Energy Eigenvalue Equation Research Articles

Analytical solution of the proca equation for the modified posch teller potential

We study the solution of the Proca equation for the modified Posch Teller potential. Proca equation is a relativistic wave equation for a massive spin-1 particle. Proca equation also describes the imaginary mass of photons. Vector fields are used to describe spin-1 mesons. The hypergeometric method was used to solve the Proca equation to obtain the relativistic energy and wave function of the particle. This work is limited to radial part of the spherical coordinate system. Using the improved approximation scheme to deal with the centrifugal term, we solve approximately the Proca equation for the modified Posch Teller potential. The relativistic energy eigenvalue and radial wave function equations are obtained.

Magnetic Hofstadter butterfly and its topologically quantized Hall conductance

The energy spectrum of massless Dirac fermions in graphene under two dimensional periodic magnetic modulation having square lattice symmetry is calculated. We show that the translation symmetry of the problem is similar to that of the Hofstadter or TKNN problem and in the weak field limit the tight binding energy eigenvalue equation is indeed given by Harper Hofstadter hamiltonian. We show that due to its magnetic translational symmetry the Hall conductivity can be identified as a topological invariant and hence quantized. We thus extend the idea of Quantum Hall Effect to magnetically modulated two dimensional electron system. Finally we indicate possible experimental systems where this may be verified.

Bound state solutions of Schrödinger equation with modified Mobius square potential (MMSP) and its thermodynamic properties.

We solved the Schrödinger equation with the modified Mobius square potential model using the modified factorization method. Within the framework of the Greene-Aldrich approximation for the centrifugal term and using a suitable transformation scheme, we obtained the energy eigenvalues equation and the corresponding eigenfunction in terms of the hypergeometric function. Using the resulting eigenvalues equation, we calculated the vibrational partition function and other relevant thermodynamic properties. We also showed that the modified Mobius square potential can be reduced to the Hua potential model using appropriate potential constant values.

Solutions to the Dirac Equation for Manning-Rosen Plus Shifted Deng-Fan Potential and Coulomb-Like Tensor Interaction Using Nikiforov-Uvarov Method

We solve the Dirac equation for the Manning-Rosen plus shifted Deng-Fan potential including a Coulomb-like tensor potential with arbitrary spin–orbit coupling quantum number κ. In the framework of the spin and pseudospin (pspin) symmetry, we obtain the energy eigenvalue equation and the corresponding eigenfunctions in closed form by using the Nikiforov–Uvarov method. Also Special cases of the potential as been considered and their energy eigen values as well as their corresponding eigen functions are obtained for both relativistic and non-relativistic scope.

Information-theoretic measure of the hyperbolical exponential-type potential

The approximate analytical solution of the 3-dimensional radial Schrödinger equation in the framework of the parametric Nikiforov-Uvarov method was obtained with a hyperbolical exponential-type potential. The energy eigenvalue equation and the corresponding wave function have been obtained explicitly. Using the integral method, we calculated Shannon entropy, information energy, Fisher information, and complexity measure. It was deduced that the complexity measure calculated using Shannon entropy with information energy and that calculated using Shannon entropy with Fisher information were similar.

Effect of the Wigner–Dunkl algebra on the Dirac equation and Dirac harmonic oscillator

In this work, we study the Dirac equation and Dirac harmonic oscillator in one-dimensional via the Dunkl algebra. By using Dunkl derivative, we solve the momentum operator and Hamiltonian that include the reflection symmetry. Based on the concept of the Wigner–Dunkl algebra and the functional analysis method, we have obtained the energy eigenvalue equation and the corresponding wave function for Dirac harmonic oscillator and Dirac equation, respectively. It is shown all results in the limit state satisfied what we had expected before.

L-state Solutions of the Relativistic and Non-Relativistic Wave Equations for Modified Hylleraas-Hulthen Potential Using the Nikiforov-Uvarov Quantum Formalism

An exact analytical and approximate solution of the relativistic and non-relativistic wave equations for central potentials has attracted enormous interest in recent years. By using the basic Nikiforov-Uvarov quantum mechanical concepts and formalism, the energy eigenvalue equations and the corresponding wave functions of the Klein–Gordon and Schrodinger equations with the interaction of Modified Hylleraas-Hulthen Potentials (MHHP) were obtained using the conventional Pekeris-type approximation scheme to the orbital centrifugal term. The corresponding unnormalized eigen functions are evaluated in terms of Jacobi polynomials.

Investigation of the Dirac Equation by Using the Conformable Fractional Derivative

In this paper,the Dirac equation is constructed using the conformable fractional derivative so that in its limit for the fractional parameter, the normal version is recovered. Then, the Cornell potential is considered as the interaction of the system. In this case, the wave function and the energy eigenvalue equation are derived with the aim of the bi-confluent Heun functions. use of the conformable fractional derivative is proven to lead to a branching treatment for the energy of the system. Such a treatment is obvious for small values of the fractional parameter, and a united value as the fractional parameter approaches unity.

A survey on the Klein–Gordon equation in the Gödel-type space-times

In this paper, the Gödel-type space-times have been considered. We are interested in the investigation of relativistic free bosons in the Gödel-type space-times. Therefore, the Klein–Gordon equation considering the Gödel-type space-times has been constructed in a general manner. Then to accomplish our goal, two well-known special cases have been considered for the unknown function in the tensor metric of the Gödel-type space-times. The explicit forms of the wave functions for each case as well as the energy eigenvalue equations have been derived in detail.

