Exact Solutions of the Klein-Gordon Equation with a New Anharmonic Oscillator Potential

We solve the Klein-Cordon equation with a new anharmonic oscillator potential and present the exact solutions. It is shown that under the condition of equal scalar and vector potentials, the Klein-Cordon equation could be separated into an angular equation and a radial equation. The angular solutions are the associated-Legendre polynomial and the radial solutions are expressed in terms of the confluent hypergeometric functions. Finally, the energy equation is obtained from the boundary condition satisfied by the radial wavefunctions.     

Infinite-Parameter Potential Symmetries and a New Exact Solution for the Particle-Cluster Dynamic Equation

The master equation of a one-dimensional lattice-gas model with order preservation where the occupation probabilities of sites corresponding to Bose statistics as a consequence of the prescribed dynamics is studied with the potential symmetry method.The infinite-parameter potential symmetry and a new exact solution are obtained.The result illustrates that there remains the possibility of the above nonlinear equation to a linear partial differential equation by a non-invertible mapping.     

Exact Solutions of the Klein-Gordon Equation for the Scarf-Type Potential via Nikiforov-Uvarov Method

In this paper we present the exact solutions of the one-dimensional Klein-Gordon equation for the Scarf-type potential with equal scalar and vector potentials. Exact solutions and corresponding energy eigenvalues equation are obtained using Nikiforov-Uvarov mathematical method for the s-wave bound state. The PT-symmetry and Hermiticity for this potential are also considered. It will be shown that the obtained results of the Scarf-type potential are reduced to the results of the well-known potentials in the special cases.     

Exact Solutions of Klein-Gordon Equation with Exponential Scalar and Vector Potentials

We obtain the exact analytical solution of the Klein-Gordon equation for the exponential vector and scalar potentials by using the asymptotic iteration method. For the scalar potential greater than the vector potential case, the exact bound state energy eigenvalues and corresponding eigenfunctions are presented. The bound state eigenfunction solutions are obtained in terms of the confluent hypergeometric functions.     

Exact Solutions of the Schrödinger Equation with Inverse-Power Potential

The Schrödinger equation for stationary states is studied in a central potential V(r) proportional to r ^– in an arbitrary number of spatial dimensions. The presence of a single term in the potential makes it impossible to use previous algorithms, which only work for quasi-exactly-solvable problems. Nevertheless, the analysis of the stationary Schrödinger equation in the neighbourhood of the origin and of the point at infinity is found to provide relevant information about the desired solutions for all values of the radial coordinate. The original eigenvalue equation is mapped into a differential equation with milder singularities, and the role played by the particular case = 4 is elucidated. In general, whenever the parameter is even and larger than 4, a recursive algorithm for the evaluation of eigenfunctions is obtained. Eventually, in the particular case of two spatial dimensions, the exact form of the ground-state wave function is obtained for a potential containing a finite number of inverse powers of r, with the associated energy eigenvalue.     

Bound States of the S-Wave Equation with Equal Scalar and Vector Standard Eckart Potential

A supersymmetric technique for the bound-state solutions of the s-wave Klein-Gordon equation with equal scalar and vector standard Eckart-type potential is proposed. Its exact solutions are obtained. Possible generalization of our approach is outlined.     

Solution of the Dirac Equation with Position-Dependent Mass for q-Parameter Modified P?schl–Teller and Coulomb-like Tensor Potential

In this research, we have been obtained the Dirac equation for q-parameter modified P?schl–Teller potential including a Coulomb-like tensor interaction with arbitrary spin-orbit coupling quantum number by choosing a position-dependent mass. The energy eigenvalues equation and the corresponding unnormalized wave functions have been obtained. The Nikiforov-Uvarov method has been used in the calculations.     

Exact Bound State Solutions of the Schr?dinger Equation for Noncentral Potential via the Nikiforov-Uvarov Method

Exact bound state solutions of the Schr?dinger equation for generalized noncentral potential are examined by means of the Nikiforov-Uvarov method. The wavefunctions and the corresponding energy eigenvalues of the system are obtained analytically. The results examined for the potential are compatible with those obtained by the other methods.     

Exact Analytical Solution of the Klein–Gordon Equation in the Generalized Woods–Saxon Potential

The exact analytical solution of the Klein–Gordon equation for the spin-0 particles in the generalized Woods–Saxon potential is presented. The bound state energy eigenvalues and corresponding wave functions are obtained in the closed forms. The correlations between the potential parameters and energy eigenvalues are examined for π0particles.     

Solution of the Dirac Equation on the Bertotti–Robinson Metric

It is shown that each component of the Dirac field satisfies a decoupled equation, which admits separable solutions, when the background spacetime is the Bertotti–Robinson metric, which is a solution of the Einstein vacuum field equations with a cosmological constant. Furthermore, the seperated functions appearing in the solutions are shown to obey identities of the Teukolsky–Starobinsky type and the separable solutions are shown to be eigenfunctions of a certain differential operator.     

