Scattering State Solutions Research Articles

Scattering State Solutions of Vector Bosons Interacting with Sun Potential

For vector bosons with spin-1, scattering state solutions have been attained by considering the Duffin-Kemmer-Petiau equation with the Sun interaction field. Based on the obtained solution, relations for phase shift and scattering amplitude have been derived. Furthermore, the bound state energy eigenvalue relation has been derived by taking the scattering amplitude to infinity. The results obtained through the Mathematica software program are presented graphically and numerically. In addition, the effects of the variables in the interaction function on the obtained results are discussed.

Linearly stable and unstable complex soliton solutions with real energies in the Bullough-Dodd model

We investigate different types of complex soliton solutions with regard to their stability against linear perturbations. In the Bullough-Dodd scalar field theory we find linearly stable complex PT-symmetric solutions and linearly unstable solutions for which the PT-symmetry is broken. Both types of solutions have real energies. The auxiliary Sturm-Liouville eigenvalue equation in the stability analysis for the PT-symmetric solutions can be solved exactly by supersymmetrically mapping it to an isospectral partner system involving a shifted and scaled inverse cosh-squared potential. We identify exactly one shape mode in form of a bound state solution and scattering states which when used as linear perturbations leave the solutions stable. The auxiliary problem for the solutions with broken PT-symmetry involves a complex shifted and scaled inverse sin-squared potential. The corresponding bound and scattering state solutions have complex eigenvalues, such that when used as linear perturbations for the corresponding soliton solutions lead to their decay or blow up as time evolves.

Completeness and orthonormality of the energy eigenfunctions of the Dirac delta derivative potential

The orthonormality and completeness of the energy eigenfunctions solution for the Schrödinger equation are usually hard to prove explicitly. There are few examples treated in the literature that give an explicit demonstration of these properties. In this work, we are going to orthonormalize the energy eigenfunctions of the solutions of the Schrödinger equation with contact potential given by the Dirac delta derivative potential λδ′(x), where λ is the coupling constant and prove that they are complete. The solutions for this problem include both continuum and discrete energy spectra. This makes it a perfect choice in the sense that it covers all possible cases of the solutions in quantum mechanics. The bound states solutions are normalized as usual, while the scattering states are normalized in continuum sense. The scattering solutions are shown to be orthogonal to each other and to the bound states. Using these orthonormal eigenfunctions an explicit demonstration of the completeness is given. It is found that both the bound and scattering solutions should be considered to ensure completeness. The calculation shows that the scattering state solutions are complete by themselves except for a certain value of the energy eigenvalue. This energy eigenvalue was shown to be precisely consistent with the energy eigenvalue of the bound state.

Generation of Exactly Solvable Potential from Trigonometric Scarf Potential in D-dimensional Space

Exact bound state as well as scattering state solutions of Schrodinger Green’s function equation is presented for constructing hyperbolic class of exactly solvable quantum systems are obtained in D-dimensional space, using extended transformation method. In the current literature the exact solution to the hyperbolic Scarf are written in terms of Jacobi polynomials and these polynomials are referred to Romanovski polynomials. The normalizability of bound state solutions of the generated exactly solvable potential are discussed.

Bound and Scattering State Solutions of the Klein–Gordon Equation with Deng–Fan Potential in Higher Dimensions

Bound and Scattering State Solutions of the Klein–Gordon Equation with Deng–Fan Potential in Higher Dimensions

Approximate Scattering State Solutions of DKPE and SSE with Hellmann Potential

We study the approximate scattering state solutions of the Duffin-Kemmer-Petiau equation (DKPE) and the spinless Salpeter equation (SSE) with the Hellmann potential. The eigensolutions, scattering phase shifts, partial-waves transitions, and the total cross section for all the partial waves are obtained and discussed. The dependence of partial-waves transitions on total angular momentum number, angular momentum number, mass combination, and potential parameters was presented in the figures.

