Bound State Energy Equation Research Articles

Approximate analytical solutions of the Schrodinger equation in central potential field

We investigate the approximate l-state solutions of the Schrodinger equation for Hulthen plus a class of Yukawa potential. In this context, we construct the bound-state energy equation and the wave-function expressed by the Gauss hypergeometric function by means of asymptotic iteration approach in detail.

Bound and scattering states solutions of the Klein–Gordon equation with the attractive radial potential in higher dimensions

In this study, the Klein–Gordon equation (KGE) is solved with the attractive radial potential using the Nikiforov–Uvarov-functional-analysis (NUFA) method in higher dimensions. By employing the Greene–Aldrich approximation scheme, the approximate bound state energy equations as well as the corresponding radial wave function are obtained in closed form. Also, the expression for the scattering phase shift is obtained in D-dimensions. The effects of the screening parameter and the total angular momentum quantum number on the bound state energy and the scattering states’ phase shift are also studied numerically and graphically at different dimensions. An interesting result of this study is the inter-dimensional degeneracy symmetry for scattering phase shift. Hence, this concept is applicable in the areas of nuclear and particle physics.

Effect of the deformation parameter on the nonrelativistic energy spectra of the q-deformed Hulthen-quadratic exponential-type potential

In this study, an approximate solution of the Schr�dinger equation for the q-deformed Hulthen-quadratic exponential-type potential model within the framework of the Nikiforov�Uvarov method was obtained. The bound state energy equation and the corresponding eigenfunction was obtained. The energy spectrum is applied to study H2, HCl, CO and LiH diatomic molecules. The effect of the deformation parameters and other potential parameters on the energy spectra of the system were graphically and numerically analyzed in detail. Special cases were considered when the potential parameters were altered, resulting in deformed Hulthen potential, Hulthen potential, deformed quadratic exponential-type potential and quadratic exponential-type potential. The energy eigenvalues expressions agreed with what obtained in literature. Finally, the results can find many applications in quantum chemistry, atomic and molecular physics.

Approximate Bound State Solutions of the Hellmann Plus Kratzer Potential in N-dimensional Space

We have examined the approximate l_(N-1)-state solutions of the N-dimensional Schrödinger equation for a particle interacting with the Hellmann plus Kratzer potential. In hyperspherical coordinate system, we have constructed the bound state energy equation and the wavefunctions expressed by the hypergeometric function via the asymptotic iteration approach in detail. When considered the special cases of parameters in Hellmann plus Kratzer potential, this potential turns into several potential models. In this connection, the non-relativistic energy spectra for the modified Kratzer, Yukawa, Coulomb and Hellmann potentials in approximate analytic form have been obtained in hyperspherical coordinates. We have presented the numerical energy eigenvalues for the Hellmann, Yukawa and Coulomb potentials in N=3 dimensions. Our present results provide an appropriate test of the accuracy of asymptotic iteration formalism.

Solutions of the Klein Gordon equation with generalized hyperbolic potential in D-dimensions

We solve the D- dimensional Klein–Gordon equation with a newly proposed generalized hyperbolic potential model, under the condition of equal scalar and vector potentials. The relativistic bound state energy equation has been obtained via the functional analysis method. We obtained the relativistic and non-relativistic ro-vibrational energy spectra for different diatomic molecules. The numerical results for these diatomic molecules tend to portray inter-dimensional degeneracy symmetry. Variations of the energy eigenvalues obtained with the potential parameters have been demonstrated graphically. Our studies will find relevant applications in the areas of chemical physics and high-energy physics.

Generalized uncertainty principle and the asymmetrical spinless Salpeter Coulomb problem

The notion of minimal length is a common feature of various frameworks which aim to unify quantum mechanics with general relativity. As a consequence, a new generalized uncertainty principle has been proposed to replace the standard Heisenberg principle and this is likely to affect all quantum systems. In this paper, we study the effects of a minimal length on the quantum system of two different particles interacting via a Coulomb-type potential and described by the asymmetrical spinless Salpeter equation. We use the momentum representation to derive the exact energy equation for bound-states and work out the associated wave functions. In particular, we find that the minimal length regularizes the singularity of the problem at the position origin.