Effective Mass and Pseudoscalar Interaction in the Dirac Equation with Woods–Saxon Potential

We consider the one-dimensional Dirac equation with the Woods–Saxon potential in the Framework of position dependent mass and pseudoscalar interaction. By imposing appropriate constraints on the mass function and the pseudoscalar term new exact solvable models are obtained. A detailed study of the scattering and bound-states problems for these models is presented. Meanwhile, we work out the exact expressions for the transmission and reflection probabilities of scattered states and obtain the exact equation for the energy eigenvalues associated to bound states. In particular, transmission resonance at zero-momentum is observed for supercritical states.     

Solution of Gauss-Codazzi Equation with Applications in the Tzitzeica Equation

The surface in R3 associated with the Tzitzeica equation ＆ considered. By curvature coordinate transformation and surface imbedding, the Gauss-Codazzi equation is presented. Resorting to the solutions of the Gauss-Codazzi equation, the solution of the Tzitzeica equation ＆ obtained under a restrictive condition.     

Spin Symmetry for Dirac Equation with the Trigonometric Pöschl-Teller Potential

Within the framework of the Dirac theory, the relativistic bound states for the trigonometric Pöschl-Teller (PT) potential are obtained in the case of spin symmetry. It is found from the numerical results that there exist only positive energy states for bound states in the case of spin symmetry. Also, the energy levels approach a constant when the potential parameter α goes to zero. The special case for equally scalar and vector trigonometric PT potential is also studied briefly.     

Rosen-MorseⅡ势函数的Klein-Gordon方程和Dirac方程束缚态的精确解

在标量势和矢量势相等的情形, 研究了Rosen-MorseⅡ势的相对论效应, 应用超对称和形状不变势方法通过求解Klein-Gordon方程和Dirac方程得到了束缚态能量本征值, 最后, 讨论了Rosen-MorseⅡ势的一种特殊情况.     

Exact Solutions of Schr¨odinger Equation with Improved Ring-Shaped Non-Spherical Harmonic Oscillator and Coulomb Potential

We propose improved ring shaped like potential of the form,2V(r,θ)=V(r)+(2/2M r)[(βsin2θ+γcos2θ+2λ)/sinθcosθ]and its exact solutions are presented via the Nikiforov–Uvarov method.The angle dependent part V(θ)=(2/22M r)[(βsin22θ+γcos2θ+λ)/sinθcosθ],which is reported for the first time embodied the novel angle dependent(NAD)potential and harmonic novel angle dependent potential(HNAD)as special cases.We discuss in detail the effects of the improved ring shaped like potential on the radial parts of the spherical harmonic and Coulomb potentials.     

Approximate Solutions of the Schr?dinger Equation for?the Rosen-Morse Potential Including Centrifugal Term

Analytical solutions of the Schr?dinger equation for the Rosen-Morse potential are presented for arbitrary orbital angular momentum quantum number by using an approximation for the centrifugal term. The energy eigenvalues and the corresponding wavefunctions are approximately obtained. Three special cases the s-wave, the Eckart potential and the PT-symmetric Rosen-Morse potential are also investigated.     

Bounded Solutions of the Dirac Equation with a PT-symmetric Kink-Like Vector Potential in Two-Dimensional Space-Time

The (1+1)-dimensional Dirac equation with a PT-symmetric kink-like vector potential is investigated. By using the basic concepts of the supersymmetric WKB formalism and the function analysis method, we solve exactly the Dirac equation and obtain the bound-state energy levels and two-component spinor components. The PT-symmetric kink-like potential is not Hermitian and absent of bound states in the context of non-relativistic Schrödinger equation, but it possesses two sets of real discrete relativistic energy spectra in the context of the Dirac theory. When the PT symmetry is spontaneously broken, two sets of real energy spectra come into complex conjugate.     

Approximate Analytical Solutions of the Dirac Equation for Yukawa Potential Plus Tensor Interaction with Any κ-Value

Approximate analytical solutions of the Dirac equation are obtained for the Yukawa potential plus a tensor interaction with any κ-value for the cases having the Dirac equation pseudospin and spin symmetry. The potential describing tensor interaction has a Yukawa-like form. Closed forms of the energy eigenvalue equations and the spinor wave functions are computed by using the Nikiforov–Uvarov method. It is observed that the energy eigenvalue equations are consistent with the ones obtained before. Our numerical results are also listed to see the effect of the tensor interaction on the bound states.     

A General Approach for the Exact Solution of the Schrödinger Equation

The Schrödinger equation is solved exactly for some well known potentials. Solutions are obtained reducing the Schrödinger equation into a second order differential equation by using an appropriate coordinate transformation. The Nikiforov-Uvarov method is used in the calculations to get energy eigenvalues and the corresponding wave functions.