Bound State Energy Eigenvalues Research Articles

Relativistic energies for the q-deformed Scarf potential with Feynman path integrals formulation

Abstract In this paper, the Dirac equation with the q-deformed Scarf potential for spin symmetry is solved for the arbitrary spin-orbit quantum number k, in the presence of Coulomb-like potential tensor. Using the Feynman path integral formalism and the Pekeris approximation of the centrifugal term, we obtain the bound state energy eigenvalues and the associated spinor of the Dirac particle. Furthermore, this
method is used to determine the spectrum of two diatomic molecules Li2 (61 Πu ) and KRb(B −1 Π). The obtained results are compared to the experimental and numerical ones.

Solutions of the Schrödinger equation with velocity-dependent potentials:a study on Kratzer, Mie and Hulthen potentials

The effect of isotropic velocity-dependent potentials on the bound state energy eigenvalues of the Kratzer, Mie and Hulthen potentials is obtained for any quantum states in the presence of constant form factor εr=γρ0. The corresponding energy eigenvalues and eigenfunctions are determined and tabulated for a set of finite quantum numbers n and l in the framework of the well-known Nikiforov-Uvarov method.

Investigation of bound state energy spectra for fermionic particles in the presence of ultra generalized exponential hyperbolic potential model

By considering the ultra generalized exponential hyperbolic potential, which covers many potential types, the solutions of the Dirac equation with spin/pseudo-spin symmetric limits are achieved. In both approaches, the relation giving the bound state energy eigenvalues is obtained in a closed form. By using these relations, the energy values are calculated numerically for both symmetry cases via the software program. In addition, it has been identified how the bound state energy eigenvalues depend on the parameters. Besides, the energy equations for the Schrödinger and Klein–Gordon particles in the limit states are derived.

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.

The calculation of ro-vibrational energies, franck-condon factors and r-centroids for Yttrium/Scandium oxides in D-dimension

The spectroscopic data as ro-vibrational energy values, Franck-Condon Factors and r-centroids are obtained for Yttrium/Scandium oxides with (B 2Σ+, X 2Σ+) different electronic states. Ro-vibrational energy values are calculated by bound state solution of the Schrödinger equation for the general molecular potential by applying Pekeris-type approximation to the centrifugal term in D-dimensional. Using these solutions, the relation giving the bound state energy eigenvalues and the corresponding wave functions expression are derived. The Franck-Condon factors and r-centroids values are determined by using Fraser-Jarmain method for the B 2Σ+ → X 2Σ+ electronic transitions in both molecules. All the results obtained are given numerically in the tables. The obtained values are compared with the previously get data in the literature and it is seen that the results are consistent.

New bound state solutions of diatomic molecules and mass spectrum of heavy quarkonium with improved modified Kratzer potential plus screened Coulomb potential model in the framework of three-dimensional nonrelativistic noncommutative phase-space symmetries

Approximate analytical solutions of the modified Schrodinger equation have been obtained with a combination of two proposed potentials, called the improved modified Kratzer potential plus screened Coulomb (IMKSC) potential, using the approximation scheme to the centrifugal term for any l-states. We applied the generalized Bopp’s shift method and standard perturbation theory. IMKSC is a superposition of modified Kratzer potential plus screened Coulomb (MKSC) potential, and new terms appear as a result of the effects of the noncommutativity properties of space and phase potential on the MKSC potential. The new bound state energy eigenvalues for some diatomic molecules were obtained in terms of the discrete atomic quantum numbers (j,n,s,m), the infinitesimal NC phase-space parameters, the dissociation energy, the equilibrium internuclear separation, the depth of the potential, and the range of the potential. Furthermore, we have shown that the NC-Hamiltonian operator is the sum of the Hamiltonian operator of the MKSC potential, the perturbed spin-orbit, and the modified Zeeman operator. The new mass spectra of heavy quarkonium systems are calculated using the new energy spectrum expression. It appears as a combination of corresponding values in the literature plus an infinitesimal part that describes the deformation phase-space.

