Solving Dirac Equation with New Ring-Shaped Non-Spherical Harmonic Oscillator Potential

A new ring-shaped non-harmonic oscillator potential is proposed. The precise bound solution of Dirac equation with the potential is gained when the scalar potential is equal to the vector potential. The angular equation and radial equation are obtained through the variable separation method. The results indicate that the normalized angle wave function can be expressed with the generalized associated-Legendre polynomial, and the normalized radial wave function can be expressed with confluent hypergeometric function. And then the precise energy spectrum equations are obtained. The ground state and several low excited states of the system are solved. And those results are compared with the non-relativistic effect energy level in Phys. Lett. A 340 （2005） 94. The positive energy states of system are discussed and the conclusions are made properly.

Supersymmetry and Shape Invariance of Hartmann Potential and Ring-Shaped Oscillator Potential in the γ and θ Dimensions of Spherical Polar Coordinates

This article shows that in spherical polar coordinates, some noncentral separable potentials have super-symmetry and shape invariance in the r and θ dimensions, we choose Hartmann potential and ring-shaped oscillator astwo important examples, thus in principle the energy eigenvalues and energy eigenfunctions of such the potentials in ther and θ dimensions can be obtained by the method of supersymmetric quantum mechanics. Here we use an alternativemethod to get the required results.

The relativistic bound states for a new ring-shaped harmonic oscillator

In this paper a new ring-shaped harmonic oscillator for spin 1/2 particles is studied, and the corresponding eigenfunctions and eigenenergies are obtained by solving the Dirac equation with equal mixture of vector and scalar potentials. Several particular cases such as the ring-shaped non-spherical harmonic oscillator, the ring-shaped harmonic oscillator, non-spherical harmonic oscillator, and spherical harmonic oscillator are also discussed.

Photodetachment dynamics of H^- ion in a harmonic potential plus a time-dependent oscillating electric field

The photodetachment dynamics of H^- ion in a harmonic potential plus an oscillating electric field is studied using the time-dependent closed orbit theory. An analytical formula for calculating the photodetachment cross section of this system is put forward. It is found that the photodetachment cross section of this system is nearly unaffected for the weak oscillating electric field strength, but oscillates complicatedly when the oscillating electric field strength turns strong. In addition, the frequency of the harmonic potential and the oscillating electric field （the frequency of the harmonic potential and the frequency of the oscillating electric field are the same in the paper, unless otherwise stated.） can also affect the photodetachment dynamics of this system. With the increase of the frequency in the harmonic potential and the oscillating electric field, the number of the closed orbits for the detached electrons increased, which makes the oscillatory structure in the photodetachment cross section much more complex. Our study presents an intuitive understanding of the photodetachment dynamics driven by a harmonic potential plus an oscillating electric field from a space and time dependent viewpoint. This study is very useful in guiding the future experimental research for the photodetachment dynamics in the electric field both changing with space and time.

Oscillation properties of matter–wave bright solitons in harmonic potentials

We investigate the oscillation periods of bright soliton pair or vector bright soliton pair in harmonic potentials. We demonstrate that periods of low-speed solitons are greatly affected by the position shift during their collisions. The modified oscillation periods are described by defining a characterized speed, with the aid of asymptotic analysis on related exact analytic soliton solutions in integrable cases. The oscillation period can be used to distinguish the inter-and intra-species interactions between solitons. However, a bright soliton cannot oscillate in a harmonic trap, when it is coupled with a dark soliton(without any trapping potentials). Interestingly, it can oscillate in an anti-harmonic potential, and the oscillation behavior is explained by a quasi-particle theory. The modified period of two dark-bright solitons can be also described well by the characterized speed. These results address well the effects of position shift during soliton collision, which provides an important supplement for previous studies without considering phase shift effects.

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 param- eter, 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 semiclassical studies on photodetachment cross sections of H^- in a harmonic potential

The photodetachment cross section of H- in a linear harmonic oscillator potential is investigated. This system pro- vides a rare example that can be studied analytically by both quantum and semiclassical methods with some approxi- mations. The formulas of the cross section for different laser polarization directions are explicitly derived by both the traditional quantum approach and closed-orbit theory. In the traditional quantum approach, we calculate the cross sections in coordinate representation and momentum representation, and get the same formulas. We compare the quantum formulas with closed-orbit theory formulas, and find that when the detachment electron energy is larger than hco, where co is the frequency of the oscillator potential, the quantum results are shown to be in good agreement with the semiclassical results.

