Exact solutions of the Klein—Gordon equation with ring-shaped oscillator potential by using the Laplace integral transform

We present exact solutions for the Klein-Gordon equation with a ring-shaped oscillator potential. The energy eigenvalues and the normalized wave functions are obtained for a particle in the presence of non-central oscillator potential. The angular functions are expressed in terms of the hypergeometric functions. The radial eigenfunctions have been obtained by using the Laplace integral transform. By means of the Laplace transform method, which is efficient and simple, the radial Klein-Gordon equation is reduced to a first-order differential equation.     

Crystalline undulator radiation and sub-harmonic bifurcation of system

Looking for new light sources, especially short wavelength laser light sources has attracted widespread attention. This paper analytically describes the radiation of a crystalline undulator field by the sine-squared potential. In the classical mechanics and the dipole approximation, the motion equation of a particle is reduced to a generalized pendulum equation with a damping term and a forcing term. The bifurcation behavior of periodic orbits is analyzed by using the Melnikov method and the numerical method, and the stability of the system is discussed. The results show that, in principle, the stability of the system relates to its parameters, and only by adjusting these parameters appropriately can the occurrence of bifurcation be avoided or suppressed.     

Exact bound state solutions of the Klein--Gordon particle in Hulthén potential

In this paper, the Klein-Gordon equation with the spherical symmetric Hulthén potential is turned into a hypergeometric equation and is solved in the framework of function analysis exactly. The corresponding bound state solutions are expressed in terms of the hypergeometric function, and the energy spectrum of the bound states is obtained as a solution to a given equation by boundary constraints.     

Properties of bottomonium in a semi-relativistic model

Using a semi-relativistic potential model we investigate the spectra and decays of the bottomonium (bb-) system. The Hamiltonian of our model consists of a relativistic kinetic energy term, a vector Coulomb-like potential and a scalar confining potential. Using this Hamiltonian, we obtain a spinless wave equation, which is then reduced to the form of a single particle Schrodinger equation. The spin dependent potentials are introduced as a perturbation. The three-dimensional harmonic oscillator wave function is employed as a trial wave function and the bb- mass spectrum is obtained by the variational method. The model parameters and the wave function that reproduce the the bb- spectrum are then used to investigate some of their decay properties. The results obtained are then compared with the experimental data and with the predictions of other theoretical models.     

A Particle Interacting with V-shaped Potential Decorated by a Dirac Delta Function Interaction at Center

The Schrodinger equation for a particle in the V-shaped potential decorated by a repulsive or attractive Dirac delta function interaction at the center is solved, demonstrating the crucial influence of point interaction on the even-parity states of the original system without decoration. As strength of the attraction increases, the ground state energy falls down without limit; and in limit of infinitely large attraction, the ground state approaches a singular state. Our analysis and conclusion can be readily generalized to any one-dimensional system a particle interacts with symmetrical potential plus the Dirac delta function interaction at the center.     

Effects of Lvy noise and immune delay on the extinction behavior in a tumor growth model

The combined effects of Lvy noise and immune delay on the extinction behavior in a tumor growth model are explored. The extinction probability of tumor with certain density is measured by exit probability. The expression of the exit probability is obtained using the Taylor expansion and the infinitesimal generator theory. Based on numerical calculations, it is found that the immune delay facilitates tumor extinction when the stability index α < 1, but inhibits tumor extinction when the stability index α > 1. Moreover, larger stability index and smaller noise intensity are in favor of the extinction for tumor with low density. While for tumor with high density, the stability index and the noise intensity should be reduced to promote tumor extinction.     

Chaotic Behavior of a Brownian Particle in a Periodic Potential

The classical deterministic dynamics of a Brownian particle with a time-dependent periodic perturbation in a spatially periodic potential is investigated. We have constructed a perturbed chaotic solution near the heteroclinic orbit of the nonlinear dynamics system by using the Constant-Variation method. Theoretical analysis and numerical result show that the motion of the Brownian particle is a kind of chaotic motion. The corresponding chaotic region in parameter space is obtained analytically and numerically.     

