME-DEPENDENT SCHRODINGER EQUATION

A symplectic pseudospectral time-domain (SPSTD) scheme is developed to solve Schrodinger equation. Instead of spatial finite differences in conventional finite-difference time-domain (FDTD) method, the fast Fourier transform is used to calculate the spatial derivatives. In time domain, the scheme adopts high-order symplectic integrators to simulate time evolution of Schrodinger equation. A detailed numerical study on the eigenvalue problems of 1D quantum well and 3D harmonic oscillator is carried out. The simulation results strongly confirm the advantages of the SPSTD scheme over the traditional PSTD method and FDTD approach. Furthermore, by comparing to the traditional PSTD method and the non-symplectic Runge-Kutta (RK) method, the explicit SPSTD scheme which is an infinite order of accuracy in space domain and energy-conserving in time domain, is well suited for a long-term simulation.

Spin force and torque in non-relativistic Dirac oscillator on a sphere

The spin force operator on a non-relativistic Dirac oscillator (in the non-relativistic limit the Dirac oscillator is a spin one-half 3D harmonic oscillator with strong spin–orbit interaction) is derived using the Heisenberg equations of motion and is seen to be formally similar to the force by the electromagnetic field on a moving charged particle. When confined to a sphere of radius R, it is shown that the Hamiltonian of this non-relativistic oscillator can be expressed as a mere kinetic energy operator with an anomalous part. As a result, the power by the spin force and torque operators in this case are seen to vanish. The spin force operator on the sphere is calculated explicitly and its torque is shown to be equal to the rate of change of the kinetic orbital angular momentum operator, again with an anomalous part. This, along with the conservation of the total angular momentum, suggests that the spin force exerts a spin-dependent torque on the kinetic orbital angular momentum operator in order to conserve total angular momentum. The presence of an anomalous spin part in the kinetic orbital angular momentum operator gives rise to an oscillatory behavior similar to the Zitterbewegung. It is suggested that the underlying physics that gives rise to the spin force and the Zitterbewegung is one and the same in NRDO and in systems that manifest spin Hall effect.