Quantum photodetachment of hydrogen negative ion in a harmonic potential subjected to static electric field

Photodetachment of negative ions has attracted immense interest owing to its fundamental nature and practical implications with regard to technology. In this study, we explore the quantum dynamics of the photodetachment cross section of negative ion of hydrogen H-in the perturbed one dimensional linear harmonic potential via static electric field. To this end,the quantum formula for total photodetachment cross section of the H-ion is derived by calculating the dipole matrix element in spherical coordinates. In order to obtain the detached electron wave function, we have solved the time-independent Schr¨odinger wave equation for the perturbed Hamiltonian of the harmonic oscillator in momentum representation. To acquire the corresponding normalized final state detached electron wave function in momentum space, we have employed an approach analogous to the WKB(Wenzel–Kramers–Brillouin) approximation. The resulting analytical formula of total photodetachment cross section depicts interesting oscillator structure that varies considerably with incident-photon energy,oscillator potential frequency, and electric field strength as elucidated by the numerical results. The current problem having close analogy with the Stark effect in charged harmonic oscillator may have potential implications in atomic and molecular physics and quantum optics.

Exact Solution of Dirac Equation with Charged Harmonic Oscillator in Electric Field: Bound States

In some quantum chemical applications, the potential models are linear combination of single exactly solvable potentials. This is the case equivalent of the Stark effect for a charged harmonic oscillator (HO) in a uniform electric field of specific strength (HO in an external dipole field). We obtain the exact s-wave solutions of the Dirac equation for some potential models which are linear combination of single exactly solvable potentials (ESPs). In the framework of the spin and pseudospin symmetric concept, we calculate analytical expressions for the energy spectrum and the corresponding two-component upper- and lower-spinors of the two Dirac particles by using the Nikiforov-Uvarov (NU) method, in closed form. The nonrelativistic limit of the solution is also studied and compared with the other works.

New Possibilities of Harmonic Oscillator Basis Application for Quantum System Description.Two Particles with Coulomb Interaction

This paper is addressed to the problem of Galilei invariant basis construction for identical fermions systems. The recently introduced method for spurious state elimination from expansions in harmonic oscillator basis [1] 08D0C9EA79F9BACE118C8200AA004BA90B02000000080000000E0000005F005200650066003400310034003400340037003600360038000000 is adopted and applied to bound states of two particles system with Coulomb potential description. Traditional expansions in this case demonstrate the extremely well-known slow convergence, and hence this is the best problem with known exact solutions for the test of the method. Obtained results demonstrate the significant simplification of the problem and fast convergence of expansions. We show that the application of this general method is very efficient in a test case of the energy spectrum calculation problem of two particles with different masses interacting with Coulomb potential.

Solutions of the Schrdinger Equation with Quantum Mechanical Gravitational Potential Plus Harmonic Oscillator Potential

The solutions of the Schrodinger equation with quantum mechanical gravitational potential plus harmonic oscillator potential have been presented using the parametric Nikiforov-Uvarov method. The bound state energy eigen values and the corresponding un-normalized eigen functions are obtained in terms of Laguerre polynomials. Also a special case of the potential has been considered and its energy eigen values are obtained.

Spin Supercurrent in Phenomena of Quantum Non-Locality (Quantum Correlations, Magnetic Vector Potential) and in Near-Field Antenna Effect

It is shown that such phenomena as quantum correlations (interaction of space-separated quantum entities), the action of magnetic vector potential on quantum entities in the absence of magnetic field, and near-field antenna effect (the existence of superluminally propagating electromagnetic fields) may be explained by action of spin supercurrents. In case of quantum correlations between quantum entities, spin supercurrent emerges between virtual particles pairs (virtual photons) created by those quantum entities. The explanation of magnetic vector potential and near-field antenna effect is based on contemporary principle of quantum mechanics: the physical vacuum is not an empty space but the ground state of the field consisting of quantum harmonic oscillators (QHOs) characterized by zero-point energy. Using the properties of the oscillators and spin supercurrent, it is proved that magnetic vector potential is proportional to the moment causing the orientation of spin of QHO along the direction of magnetic field. The near-field antenna effect is supposed to take place as a result of action of spin supercurrent causing secondary electromagnetic oscillations. In this way, the electromagnetic field may spread at the speed of spin supercurrent. As spin supercurrent is an inertia free process, its speed may be greater than that of light, which does not contradict postulates of special relativity that sets limits to the speed of inertial systems only.

ANALYTIC SOLUTION TO THE SCHRDINGER EQUATION WITH A HARMONIC OSCILLATOR POTENTIAL PLUS δ-POTENTIAL

The analytic solution to the Schrodinger equation with a harmonic oscillator potentialplus δ-potential is presented in the coordinate representation. Its algebraic expression in theenergy representation is also given. The degeneracy of the energy levels is discussed and acomparison between the energy spectra of a harmonic oscillator and a particle in a δ-poten-tial is made.

Uncertainty Relations for Some Central Potentials in N-Dimensional Space

We study the uncertainty relation for three quantum systems in the N-dimensional space by using the virial theorem (VT). It is shown that this relation depends on the energy spectrum of the system as well as on the space dimension N. It is pointed out that the form of lower bound of the inequality, which is governed by the ground state, depends on the system and on the space dimension N. A comparison between our result for the lower bound and recent results, based on information-theoretic approach, is pointed out. We examine and analyze these derived uncertainties for different angular momenta with a special attention made for the large N limit.