Analytical Solutions of the Manning-Rosen Potential In the Tridiagonal Program

The Schr？dinger equation with the Manning-Rosen potential is studied by working in a complete square integrable basis that carries a tridiagonal matrix representation of the wave operator. In this program, solving the Schr？dinger equation is translated into finding solutions of the resulting three-term recursion relation for the expansion coefficients of the wavefunction. The discrete spectrum of the bound states is obtained by diagonalization of the recursion relation with special choice of the parameters and the wavefunctions is expressed in terms of the Jocobi polynomial.     

Collapse arrest in the space-fractional Schrödinger equation with an optical lattice

The soliton solution and collapse arrest are investigated in the one-dimensional space-fractional Schr?dinger equation with Kerr nonlinearity and optical lattice. The approximate analytical soliton solutions are obtained based on the variational approach, which provides reasonable accuracy. Linear-stability analysis shows that all the solitons are linearly stable. No collapses are found when the Lévy index 1 α≤ 2. For α = 1, the collapse is arrested by the lattice potential when the amplitude of perturbations is small enough. It is numerically proved that the energy criterion of collapse suppression in the two-dimensional traditional Schr?dinger equation still holds in the one-dimensional fractional Schr?dinger equation. The physical mechanism for collapse prohibition is also given.     

Transmission time in the reflectionless complex potential

A phase time definition directly obtained from the Schr6dinger equation is used to investigate the time delay of a particle scattered by complex reflectionless potential. The artifacts introduced by truncating in the numerical simulation are clarified. The time delay of the transmitted wave packet is found to be equal to the reflection time of the truncated potential. Both time delays are the same as the traversal time in the free space, but shorter than the time taken by a classical particle to pass the same potential.     

Exact Polynomial Solutions of SchrSdinger Equation with Various Hyperbolic Potentials

The Schrodinger equation with hyperbolic potential V （ x ）=- Vosinh 2q （ x / d） / cosh 6 （ x / d） （q= 0, 1, 2, 3） is studied by transforming it into the confluent Heun equation. We obtain genera/symmetric and antisymmetric polynomial solutions of the SchrSdinger equation in a unified form via the Functional Bethe ansatz method. Furthermore, we discuss the characteristic of wavefunction of bound state with varying potential strengths. Particularly, the number of wavefunction＇s nodes decreases with the increase of potentiaJ strengths, and the particle tends to the bottom of the potential well correspondingly.     

Exact Polynomial Solutions of Schrdinger Equation with Various Hyperbolic Potentials

The Schrdinger equation with hyperbolic potential 2V(x) =-V0sinhq(x/d)/cosh6(x/d)(q = 0, 1, 2, 3) is studied by transforming it into the confluent Heun equation. We obtain general symmetric and antisymmetric polynomial solutions of the Schrdinger equation in a unified form via the Functional Bethe ansatz method. Furthermore, we discuss the characteristic of wavefunction of bound state with varying potential strengths. Particularly, the number of wavefunction's nodes decreases with the increase of potential strengths, and the particle tends to the bottom of the potential well correspondingly.     

Solution of Dirac equation around a charged rotating black hole

The aim of this paper is to solve the radial parts of a Dirac equation in Kerr-Newman （KN） geometry. The potential is replaced by a collection of step functions, then the reflection and transmission coefficients as well as the solution of the wave equation are obtained by using a quantum mechanical method. The result shows that the waves with different values of mass will be scatted off very differently.     

Analytical Solutions of a Model for Brownian Motion in the Double Well Potential

In this paper, the analytical solutions of Schr¨odinger equation for Brownian motion in a double well potential are acquired by the homotopy analysis method and the Adomian decomposition method. Double well potential for Brownian motion is always used to obtain the solutions of Fokker–Planck equation known as the Klein–Kramers equation, which is suitable for separation and additive Hamiltonians. In essence, we could study the random motion of Brownian particles by solving Schr¨odinger equation. The analytical results obtained from the two different methods agree with each other well. The double well potential is affected by two parameters, which are analyzed and discussed in details with the aid of graphical illustrations. According to the final results, the shapes of the double well potential have significant influence on the probability density function.     

