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

In this paper, time-independent Schrödinger 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 δ ∈ [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 Schrödinger equation for a charged particle in parallel electric and magnetic fields.

The noncommutative harmonic oscillator, with noncommutativity not only in position space but also in phase space, in arbitrary dimension is examined. It is shown that the ⋆-genvalue problem, which replaces the Schrödinger problem in this case, can be decomposed into separate harmonic oscillator equations for each dimension. The two-dimensional noncommutative harmonic oscillator (four noncommutative phase-space dimensions) is investigated in greater detail. The requirement of the existence of rotationally symmetric solutions leads to a two parameter harmonic oscillator which is completely solved in this case. The angular momentum operator is derived and its ⋆-genvalue problem is shown to be equivalent to the usual eigenvalue problem of the ⋆-genfunction related wave function. The ⋆-genvalues of the angular momentum are found to depend on the energy difference of the oscillations in the two dimensions. Furthermore two examples of a symmetric noncommutative harmonic oscillators are analyzed. The first is the noncommutative two-dimensional Landau problem with harmonic oscillator potential, which shows degeneracy in the energy levels for certain critical values of the noncommutativity parameters, and the second is the three-dimensional harmonic oscillator with noncommuting coordinates and momenta.

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.

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.

This work provides analytical study of a two-dimensional quantum harmonic oscillator (HO) coupled with a perpendicular magnetic field (B) in the x-direction and an inverse-square potential in the y-direction. The system’s Hamiltonian is exactly solved via variable separation, to obtain the energy levels and wavefunctions composed of Hermite and associated Laguerre polynomials. Fisher information (FI) and Shannon entropy (SE) for both position space (PS) and momentum space (MS) are fully examined under the magnetic field strength and anisotropic potential parameters. The canonical partition function is computed and thermodynamic quantities such as internal energy, entropy, and Gibbs free energy are evaluated. The result shows that the presence of magnetic field enhances spatial localization and reduces uncertainty in PS but has a revised effect in the MS. In the thermodynamic properties, there is asymmetric thermodynamic responses in the x- and y-directions due to the presence of the magnetic field and the inverse-square potential. This study highlights the deep relationship between quantum information theory and thermodynamics in low-dimensional systems with structured interactions.

A charged harmonic oscillator in a magnetic field, Landau problems, and an oscillator in a noncommutative space share the same mathematical structure in their Hamiltonians. We have considered a two-dimensional anisotropic harmonic oscillator with arbitrarily time-dependent parameters (effective mass and frequencies), placed in an arbitrarily time-dependent magnetic field. A class of quadratic invariant operators (in the sense of Lewis and Riesenfeld) have been constructed. The invariant operators (Î) have been reduced to a simplified representative form by a linear canonical transformation [the group Sp(4,R)]. An orthonormal basis of the Hilbert space consisting of the eigenvectors of Î is obtained. In order to obtain the solutions of the time-dependent Schrödinger equation corresponding to the system, both the geometric and dynamical phase-factors are constructed. A generalized Peres–Horodecki separability criterion (Simon’s criterion) for the ground state corresponding to our system has been demonstrated.

The non-relativistic motion of a particle in a central field with 1/r potential, e.g. the motion of an electron in the Coulomb field of a charged nucleus at rest, is described by the equation of motion (non-relativistic Kepler problem) m x″ = α · x /r3 with α = ez e (product of the charges of the central body ez and the electron e). From this equation of motion, three statements of conservation can be derived: in respect of the energy E, of the angular momentum L and of the Lenz vector Λ = m {x′× L + α ·x/r}. The geometric meaning of Λ is that of a vector pointing in the direction of the perihelion of the particle orbits (conic sections). It will be demonstrated that also at the relativistic Kepler problem, which is based on the equation of motion an analogous Lenz vector exists. It represents a quantity of conservation - in the same way as the relativistic energy and the relativistic angular momentum. For the transitional case → ∞, where the relativistic problem turns into the non-relativistic problem, the relativistic Lenz vector also turns into the non-relativistic Lenz vector. The generalised (relativistic) Lenz vector has also a geometric meaning. Its direction coincides with the oriented axis of symmetry of the orbits (rosettes, spirals, hyperbola-type curves etc.). The quantity of conservation Λ occupies a special position in respect of the quantities of conservation energy and angular momentum. Whereas the energy and the angular momentum correspond with a symmetry of time and space, the Lenz quantity of conservation corresponds with a symmetry of the orbits. The fact that the Lenz vector can relativistically be generalised touches thereby on principal aspects.

The connection between the wavefunctions and the classical periodic orbits in a 2D harmonic oscillator is analytically constructed by using the representation of SU(2) coherent states. It is found that the constructed wavefunction generally corresponds to an ensemble of classical trajectories and its localization is extremely efficient. With the constructed wavefunction, we also analyse the property of the probability current density associated with the classical periodic orbit. The appearance of vortex structure in the quantum flow is clearly found to arise from the wave interference.

Harmonic oscillators from displacement operators and thermodynamics

Making use of the Bloch density matrix technique, we derive exact analytical expressions for the density profile in Fourier space, for the current density and the so-called integrated current for fermionic systems confined by a two-dimensional harmonic oscillator, in the presence of a magnetic field or in a rotating trap of arbitrary strength. We present numerical, illustrative examples with or without magnetic field (with or without rotation).

It is proposed to train a deep convolution neural network (CNN) to learn the mapping relation between two-dimensional electrostatic potential and ground state energy eigenvalue by using the feature extraction and function fitting ability of machine learning (ML), so as to avoid the difficulty of strict solution of Schrodinger equation. The potential energy function is processed by CNN image processing, and the projection of the two-dimensional simple harmonic oscillator potential energy function on the two-dimensional plane is taken as input. Experimental results verify the validity of the method: the average absolute error is 0.0372eV, The Standard Deviation was 0.429eV, and the Relative Error was 1.042%.

The first-order approximate Lie symmetries and approximate conserved quantities of the weak nonlinear coupled two-dimensional anisotropic harmonic oscillator are studied. When the 1/2 is equal to 2/1, the system possesses six first-order approximate Lie symmetries and approximate conserved quantities, one of them is an exact conserved quantity, four of them are trivial conserved quantities, only one of them is a stable conserved quantity.

A new kind of representations on noncommutative phase space

We map the problem of a Morse potential in one dimension into a two-dimensional harmonic oscillator. The symmetry group for this problem is U(2). Starting from the dynamical group Sp(4), we use two different chains of groups including SU(1,1) and U(2) respectively.

We present a microscopic theory of the optical properties of self-assembled quantum dots doped with a single magnetic manganese (Mn) impurity and containing a controlled number of electrons. The single-particle electron and heavy-hole electronic shells are described by two-dimensional harmonic oscillators. The electron-electron, electron-hole Coulomb as well as the short-range electron spin-Mn spin and hole spin-Mn spin contact exchange interactions are included. The electronic states of the photoexcited electron-hole-Mn complex and of the final electron-Mn complex are expanded in a finite number of configurations and the full interacting Hamiltonian is diagonalized numerically. The emission spectrum is predicted as a function of photon energy for a given number of electrons and different number of confined electronic quantum dot shells. We show how emission spectra allow to identify the number of electronic shells, the number of electrons populating these shells and, most importantly, their spin. We show that electrons not interacting directly with the spin of Mn ion do so via electron-electron interactions. This indirect interaction is a strong effect even when Mn impurity is away from the quantum dot center.