二维旋转谐振子势中单粒子的跳频压缩及演化

本文主要研究二维旋转谐振子势中单粒子的动力学行为,通过跳频的方式分析该粒子的压缩演化过程,并分析了相应的物理机制.一方面,研究跳频过程对回旋半径模式的压缩演化,通过选择适当的跳频时刻,分析跳频过程对压缩的影响.研究表明,回旋半径坐标的压缩程度并未在跳频时刻发生改变,但可在后续的演化中出现更强的压缩现象.另一方面,主要研究跳频过程中心导向模式的压缩演化.通过参数的选择,分析了两种压缩模式,即发散模式和振荡模式的压缩及演化.有趣的是,在中心导向模式压缩中,外势存在一个由旋转角速度决定的临界势阱纵横比.压缩模式在此处发生突变,且在振荡模式中,势阱纵横比趋于该临界值时,将出现明显的压缩.

Observation about the Classical Electromagnetic Gauge Transformation and Its Quantum Correspondence

Using the Landau and symmetric gauges for the vector potential of a constant magnetic field and the quantum problem of a charged particle moving on a flat surface, we show the classical electromagnetic gauge transformation does not correspond to a one-dimensional unitary group transformation U(1) of the wave function for the quantum case. In addition, with the re-examination of the relation between the magnetic field B and its vector potential A, we found that, in order to have a consistent formulation of the dynamics of the charged particle with both expressions, we must have that B=∇×A if and only if B≠0.

Mathematical Wave Functions and 3D Finite Element Modelling of the Electron and Positron

The wave/particle duality of particles in Physics is well known. Particles have properties that uniquely characterize them from one another, such as mass, charge and spin. Charged particles have associated Electric and Magnetic fields. Also, every moving particle has a De Broglie wavelength determined by its mass and velocity. This paper shows that all of these properties of a particle can be derived from a single wave function equation for that particle. Wave functions for the Electron and the Positron are presented and principles are provided that can be used to calculate the wave functions of all the fundamental particles in Physics. Fundamental particles such as electrons and positrons are considered to be point particles in the Standard Model of Physics and are not considered to have a structure. This paper demonstrates that they do indeed have structure and that this structure extends into the space around the particle’s center (in fact, they have infinite extent), but with rapidly diminishing energy density with the distance from that center. The particles are formed from Electromagnetic standing waves, which are stable solutions to the Schrödinger and Classical wave equations. This stable structure therefore accounts for both the wave and particle nature of these particles. In fact, all of their properties such as mass, spin and electric charge, can be accounted for from this structure. These particle properties appear to originate from a single point at the center of the wave function structure, in the same sort of way that the Shell theorem of gravity causes the gravity of a body to appear to all originate from a central point. This paper represents the first two fully characterized fundamental particles, with a complete description of their structure and properties, built up from the underlying Electromagnetic waves that comprise these and all fundamental particles.

Multiquark cluster form factors in the relativistic harmonic oscillator model

A QCD multiquark cluster system is studied in the relativistic harmonic oscillator potential model （RHOPM）, and the electromagnetic form factors of the pion, proton and deuteron in the RHOPM are predicted. The calculated theoretical results are then compared with existing experimental data, finding very good agreement between the theoretical predictions and experimental data for these three target particles. We claim that this model can be applied to study QCD hadronic properties, particularly neutron properties, and to find six-quark cluster and/or nine-quark cluster probabilities in light nuclei such as helium 3He and tritium 3H. This is a problem of particular importance and interest in quark nuclear physics.

Bound States Energies of a Harmonic Oscillator Perturbed by Point Interactions

We determine explicitly the exact transcendental bound states energies equation for a one-dimensional harmonic oscillator perturbed by a single and a double point interactions via Green's function techniques using both momentum and position space representations. The even and odd solutions of the problem are discussed. The corresponding limiting cases are recovered. For the harmonic oscillator with a point interaction in more than one dimension,divergent series appear. We use to remove this divergence an exponential regulator and we obtain a transcendental equation for the energy bound states. The results obtained here are consistent with other investigations using different methods.

非线性阻尼下光滑不连续振子极限环的全局演化

针对光滑不连续振子,提出了一种优化的广义谐波函数摄动法,得到其极限环的振幅与系统参数之间的解析关系式以及极限环的解析近似解。同时,基于微分方程定性理论,建立了该振子极限环特征量的解析计算公式。利用上述结果,可围绕极限环何时产生、如何分岔、在何处消失以及稳定性如何等问题,对具有复杂非线性阻尼项的光滑不连续振子极限环的全局演化过程展开定量分析。通过将本文所得之结果与龙格-库塔法之结果进行对比,验证了所提优化方法的可行性和可靠性,为研究强非线性振动系统解的全局演化问题,提供了新的参考方法。