Quantization of Space in the Presence of a Minimal Length

In this article,we apply the Generalized Uncertainty Principle(GUP),which is consistent with quantum gravity theories to an elementary particle in a finite potential well,and study the quantum behavior in this system.The generalized Hamiltonian contains two additional terms,which are proportional to αp~3(the result of the maximum momentum assumption) and α~2p~4(the result of the minimum length assumption),where α ~ 1/M_(PIC) is the GUP parameter.On the basis of the work by Ali et al.,we solve the generalized Schrodinger equation which is extended to include the α~2 correction term,and find that the length L of the finite potential well must be quantized.Then a generalization to the double-square-well potential is discussed.The result shows that all the measurable lengths especially the distance between the two potential wells are quantized in units of α_0l_(PI) in GUP scenario.     

Schrdinger Equation of a Particle on a Rotating Curved Surface

We derive the Schrdinger equation of a particle constrained to move on a rotating curved surface S.Using the thin-layer quantization scheme to confine the particle on S,and with a proper choice of gauge transformation for the wave function,we obtain the well-known geometric potential V_g and an additive Coriolis-induced geometric potential in the co-rotational curvilinear coordinates.This novel effective potential,which is included in the surface Schrdinger equation and is coupled with the mean curvature of S,contains an imaginary part in the general case which gives rise to a non-Hermitian surface Hamiltonian.We find that the non-Hermitian term vanishes when S is a minimal surface or a revolution surface which is axially symmetric around the rolling axis.     

Depletion interactions in binary mixtures of repulsive colloids

Acceptance ratio method, which has been used to calculate the depletion potential in binary hard-sphere mixtures, is extended to the computation of the depletion potential of non-rigid particle systems. The repulsive part of the Lennard-Jones pair potential is used as the direct pair potential between the non-rigid particles. The depletion potential between two big spheres immersed in a suspension of small spheres is determined with the acceptance ratio method through the application of Monte Carlo simulation. In order to check the validity of this method, our results are compared with those obtained by the Asakura-Oosawa approximation, and by Varial expansion approach, and by molecular dynamics simulation. The total effective potential and the depth of its potential well are computed for various softness parameters of the direct pair potential.     

Solutions of Dirac Equation for Makarov Potential with Pseudospin Symmetry

The pseudospin symmetry in the Makarov potential is investigated systematically by solving the Dirac equation. The analytical solution for the Makarov potential with pseudospin symmetry is obtained by Nikiforov-Uvarov （N-U） method. The eigenfunctions and eigenenergies are presented with equal mixture of vector and scalar potentials in opposite signs, for which is exact.     

Bose-Einstein Condensate in a Linear Trap with a Dimple Potential

We study Bose-Einstein condensation in a linear trap with a dimple potential where we model dimple potentials by Dirac δ function. Attractive and repulsive dimple potentials are taken into account. This model allows simple, explicit numerical and analytical investigations of noninteracting gases. Thus, the Schrdinger equation is used instead of the Gross-Pitaevski equation. We calculate the atomic density, the chemical potential, the critical temperature and the condensate fraction. The role of the relative depth of the dimple potential with respect to the linear trap in large condensate formation at enhanced temperatures is clearly revealed. Moreover, we also present a semi-classical method for calculating various quantities such as entropy analytically. Moreover, we compare the results of this paper with the results of a previous paper in which the harmonic trap with a dimple potential in 1D is investigated.     

Thermal transport property of Ge34 and d-Ge investigated by molecular dynamics and the Slack’s equation

In this study,we evaluate the values of lattice thermal conductivity κ L of type II Ge clathrate (Ge 34) and diamond phase Ge crystal (d-Ge) with the equilibrium molecular dynamics (EMD) method and the Slack's equation.The key parameters of the Slack's equation are derived from the thermodynamic properties obtained from the lattice dynamics (LD) calculations.The empirical Tersoff's potential is used in both EMD and LD simulations.The thermal conductivities of d-Ge calculated by both methods are in accordance with the experimental values.The predictions of the Slack's equation are consistent with the EMD results above 250 K for both Ge34 and d-Ge.In a temperature range of 200-1000 K,the κ L value of d-Ge is about several times larger than that of Ge 34.