Bound State Solutions Research Articles

Rosen-Morse potentials for relativistic spinless particles; approximate solutions

We deal with the D-dimensional radial Klein-Gordon equation for Rosen-Morse type potential with unequal scalar and vector potentials. We apply a proper approximation to the centrifugal term and, by proposing an ansatz solution to the resulting equation, we obtain the bound-state solution. To check the accuracy of our results, we compare our obtained quasi-analytical energies with the exact numerical ones.

Fermion in a multi-kink-antikink soliton background, and exotic supersymmetry

We construct a fermion system in a multi-kink-antikink soliton background, and present in an explicit form all its trapped configurations (bound state solutions) as well as scattering states. This is achieved by exploiting an exotic N=4 centrally extended nonlinear supersymmetry of completely isospectral pairs of reflectionless Schrodinger systems with potentials to be n-soliton solutions for the Korteweg-de Vries equation. The obtained reflectionless Dirac system with a position-dependent mass is shown to possess its own exotic nonlinear supersymmetry associated with the matrix Lax-Novikov operator being a Darboux-dressed momentum. In the process, we get an algebraic recursive representation for the multi-kink-antikink backgrounds, and establish their relation to the the modified Korteweg-de Vries equation. We also indicate how the results can be related to the physics of self-consistent condensates based on the Bogoliubov-de Gennes equations.

Stationary states of fermions in a sign potential with a mixed vector–scalar coupling

The scattering of a fermion in the background of a sign potential is considered with a general mixing of vector and scalar Lorentz structures with the scalar coupling stronger than or equal to the vector coupling under the Sturm–Liouville perspective. When the vector coupling and the scalar coupling have different magnitudes, an isolated solution shows that the fermion under a strong potential can be trapped in a highly localized region without manifestation of Klein’s paradox. It is also shown that the lonely bound-state solution disappears asymptotically as one approaches the conditions for the realization of spin and pseudospin symmetries.

Dirac equation for the generalized Deng-Fan potential with coulomb and Yukawa tensor interactions

In this paper, we investigate the bound-state solutions of the Dirac equation with the generalized Deng-Fan potential within the framework of spin and pseudospin symmetries and with Coulomblike and Yukawa-like tensor interactions by using a supersymmetric quantum-mechanics (SUSYQM) formulation. We obtain the energy eigenvalue equations and the corresponding upper and lower spinor wave functions for both the spin and the pseudospin cases. We also report some numerical results and figures to show the effect of the tensor interaction.

Missing solution in a Cornell potential

Missing bound-state solutions for fermions in the background of a Cornell potential consisting of a mixed scalar–vector–pseudoscalar coupling is examined. Charge-conjugation operation, degeneracy and localization are discussed.

Compact submanifolds supporting singular interactions

A quantum particle moving under the influence of singular interactions on embedded surfaces furnish an interesting example from the spectral point of view. In these problems, the possible occurrence of a bound-state is perhaps the most important aspect. Such systems can be introduced as quadratic forms and generically they do not require renormalization. Yet an alternative path through the resolvent is also beneficial to study various properties. In the present work, we address these issues for compact surfaces embedded in a class of ambient manifolds. We discover that there is an exact bound state solution written in terms of the heat kernel of the ambient manifold for a range of coupling strengths. Moreover, we develop techniques to estimate bounds on the ground state energy when several surfaces, each of which admits a bound state solution, coexist.

Approximate Solutions of the Schrödinger Equation with the Hyperbolical Potential: Supersymmetric Approach

The bound state solution of the Schrödinger equation with the hyperbolical potential is obtained by using supersymmetric approach. By applying proper approximation scheme to deal with the centrifugal barrier, we obtain the energy eigenvalues and the corresponding wave functions are obtained in terms of generalized hypergeometric functions. Comparison of our computed numerical results with the ones obtained by findings of other methods reveals that supersymmetric approach is reliable, efficient and accurate.

Scattering States of Schrödinger Equation under the Modified Cusp Potential

We solve the one-dimensional time-independent Schrödinger equation in the presence of the modified Cusp potential and report the solutions in terms of the Whittaker functions. We obtain the reflection and transmission coefficients as well as the bound-state solutions in terms of the Whittaker functions. We comment on the solutions and discuss them in terms of the engaged parameters.

Approximate Bound States Solution of the Hellmann Potential

The Hellmann potential, which is a superposition of an attractive Coulomb potential −a/r and a Yukawa potential b e−δr/r, is often used to compute bound-state normalizations and energy levels of neutral atoms. By using the generalized parametric Nikiforov—Uvarov (NU) method, we have obtained the approximate analytical solutions of the radial Schrödinger equation (SE) for the Hellmann potential. The energy eigenvalues and corresponding eigenfunctions are calculated in closed forms. Some numerical results are presented, which show good agreement with a numerical amplitude phase method and also those previously obtained by other methods. As a particular case, we find the energy levels of the pure Coulomb potential.

Rotating effects on the Dirac oscillator in the cosmic string spacetime

In this contribution, we study the Dirac oscillator under the influence of noninertial effects of a rotating frame in the cosmic string spacetime. We show that both noninertial effects and the topology of the cosmic string spacetime restrict the physical region of the spacetime where the quantum particle can be placed, and discuss two different cases of bound states solutions of the Dirac equation by analysing the behaviour of the Dirac oscillator frequency.

Bound state solutions of Klein–Gordon equation with Mobius square plus Yukawa potentials

We have solved approximately Klein–Gordon equation with equal scalar and vector Mobius square plus Yukawa potentials in D-dimensions using the parametric form of Nikiforov–Uvarov method. Energy eigenvalues and corresponding wave functions in terms of Jacobi polynomials are obtained. We have also discussed some special cases of our potential.