Approximate Treatment of the Dirac Equation with Hyperbolic Potential Function

The time independent Dirac equation is solved analytically for equal scalar and vector hyperbolic potential function in the presence of Greene and Aldrich approximation scheme. The bound state energy equation and spinor wave functions expressed by the hypergeometric function have been obtained in detail with asymptotic iteration approach. In order to indicate the accuracy of this different approach proposed to solve second order linear differential equations, we present that in the non-relativistic limit, analytical solutions of the Dirac equation converge to those of the Schrodinger one. We introduce numerical results of the theoretical analysis for hyperbolic potential function. Bound states corresponding to arbitrary values of n and l are reported for potential parameters covering a wide range of interaction. Also, we investigate relativistic vibrational energy spectra of alkali metal diatomic molecules in the different electronic states. It is observed that theoretical vibrational energy values are consistent with experimental Rydberg–Klein–Rees (RKR) results and vibrational energies of NaK, $$K_2$$ and KRb diatomic molecules interacting with hyperbolic potential smoothly converge to the experimental dissociation limit $$D_e=2508cm^{-1}$$ , $$254cm^{-1}$$ and $$4221cm^{-1}$$ , respectively.

Relativistic energies of the SiC radical in higher spatial dimensions

We solve the Dirac equation with the modified Rosen-Morse potential energy model in higher spatial dimensions. The bound state energy equation has been presented. We find that the inter-dimensional degeneracy symmetry exists for the molecular system represented by the modified Rosen-Morse potential. We observe that the behavior of the relativistic vibrational energies in higher dimensions remains similar to that of the three-dimensional system for the \( X^{3}\Pi\) state of the SiC radical. This symmetry phenomenon will breaks upon the critical point, through which the system undergoes a phase transition from a stable to an unstable state.

Bound state solutions of the SchrӦdinger equation with second PӦschl-Teller like potential model and the vibrational partition function, mean energy and mean free energy

By employing two different methods and suitable approximation schemes to deal with the centrifugal term, we solved approximately the Schrȍdinger equation with the Second Pȍschl-Teller like potential model. The bound state energy equation and the unnormalized radial wave functions have been obtained. The vibrational partition function, mean energy and vibrational mean free energy are also obtained.

Solutions of the Dirac equation with the Morse potential energy model in higher spatial dimensions

Analytical solutions of the Dirac equation with the Morse potential energy model in higher spatial dimensions have been explored. We present the bound-state energy equation and the corresponding upper and lower radial wave functions. We find that the behavior of the higher-dimensional relativistic vibrational energies remains similar to that of the three-dimensional molecular system for the \( {\rm X}^2\sum^+\) state of the CP molecule. This symmetry phenomenon will break at the critical point through which the system undergoes a phase transition from a stable to an unstable state.

Stability analysis of the solution of the Dirac equation for the vibrational energies of the SiF+ molecule

Analytical solutions of the Dirac equation with the spin symmetry in higher spatial dimensions have been explored for the improved Manning-Rosen potential energy model. We present the bound-state energy equation and the corresponding upper and lower radial wave functions. It has been found that the behavior of the higher-dimensional relativistic vibrational energies remains similar to that of the three-dimensional molecular system for the $ X^{1}\Sigma^{+}$ state of the SiF+ molecule. This symmetry phenomenon will break upon the critical point through which the system undergoes a phase transition from a stable to an unstable state.

Relativistic energies for the SiC radical

The analytical solutions of the Dirac equation with the modified Rosen-Morse potential energy model have been explored. Under the condition of the spin symmetry, we present the bound state energy equation. In the nonrelativistic limit, the relativistic energy equation becomes the nonrelativistic energy form deduced within the framework of the Schrodinger equation. We find that the relativistic effect of the relative motion of the ions leads to a little decrease in the vibrational energies when the vector potential is equal to the scalar potential for the electronic ground state of the SiC radical, while to an increase in those if the vector potential is greater than the scalar potential.

D-Dimensional relativistic energies for silver dimer

We study solutions of the Dirac equation with the improved Tietz potential model in higher spatial dimensions. Under the condition of the spin symmetry, we obtain the bound state energy equation. It has been found that there exists a relativistic inter-dimensional degeneracy: under the transformation of a decrease in the spatial dimension by two and an increase in the orbital angular momentum by one, the relativistic rotation-vibrational energy for a diatomic molecule is invariant. We observe that the behavior of the relativistic vibrational energies for the Ag2 B-11Πu state in higher dimensions remains similar to that of the three-dimensional system.