Scattering states solutions of Klein–Gordon equation with three physically solvable potential models

The scattering state solutions of the Klein–Gordon equation with equal scalar and vector Varshni, Hellmann and Varshni–Shukla potentials for any arbitrary angular momentum quantum number l are investigated within the framework of the functional analytical method using a suitable approximation. The asymptotic wave functions, approximate scattering phase shifts, normalization constants and bound state energy equations were obtained. The non-relativistic limits of the scattering phase shifts and the bound states energy equations for the three potentials were also obtained. Our bound states energy equations are in excellent agreement with the available ones in the literature. Our numerical and graphical results indicate the dependence of phase shifts on the screening parameter β, the potential parameter b and angular momentum quantum number l.

The semi-relativistic scattering states of the two-body spinless Salpeter equation with the Varshni potential model

In the present work, the scattering state solutions of the spinless Salpeter equation with the Varshni potential model are investigated. The approximate scattering phase shift, normalization constant, bound state energy, wave number and wave function in the asymptotic region are obtained. The behaviour of the orbital angular momentum number dependent phase shift with the equal and unequal masses combination are discussed. An expression for the total cross-section is also derived.

Approximate bound and scattering solutions of Dirac equation for the modified deformed Hylleraas potential with a Yukawa-type tensor interaction

Analytical bound and scattering state solutions of Dirac equation are investigated for the modified deformed Hylleraas potential with a Yukawa-type tensor interaction. The energy equation, phase shifts and normalization constants of the pseudospin and spin symmetry limits are represented. Since the modified deformed Hylleraas potential reduces to the Poschl–Teller, Hulthen and deformed Hylleraas potential, we have obtained energy equation and scattering properties of the Dirac equation for these potentials within a Yukawa-type tensor interaction. We have also reported some numerical results to show the effect of tensor interaction.

Scattering state solutions of the Duffin-Kemmer-Petiau equation with the Varshni potential model

The scattering state of the Duffin-Kemmer-Petiau equation with the Varshni potential was studied. The asymptotic wave function, the scattering phase shift and normalization constant were obtained for any J states by dealing with the centrifugal term using a suitable approximation. The analytical properties of the scattering amplitude and the bound state energy were obtained and discussed. Our numerical and graphical results indicate that the scattering phase shift depends largely on total angular momentum J , screening parameter $\beta$ and potential strengths a and b.

Relativistic Energies and Scattering Phase Shifts for the Fermionic Particles Scattered by Hyperbolical Potential with the Pseudo(spin) Symmetry

In this paper, we studied the approximate scattering state solutions of the Dirac equation with the hyperbolical potential with pseudospin and spin symmetries. By applying an improved Greene-Aldrich approximation scheme within the formalism of functional analytical method, we obtained the spin-orbit quantum numbers dependent scattering phase shifts for the spin and pseudospin symmetries. The normalization constants, lower and upper radial spinor for the two symmetries, and the relativistic energy spectra were presented. Our results reveal that both the symmetry constants (Cps and Cs) and the spin-orbit quantum number κ affect scattering phase shifts significantly.

The scattering phase shifts of the Hulthén-type potential plus Yukawa potential

The Hulthen-type potential is perturbed by adding the Yukawa potential. By applying a short-range approximation to the Schrodinger equation containing this model via standard method, the scattering state solutions and the corresponding phase shifts were obtained. Our numerical calculations and graphical solutions show the dependencies of scattering phase shifts on Yukawa potential constant A, asymptotic wave number k, screening parameter \( \alpha\) and angular momentum quantum number l.

Scattering States of l-Wave Schrödinger Equation with Modified Rosen—Morse Potential**Supported by the National Natural Science Foundation of China under Grant No. 11405128, and Natural Science Basic Research Plan in Shaanxi Province of China under Grant No. 15JK2093

Within a Pekeris-type approximation to the centrifugal term, we examine the approximately analytical scattering state solutions of the l-wave Schrödinger equation with the modified Rosen—Morse potential. The calculation formula of phase shifts is derived, and the corresponding bound state energy levels are also obtained from the poles of the scattering amplitude.