Numerical solution of the Schrödinger equations by using Delta-shaped basis functions

Numerical simulations of the nonlinear Schrodinger equations are studied using Delta-shaped basis functions, which recently proposed by Reutskiy. Propagation of a soliton, interaction of two solitons, birth of standing and mobile solitons and bound state solutions are simulated. Some conserved quantities are computed numerically for all cases. Then we extend application of the method to solve some coupled nonlinear Schrodinger equations. Obtained systems of ordinary differential equations are solved via the fourth- order Runge–Kutta method. Numerical solutions coincide with the exact solutions in desired machine precision and invariant quantities are conserved sensibly. Some comparisons with the methods applied in the literature are carried out.

Confinement of spin-1 and spin-0 bosons in (1 + 1)-dimensions with a scalar linear potential

We obtain bound-state solutions of the Duffin–Kemmer–Petiau equation in one spatial dimension in the presence of a Lorentz-scalar linear potential. The exact conditions selecting the eigenvalues of the system have been derived and are shown to match their non-relativistic counterparts in the weak-coupling limit.

Effective Hamiltonian for bound states in Yukawa theory

A generalization of the Gell-Mann–Low theorem is applied to lowest nontrivial order to determine an effective Hamiltonian for two-fermion states in relativistic Yukawa theory. The consistency of the corresponding effective Schrödinger equation is thoroughly investigated in various aspects, among others the nonrelativistic and one-body limits, and the small-distance or large-momentum regime of the bound state solutions is discussed in detail.

Infinite families of (non)-Hermitian Hamiltonians associated with exceptional Xm Jacobi polynomials

Using an appropriate change of variable, the Schrödinger equation is transformed into a second-order differential equation satisfied by recently discovered Jacobi-type Xm exceptional orthogonal polynomials. This facilitates the derivation of infinite families of exactly solvable Hermitian as well as non-Hermitian trigonometric Scarf potentials and a finite number of Hermitian and an infinite number of non-Hermitian -symmetric hyperbolic Scarf potentials. The bound state solutions of all these potentials are associated with the aforesaid exceptional orthogonal polynomials. These infinite families of potentials are shown to be extensions of the conventional trigonometric and hyperbolic Scarf potentials by the addition of some rational terms characterized by the presence of classical Jacobi polynomials. All the members of a particular family of these ‘rationally extended polynomial-dependent’ potentials have the same energy spectrum and possess translational shape-invariant symmetry. The obtained non-Hermitian trigonometric Scarf potentials are shown to be quasi-Hermitian in nature ensuring the reality of the associated energy spectra.

Massive Dirac equation in asymmetric Hulthén potential

One-dimensional effective mass Dirac equation is solved in asymmetric Hulthén potential. Scattering and bound state solutions are obtained in terms of hypergeometric functions. Transmission and reflection coefficients are calculated by matching conditions on the wave function. The necessary conditions for the existence of transmission resonances and supercriticality are derived.

Spinless particles in the field of unequal scalar—vector Yukawa potentials

We present analytical bound state solutions of the spin-zero Klein—Gordon (KG) particles in the field of unequal mixture of scalar and vector Yukawa potentials within the framework of the approximation scheme to the centrifugal potential term for any arbitrary l-state. The approximate energy eigenvalues and unnormalized wave functions are obtained in closed forms using a simple shortcut of the Nikiforov—Uvarov (NU) method. Further, we solve the KG—Yukawa problem for its exact numerical energy eigenvalues via the amplitude phase (AP) method to test the accuracy of the present solutions found by using the NU method. Our numerical tests using energy calculations demonstrate the existence of inter-dimensional degeneracy amongst the energy states of the KG—Yukawa problem. The dependence of the energy on the dimension D is numerically discussed for spatial dimensions D = 2–6.

(1+1)-Dirac bound states in one dimension, with position-dependent Fermi velocity and mass

Abstract We extend Panella and Roy’s [17] work for massless Dirac particles with position-dependent (PD) velocity. We consider Dirac particles where the mass and velocity are both position-dependent. Bound states in the continuum (BIC)-like and discrete bound state solutions are reported. It is observed that BIC-like solutions are not only feasible for the ultra-relativistic (massless) Dirac particles but also for Dirac particles with PDmass and PD-velocity that satisfy the condition m(x) v F2 (x) = A, where A ≥ 0 is constant. Dirac Pöschl-Teller and harmonic oscillator models are also reported.

Arbitrary-state solutions of the Dirac equation for a Möbius square potential using the Nikiforov-Uvarov method

We inquire into spin and pseudospin symmetries of the Dirac equation under a Möbius square-type potential using the Nikiforov-Uvarov method to calculate the bound state solutions. We numerically discuss the problem and include various explanatory figures.

Relativistic symmetries in the Rosen—Morse potential and tensor interaction using the Nikiforov—Uvarov method

Approximate analytical bound-state solutions of the Dirac particle in the fields of attractive and repulsive Rosen—Morse (RM) potentials including the Coulomb-like tensor (CLT) potential are obtained for arbitrary spin-orbit quantum number κ. The Pekeris approximation is used to deal with the spin-orbit coupling terms κ (κ± 1)r−2. In the presence of exact spin and pseudospin (p-spin) symmetries, the energy eigenvalues and the corresponding normalized two-component wave functions are found by using the parametric generalization of the Nikiforov—Uvarov (NU) method. The numerical results show that the CLT interaction removes degeneracies between the spin and p-spin state doublets.