The Fractional Schrödinger Equation with the Generalized Woods-Saxon Potential

The bound state energy eigenvalues and the corresponding eigenfunctions of the generalized Woods-Saxon potential reported in [Phys. Rev. C, 72, 027001 (2005)] is extended to the fractional forms using the generalized fractional derivative and the fractional Nikiforov-Uvarov (NU) technique. Analytical solutions of bound states of the Schrodinger equation for the present potential are obtained in the terms of fractional Jacobi polynomials. It is demonstrated that the classical results are a special case of the present results at α=β=1. Therefore, the present results play important role in molecular chemistry and nuclear physics.

Pure vibrational state energies and statistical-mechanical models for the reparameterized scarf oscillator

In this work, the reparameterized Scarf II oscillator was employed to describe the internal vibration of diatomic systems. Analytical equations for bound state pure vibrational energies and canonical partition function were obtained. The equations were used to derive statistical-mechanical models for the prediction of molar entropy, enthalpy, Gibbs free energy and constant pressure (isobaric) heat capacity of gaseous substances. The obtained model equations were used to generate numerical data on bound state energy eigenvalues and, to investigate the thermodynamic properties of the ground states chloroborane (BCl), bromine fluoride (BrF), and bromine chloride (BrCl) molecules. With the aid of the expression for molar entropy of the system, average absolute deviations obtained for the molecules are 0.1878%, 0.1267%, and 0.0586% from experimental data. The isobaric heat capacity model yields average absolute deviation of 2.1608%, 1.8601%, and 1.9805%. The results obtained are in good agreement with available literature data on gaseous molecule. The work could be applicable in the fields of molecular physics, chemical physics, solid-state physics and chemical engineering.

Analytical Solutions of the Molecular Kratzer-Feus potential by means of the Nikiforov-Uvarov Method

The analytical methods for solving Schrödinger equation are essential and effective tools with which we can investigate the spectroscopic properties, the electronic structure, and the energetic properties of the diatomic molecules (DMs). Accordingly, in this work, we used the Nikiforov-Uvarov (NU) method to solve the three-dimensional nonrelativistic Schrödinger equation with the molecular Kratzer-Feus (KF) potential and obtain the exact analytical bound state energy eigenvalues as well as their corresponding normalized eigenfunctions. The effective KF diatomic molecular potential well is investigated and represented graphically for several well-known DMs. The bound state energy levels are tabulated numerically for arbitrary values of the vibrational and rotational quantum numbers. The results obtained in this work are found to be in excellent agreement with the already-existing results in the literature.

Expectation Values and Energy Spectra of the Varshni Potential in Arbitrary Dimensions

Abstract: The Klein-Gordon equation with Varshni potential was solved through the Nikiforov-Uvarov method. The Greene and Aldrich approximation schemes were employed to overcome the centrifugal barrier. The energy eigenvalues were obtained in relativistic and non-relativistic regimes, as well as the corresponding normalized wave functions. Energy spectra and expectation values of the square of inverse of position kinetic energy and the square of the momentum for five selected diatomic molecules: H2, HCl, TiH, I2 and CO, using their separate spectroscopic parameters were computed through Hellmann-Feynman Theorem. Bound-state energy eigenvalues were also computed for Varshni potential and the numerical results agree with the already existing literature. Keywords: Expectation values, Varshni potential, Nikiforov-Uvarov method, Klein-Gordon equation.

Generalized Uncertainty Principle and Delta-Function Potential

In recent studies, the Heisenberg uncertainty principle has been modified into the generalized uncertainty principle (GUP) to explain gravity from a quantum mechanical perspective. Here, we study the GUP corrections to the bound-state energy eigenvalues for a delta-function potential well and a double delta-function potential well using nonrelativistic quantum mechanical tools. Transmission probabilities for scattering states have also been derived and compared with the unmodified cases for both systems.