Bound and scattering states for a hyperbolic‐type potential in view of a new developed approximation

A new developed approximation is used to obtain the arbitrary l‐wave bound and scattering state solutions of Schrödinger equation for a particle in a hyperbolic‐type potential. For bound state, the energy eigenvalue equation and unnormalized wave functions in terms of Jacobi polynomials are achieved using the Nikiforov–Uvarov (NU) method. Besides, energy eigenvalues are calculated numerically for some states and compared with those given in the literature to check accuracy of our results. For scattering state, the wave function is found in terms of hypergeometric functions. Furthermore, scattering amplitude and phase shifts are achieved using scattering solutions. Also it is shown that the energy eigenvalue equation obtained from analytic property of scattering amplitude is same with one obtained using NU method. © 2015 Wiley Periodicals, Inc.

Self-consistent field theory of collisions: Orbital equations with asymptotic sources and self-averaged potentials

The self-consistent field theory of collisions is formulated, incorporating the unique dynamics generated by the self-averaged potentials. The bound state Hartree–Fock approach is extended for the first time to scattering states, by properly resolving the principal difficulties of non-integrable continuum orbitals and imposing complex asymptotic conditions. The recently developed asymptotic source theory provides the natural theoretical basis, as the asymptotic conditions are completely transferred to the source terms and the new scattering function is made fullyintegrable. The scattering solutions can then be directly expressed in terms of bound state HF configurations, establishing the relationship between the bound and scattering state solutions. Alternatively, the integrable spin orbitals are generated by constructing the individual orbital equations that contain asymptotic sources and self-averaged potentials. However, the orbital energies are not determined by the equations, and a special channel energy fixing procedure is developed to secure the solutions. It is also shown that the variational construction of the orbital equations has intrinsic ambiguities that are generally associated with the self-consistent approach. On the other hand, when a small subset of open channels is included in the source term, the solutions are only partiallyintegrable, but the individual open channels can then be treated more simply by properly selecting the orbital energies. The configuration mixing and channel coupling are then necessary to complete the solution. The new theory improves the earlier continuum HF model.

PT-/non-PT-symmetric and non-Hermitian Hellmann potential: approximate bound and scattering states with any ℓ-values

We investigated the approximate bound-state solutions of the Schrödinger equation for the PT-/non-PT-symmetric and non-Hermitian Hellmann potential. Exact energy eigenvalues and corresponding normalized wave functions were obtained. Numerical energy eigenvalues for the bound states were compared with ones obtained before. Scattering state solutions were also studied. Phase shifts of the potential are written in terms of the angular momentum quantum number ℓ.

Effects of tensors coupling to Dirac equation with shifted Hulthen potential via SUSYQM

The Dirac equation with shifted Hulthen potential in the presence of the Yukawa-like tensor (YLT) and generalized tensor (GLT) interactions using supersymmetric quantum mechanics (SUSYQM) is presented. The bound state energy spectra and the radial wave functions have been approximately obtained in the case of spin and pseudospin symmetries. We have also reported some numerical results and figures to show the effect of the tensor interactions. Furthermore, scattering state solution under the generalized tensor (GLT) interaction is reported.

Bound and scattering states of spinless particles under the generalized Pöschl–Teller potential

Bound and scattering state solutions of Klein–Gordon equation are obtained for equal scalar and vector Generalized Poschl–Teller potentials with angular momentum l. Energy eigenvalues, normalized wave functions and scattering phase shifts are calculated.

Bound and scattering state solutions of a hyperbolic-type potential

A hyperbolic-type potential with a centrifugal term is solved approximately using the path integral approach. The radial Green's function is expressed in closed form, from which the energy spectrum and the suitably normalized wave functions of bound and scattering states are extracted for (1/2) − [Formula: see text] < σ < (1/2) + [Formula: see text]. Besides, the phase shift and the scattering function Sl for each angular momentum l are deduced. The particular cases corresponding to the s-waves (l = 0) and the barrier potential (σ = 1) are also analyzed.

The Nonrelativistic Scattering States of the Deng-Fan Potential

The approximately analytical scattering state solution of the Schrodinger equation is obtained for the Deng-Fan potential by using an approximation scheme to the centrifugal term. Energy eigenvalues, normalized wave functions, and scattering phase shifts are calculated. We consider and verify two special cases: thel=0and thes-wave Hulthén potential.