Solutions of the Klein–Gordon equation with the Morse potential energy model in higher spatial dimensions

We solve the Klein–Gordon equation with the Morse potential energy model to obtain the relativistic bound state energy equation in D spatial dimensions. We find that the inter-dimensional degeneracy symmetry exists for the molecular system represented by the Morse potential model. For a fixed vibrational quantum number and various rotational quantum numbers, the relativistic energies for the X1Σ+ state of the ScI molecule diverge as D increases. We observe that the behavior of the relativistic vibrational energies in higher dimensions remains similar to that of the three-dimensional system.

Molecular spinless energies of the improved Rosen-Morse potential energy model in D dimensions

Under the circumstance of equal scalar and vector potentials, we solve the Klein-Gordon equation with the improved Rosen-Morse potential energy model in D spatial dimensions. The relativistic bound state energy equation has been obtained by using the supersymmetric WKB approximation approach. For fixed vibrational quantum number and various rotational quantum numbers, the relativistic energies for the 33Σg+ state of the Cs2 molecule and the 51Δg state of the Na2 molecule increase as D increases. We observe that the behavior of the relativistic vibrational energies in higher dimensions remains similar to that of the three-dimensional system.

Molecular spinless energies of the modified Rosen–Morse potential energy model in higher spatial dimensions

We solve the Klein–Gordon equation with the modified Rosen–Morse potential energy model in D spatial dimensions. The bound state energy equation has been obtained by using the supersymmetric WKB approximation approach. We find that the inter-dimensional degeneracy symmetry exists for the molecular system represented by the modified Rosen–Morse potential. For fixed vibrational and rotational quantum numbers, the relativistic energies for the 61Πu state of the 7Li2 molecule and the X3Π state of the SiC radical increase as D increases. We observe that the behavior of the relativistic vibrational energies in higher dimensions remains similar to that of the three-dimensional system.

Relativistic energies for diatomic molecule nucleus motions with the spin symmetry

Abstract The analytical solutions of the Dirac equation with the spin symmetry for the improved Manning–Rosen potential energy model have been explored. We present the bound state energy equation and the corresponding upper and lower radial wave functions. The degeneracy between the two states of the spin doublet for the nucleus motions of the X 1 Σ + state of SiF+ molecule has been observed. When the vector potential is equal to the scalar potential, the relativistic effect of the nuclear motion leads to a little decrease in the vibrational energies, while to an increase in those if the vector potential is greater than the scalar potential.

Molecular Spinless Energies of the Modified Rosen-Morse Potential Energy Model

We solve the Klein-Gordon equation with the modified Rosen-Morse potential energy model. The bound state energy equation has been obtained by using the supersymmetric shape invariance approach. The relativistic vibrational transition frequencies for the 6 1 Πu state of the 7 Li2 molecule have been computed by using the modified Rosen-Morse potential model. The calculated relativistic vibrational transition frequencies are in good agreement with the experimental RKR values.

Solutions of the Klein-Gordon equation with the improved Manning-Rosen potential energy model in D dimensions

We solve the Klein-Gordon equation with the improved Manning-Rosen potential energy model in D spatial dimensions. The relativistic bound state energy equation and the unnormalized radial wave functions have been obtained. For fixed vibrational quantum number and various rotational quantum numbers, the relativistic energies for the a 3 Σ + states of the 7Li2 molecule increase as D increases. We observe that the change behavior of the relativistic vibrational energies in higher dimensions remains similar to that of the three-dimensional system.

Laplace Transformation Approach to the Spin Symmetry of the Mie-Type Potential with a Coulomb Tensor Interaction

The Dirac equation is solved exactly under the condition of spin symmetry for a spin 1=2 particle in the field of Mie-type potential and a Coulomb-like tensor interaction via the Laplace transform approach (LTA). The Dirac bound state energy equation and the corresponding normalized wave functions are obtained in closed forms with any spin-orbit coupling term k. The effects of the tensor interaction and the potential parameters on the bound states are also studied. It is noticed that the tensor interaction removes degeneracy between two states in spin doublets. Some numerical results are given and a few special cases of interest are presented. We have demonstrated that in the nonrelativistic limit, the solutions of the Dirac system converges to that of the Schrödinger system. The nonrelativistic limits of the present solutions are compared with the ones obtained by findings of other methods. Our results are sufficiently accurate for practical purpose.