Rotation and vibration of diatomic molecule oscillator with hyperbolic potential function

The explicit expressions of energy eigenvalues and eigenfunctions of bound states for a three-dimensional diatomic molecule oscillator with a hyperbolic potential function are obtained approximately by means of the hypergeometric series method. Then for a one-dimensional system, the rigorous solutions of bound states are solved with a similar method. The eigenfunctions of a one-dimensional diatomic molecule oscillator, expressed in terms of the Jacobi polynomial, are employed as an orthonormal basis set, and the analytic expressions of matrix elements for position and momentum operators are given in a closed form.

Exact solutions of the Schrodinger equation for a class of hyperbolic potential well

We propose a new scheme to study the exact solutions of a class of hyperbolic potential well.We first apply different forms of function transformation and variable substitution to transform the Schrodinger equation into a confluent Heun differential equation and then construct a Wronskian determinant by finding two linearly dependent solutions for the same eigenstate.And then in terms of the energy spectrum equation which is obtained from the Wronskian determinant,we are able to graphically decide the quantum number with respect to each eigenstate and the total number of bound states for a given potential well.Such a procedure allows us to calculate the eigenvalues for different quantum states via Maple and then substitute them into the wave function to obtain the expected analytical eigenfunction expressed by the confluent Heun function.The linearly dependent relation between two eigenfunctions is also studied.

The nonrelativistic oscillator strength of a hyperbolic-type potential

We consider the D-dimensional SchrSdinger equation under the hyperbolic potential V0（1 -coth（ar））＋ 171 （1 - coth（ar））2. Using a Pekeris-type approximation, the approximate analytical solutions of the problem are obtained via the supersymmetric quantum mechanics. The behaviors of energy eigenvalues versus dimension are discussed for various quantum numbers. Useful expectation values as well as the oscillator strength are obtained.

Generalised tanh-shaped hyperbolic potential: Klein–Gordon equation's bound state solution

The development of potential theory heightens the understanding of fundamental interactions in quantum systems.In this paper,the bound state solution of the modified radial Klein–Gordon equation is presented for generalised tanh-shaped hyperbolic potential from the Nikiforov–Uvarov method.The resulting energy eigenvalues and corresponding radial wave functions are expressed in terms of the Jacobi polynomials for arbitrary l states.It is also demonstrated that energy eigenvalues strongly correlate with potential parameters for quantum states.Considering particular cases,the generalised tanh-shaped hyperbolic potential and its derived energy eigenvalues exhibit good agreement with the reported findings.Furthermore,the rovibrational energies are calculated for three representative diatomic molecules,namely H2,HCl and O2.The lowest excitation energies are in perfect agreement with experimental results.Overall,the potential model is displayed to be a viable candidate for concurrently prescribing numerous quantum systems.

Exact Polynomial Solutions of Schrdinger Equation with Various Hyperbolic Potentials

The Schrodinger equation with hyperbolic potential V （ x ）=- Vosinh 2q （ x / d） / cosh 6 （ x / d） （q= 0, 1, 2, 3） is studied by transforming it into the confluent Heun equation. We obtain genera/symmetric and antisymmetric polynomial solutions of the SchrSdinger equation in a unified form via the Functional Bethe ansatz method. Furthermore, we discuss the characteristic of wavefunction of bound state with varying potential strengths. Particularly, the number of wavefunction＇s nodes decreases with the increase of potentiaJ strengths, and the particle tends to the bottom of the potential well correspondingly.

Shannon information entropies for position-dependent mass Schrdinger problem with a hyperbolic well

The Shannon information entropy for the Schrodinger equation with a nonuniform solitonic mass is evaluated for a hyperbolic-type potential. The number of nodes of the wave functions in the transformed space z are broken when recovered to original space x. The position Sx and momentum S p information entropies for six low-lying states are calculated. We notice that the Sx decreases with the increasing mass barrier width a and becomes negative beyond a particular width a,while the Sp first increases with a and then decreases with it. The negative Sx exists for the probability densities that are highly localized. We find that the probability density ρ（x） for n = 1, 3, 5 are greater than 1 at position x = 0. Some interesting features of the information entropy densities ρs（x） and ρs（p） are demonstrated. The Bialynicki-Birula-Mycielski（BBM）inequality is also tested for these states and found to hold.

Exact solutions of the Duffin-Kemmer-Petiau equation with a hyperbolical potential in (1+3) dimensions

We study S-wave solutions of the Duffin-Kemmer-Petiau （DKP） equation in the presence of a hyperbolical potential in （1＋3）-dimensional space-time for spin-one particles. The exact analytical Nikiforov-Uvarov （NU） method is used in the calculations to obtain the eigenfunctions and the corresponding eigenvalues. Some figures and numerical values are included to give a better insight to the solutions.

Supersymmetry and SWKB Approach to Dirac Equation with Hyperbolic Scarf Potential

By using the supersymmetric quantum mechanics and shape invariance concept, we study the Dirac equation with the hyperbolic Scarf potential and the exact energy spectrum is obtained. Also, we calculate the bound state energy eigenvalues by using the supersymmetric WKB approximation approach so that we get the same results.