Eigensolution and Expectation Values of the Hulthen and Generalized Inverse Quadratic Yukawa Potential

Abstract: In this study, the Schrödinger equation was solved with a superposition of the Hulthen potential and generalized inverse quadratic Yukawa potential model using the Nikiforov-Uvarov (NU) method. For completeness, we also calculated the wave function. To validate our results, the numerical bound state energy eigenvalues was computed for various principal n and angular momentum l quantum numbers. With the aid of the Hellmann-Feynman theorem, the expressions for the expectation values of the square of inverse of position, r^(-2), inverse of position, r^(-1), kinetic energy, T ̂ and square of momentum, p ̂ are calculated. By adjusting the potential parameters, special cases of the potential were considered, resulting in Generalized Inverse Quadratic Yukawa potential, Hulthen potential, Coulomb potential, Kratzer potential, Inversely Quadratic Yukawa potential and Coulomb plus inverse square potential, respectively. Their energy eigenvalue expressions and numerical computations agreed with the literature. Keywords: Schrödinger equation, Hulthen potential (HP), Generalized inverse quadratic Yukawa potential (GIQYP), Nikiforov-Uvarov method. PACS: 03.65.−w, 03.65.Ca, 03.65.Ge.

Solutions of the Dirac equation with position-dependent mass for q-deformed Pöschl–Teller and Coulomb-like tensor potentials

We consider the Dirac equation with position-dependent mass for a q-deformed Pöschl–Teller potential plus a Coulomb-like tensor interaction in the limits of spin and pseudospin symmetries. Under the condition of spin symmetry, the bound state energy eigenvalue equation and the two radial components of the Dirac wave function in terms of the Jacobi polynomials are obtained approximately for an arbitrary shifted spin-orbit quantum number λκ = κ + H and for any value of the deformation parameter q ≥ 1. In the case of the pseudospin symmetry limit, there are no bound states. The Dirac equation describes a free physical system and the two components of the wave function are expressed in terms of the spherical Bessel functions.

Comparison study of bound states for diatomic molecules using Kratzer, Morse, and modified Morse potentials

In this paper, the bound-state energy eigenvalues for several diatomic molecules (O2, I2, N2, H2, CO, NO, LiH, HCl) are computed for various quantum numbers using the shifted 1/N expansion method with the Kratzer, Morse and Modified Morse potentials. Numerical results of the energy eigenvalues for the selected diatomic molecules are discussed. Our results for energy eigenvalues agree perfectly with the results obtained in the literature. A comparative study is performed for four diatomic molecules (H2, N2, CO and HCl) in their ground states using the three potentials.

The Klein–Gordon equation with equal scalar and vector Bargmann potentials in [formula omitted] dimensions

Using an improved Greene-Aldrich approximation scheme to handle the centrifugal potential, we solve the D-dimensional Klein–Gordon equation with equal scalar and vector Bargmann potentials with a non-zero angular momentum number. The equation giving the bound-state energy eigenvalues as well as the corresponding normalized radial wave functions are obtained. The particular cases of Hultén and Yukawa potentials are also discussed. A lower-bound for the energy levels is obtained and their behaviour with the parameters of the Bargmann potential is analysed.

Improved q-deformed Scarf II oscillator

In this work, the improved q-deformed Scarf II oscillator (IQSO) is formulated through the Varshni conditions for diatomic molecule potential. Equation for bound state energy eigenvalues of the IQSO is obtained by the SUSYQM method, a Pekeris-like approximation scheme is used to model the centrifugal term of the Schrödinger equation. Expressions for molar entropy and Gibbs free energy are deduced for the IQSO from equations of energy eigenvalues and partition function within the context of Poisson summation formula. The IQSO is used to model experimental Rydberg-Klein-Rees (RKR) data of six diatomic molecules including: KRb (B 1Π), CO+ (X 2Σ+), Cl2 (X 1Σg +), ICl (X 1Σg +), NbO (X 1Σ+), and K2 (X 1Σg +). The average absolute deviation and mean absolute percentage deviation are used as accuracy indicators. Computed data revealed that the IQSO is an excellent model in fitting the RKR data of the diatomic molecules examined in this work. Numerical values of molar entropy and Gibbs free energy calculated for ICl (X 1Σg +) molecule are almost indistinguishable from existing experimental data for gaseous ICl (X 1Σg +) molecules. This suggests that the IQSO is a near perfect model for representing the internal vibrations of the ICl (X 1Σg +) molecule.

