Solution Of Harmonic Oscillator Research Articles

Symmetries in Collisions as Explored through the Harmonic Oscillator

The present study explores the symmetries associated with the cluster structure of light nuclei and draws the connection between solutions of the Schrödinger equation for the harmonic oscillator and the quasi-crystalline arrangements of α-particles, which gives rise to a series of collective behaviors. The double-center harmonic oscillator is used to formulate the collisions of two nuclei described by harmonic oscillator solutions and traces out the evolution of the cluster structure in the dynamics of the collision process and demonstrates that the symmetries are preserved in this process. The connection between this study and stellar nucleosynthesis is described.

Asymptotics for Solutions of Harmonic Oscillator with Integral Perturbation

We construct the asymptotics for solutions of a harmonic oscillator with integral perturbation when the independent variable tends to infinity. The distinctive feature of the considered integral perturbation is the oscillatory decreasing character of its kernel. We assume that the integral kernel is degenerate. This makes it possible to reduce the initial integro-differential equation to an ordinary differential system. To get the asymptotic formulas for the fundamental solutions of the obtained ordinary differential system, we use a special method proposed for the asymptotic integration of linear dynamical systems with oscillatory decreasing coefficients. By the use of special transformations, we reduce the ordinary differential system to the so-called L-diagonal form. We then apply the classical Levinson’s theorem to construct the asymptotics for the fundamental matrix of the L-diagonal system. The obtained asymptotic formulas allow us to reveal the resonant frequencies, i.e., the frequencies of the oscillatory component of the kernel that give rise to unbounded oscillations in the initial integro-differential equation. It appears that these frequencies differ slightly from the resonant frequencies that occur in the adiabatic oscillator with the sinusoidal component of the time-decreasing perturbation.

Asymptotics for Solutions of Harmonic Oscillator with Integral Perturbation

We construct the asymptotics for solutions of a harmonic oscillator with integral perturbation when the independent variable tends to infinity. The specific feature of the considered integral perturbation is an oscillatory decreasing character of its kernel. We assume that the integral kernel is degenerate. This makes it possible to reduce the initial integro-differential equation to an ordinary differential system. To get the asymptotic formulas for the fundamental solutions of the obtained ordinary differential system, we use a special method proposed for the asymptotic integration of linear dynamical systems with oscillatory decreasing coefficients. By the use of the special transformations we reduce the ordinary differential system to the so called L-diagonal form. We then apply the classical Levinson’s theorem to construct the asymptotics for the fundamental matrix of the L-diagonal system. The obtained asymptotic formulas allow us to reveal the resonant frequencies, i. e., frequencies of the oscillatory component of the kernel that give rise to unbounded oscillations in the initial integro-differential equation. It appears that these frequencies differ slightly from the resonant frequencies that occur in the adiabatic oscillator with the sinusoidal component of the time-decreasing perturbation.

Effects of external fields on a two-dimensional Klein—Gordon particle under pseudo-harmonic oscillator interaction

We study the effects of the perpendicular magnetic and Aharonov—Bohm (AB) flux fields on the energy levels of a two-dimensional (2D) Klein—Gordon (KG) particle subjected to an equal scalar and vector pseudo-harmonic oscillator (PHO). We calculate the exact energy eigenvalues and normalized wave functions in terms of chemical potential parameter, magnetic field strength, AB flux field, and magnetic quantum number by means of the Nikiforov—Uvarov (NU) method. The non-relativistic limit, PHO, and harmonic oscillator solutions in the existence and absence of external fields are also obtained.

Quantum and Classical Exact Solutions ofHarmonic Oscillator in a Time-Dependent Magnetic Field

In cylindrical coordinate, exact wave functionsof the two-dimensional time-dependent harmonic oscillator ina time-dependent magnetic field are derived by using the trialfunction method. Meanwhile, the exact classical solution as wellas the classical phase is obtained too. Through the Heisenbergcorrespondence principle, the quantum solution and the classicalsolution are connected together.

An integral expression of exact solution for harmonic oscillator with time-varying frequency

Using Feynman's method of disentangling operators, an exact expression of evolution operator of harmonic oscillator with time-varying frequency is obtained. The parameters in this expression are related to the time-varying frequency directly. Some implications are further investigated, including general form of linear and quadratic invariants and evolution of Wigner function.

Theory of the singular quantum oscillator

We propose a number of arguments in favor of reevaluating the theory of a quantum oscillator described by the Hamiltonian H=−d2/dx2 + 2ω2x2 + λx−2(ħ=2m=1). We propose that functions Ψ+(x) which continuously reduce to even harmonic oscillator solutions in the λ → 0 limit be taken as the even solutions of the Hamiltonian in the −1/4 < λ < 3/4 range. In this scheme the problem becomes truly one-dimensional such that even and odd parity energy levels alternate, whereas the usual approach leads to parity degeneracy.