Bound state eigenvalues from transmission coefficients

Experts know that bound state energy eigenvalues of a potential well are poles of its transmission amplitude, t(E). The textbooks do well by solving the bound states and scattering states separately, but the connection goes unheeded. Here, we present interesting and instructive illustrations, where we extract bound state eigenvalues from the known transmission coefficients (T(E) = |t(E)|2) for six analytically solvable potential wells.

ENERGY SPECTRUM AND SOME USEFUL EXPECTATION VALUES OF THE TIETZ-HULTHÉN POTENTIAL

In this paper, concept of supersymmetric quantum mechanics has been employed to derive expression for bound state energy eigenvalues of the Tietz-Hulthén potential, the corresponding equation for normalized radial eigenfunctions were deduced by ansatz solution technique. In dealing with the centrifugal term of the effective potential of the Schrödinger equation, a Pekeris-like approximation recipe is considered. By means of the expression for bound state energy eigenvalues and radial eigenfunctions, equations for expectation values of inverse separation-squared and kinetic energy of the Tietz-Hulthén potential were obtained from the Hellmann-Feynman theorem. Numerical values of bound state energy eigenvalues and expectation values of inverse separation-squared and kinetic energy the Tietz-Hulthén potential were computed at arbitrary principal and angular momentum quantum numbers. Results obtained for computed energy eigenvalues of Tietz-Hulthén potential corresponding to Z = 0 and V0 = 0 are in excellent agreement with available literature data for Tietz and Hulthén potentials respectively. Studies have also revealed that increase in parameter Z results in monotonic increase in the mean kinetic energy of the system. The results obtained in this work may find suitable applications in areas of physics such as: atomic physics, chemical physics, nuclear physics and solid state physics

Diatomic Molecules and Mass Spectrum of Heavy Quarkonium System with Kratzer- Screened Coulomb Potential (KSCP) through the Solutions of the Schrödinger Equation

In this work, we obtain solutions of the Schrödinger equation with Kratzer-screened Coulomb potential (KSCP) model using the series expansion method. Explicitly, we compute the bound state energy eigenvalues for selected diatomic molecules of N2, CO, NO, and CH, respectively, for the various vibrational and rotational quantum states and the numerical energy eigenvalues agree with the existing literature. Three special cases were considered. The energy eigenvalues are applied to obtain the mass spectra of heavy quarkonium system such as charmonium and bottomonium. The results agree with the experimental data and other recent theoretical studies.

Arbitrary l-Solutions of the Schrodinger Equation in Arbitrary Dimensions for the Energy Dependent Generalized Inverse Quadratic Yukawa Potential

Within the framework of Nikiforov-Uvarov method, we obtained an approximate solution of the Schrodinger equation for the Energy Dependent Generalized inverse quadratic Yukawa potential model. The bound state energy eigenvalues for were computed for various vibrational and rotational quantum numbers. Special cases were considered when the potential parameters were altered, resulting into Energy Dependent Kratzer and Kratzer potential, Energy Dependent Kratzer fues and Kratzer fues potential, Energy Dependent Inverse quadratic Yukawa and Inverse quadratic Yukawa Potential, Energy Dependent Yukawa (screened Coulomb) and Yukawa (screened Coulomb) potential, and Energy Dependent Coulomb and Coulomb potential, respectively. Their energy eigenvalues expressions and numerical computations agreed with the already existing literatures.