Screw dislocation and external fields effects on the Kratzer pseudodot

In this research, we investigate the influence of the screw dislocation on the energy levels of nonrelativistic electrons on a two-dimensional solid taking into account the deformed Kratzer potential under an external magnetic and the Aharonov–Bohm (AB) fields. We obtain the bound state energy eigenvalues equation and the corresponding wave functions by using the Frobenius method. We also calculate the direct interband light absorption coefficient and threshold frequency of absorption in a quantum pseudodot. Some of the numerical results are given.

Analytic Spin and Pseudospin Solutions to the Dirac Equation for the Quadratic Exponential-Type Potential Plus Eckart Potential and Yukawa-Like Tensor Interaction

We solve the Dirac equation for the quadratic exponential-type potential plus Eckart potential including a Yukawa-like tensor potential with arbitrary spin–orbit coupling quantum number κ. In the framework of the spin and pseudospin (pspin) symmetry, we obtain the energy eigenvalue equation and the corresponding eigenfunctions in closed form by using the Nikiforov–Uvarov method. Also Special cases of the potential as been considered and their energy eigen values as well as their corresponding eigen functions are obtained for both relativistic and non-relativistic scope.

Excitons and trions in monolayer transition metal dichalcogenides: A comparative study between the multiband model and the quadratic single-band model

The electronic and structural properties of excitons and trions in monolayer transition metal dichalcogenides are investigated using both a multiband and a single-band model. In the multiband model we construct the excitonic Hamiltonian in the product base of the single-particle states at the conduction and valence band edges. We decouple the corresponding energy eigenvalue equation and solve the resulting differential equation self-consistently, using the finite element method (FEM), to determine the energy eigenvalues and the wave functions. As a comparison, we also consider the simple single-band model which is often used in numerical studies. We solve the energy eigenvalue equation using the FEM as well as with the stochastic variational method (SVM) in which a variational wave function is expanded in a basis of a large number of correlated Gaussians. We find good agreement between the results of both methods, as well as with other theoretical works for excitons, and we also compare with available experimental data. For trions the agreement between both methods is not as good due to our neglect of angular correlations when using the FEM. Finally, when comparing the two models, we see that the presence of the valence bands in the mutiband model leads to differences with the single-band model when (interband) interactions are strong.

The relativistic bound states of a non-central potential

We investigate the relativistic effects of a moving particle in the field of a pseudo-harmonic oscillatory ring-shaped potential under the spin and pseudo-spin symmetric Dirac wave equation. We obtain the bound state energy eigenvalue equation and the corresponding two-components spinor wave functions by using the formalism of Suppersymmetric Quantum Mechanics (SUSYQM). Furthermore, the non-relativistic limits are obtained by simply making a proper replacement of parameters. The thermodynamic properties are shortly studied. Our numerical results for the energy eigenvalues are presented too.

Pseudospin symmetric solution of the Dirac–Eckart problem with a Hulthén tensor interaction in the tridiagonal representation

Under the condition of the pseudospin symmetry, the approximate analytical solution of the Dirac–Eckart problem with a Hulthén tensor interaction is investigated by working in a complete square integrable basis that supports a tridiagonal matrix representation of the wave operator. The pseudo-centrifugal term is treated with Greene and Aldrich's approximation scheme. The energy eigenvalue equation is obtained by diagonalization of the recursion relation and the corresponding spinor wave functions are presented in terms of Jacobi polynomials or hypergeometric functions.

Spin (Pseudospin) doublet in view of energy-dependent potential

Behavior of Dirac particles in the presence of scalar, vector, and tensor potentials respectively represented by energy-dependent Morse and Coulomb-like potentials is examined by working out the Dirac equation under the condition of spin (pseudospin) symmetry. The closed form of the energy eigenvalue equation and corresponding wave functions in terms of hypergeometric functions are acquired by making use of the asymptotic iteration method. We investigate the effect of energy-dependent potential on both bound states and energy splitting between members of spin (pseudospin) doublets.

Bound state solutions of D-dimensional Klein–Gordon equation with hyperbolic potential

By using the basic supersymmetric quantum mechanics concepts and formalism, the energy eigenvalue equation and the corresponding wave function of the Klein–Gordon equation with vector and scalar potentials for an arbitrary dimensions are obtained together with hyperbolic potential using a suitable approximation scheme to the orbital centrifugal term. The non-relativistic limit is obtained and the numerical values for various values of D, n, α and ℓ are obtained.

Relativistic Approximate Solutions for a Two-Term Potential: Riemann-Type Equation

Approximate analytical solutions of a two-term potential are studied for the relativistic wave equations, namely, for the Klein-Gordon and Dirac equations. The results are obtained by solving of a Riemann-type equation whose solution can be written in terms of hypergeometric function 2Fl(a,b; c; z). The energy eigenvalue equations and the corresponding normalized wave functions are given both for two wave equations. The results for some special cases including the Manning-Rosen potential, the Hulthen potential and the Coulomb potential are also discussed by setting the parameters as required.

Modelling of diatomic molecules

ABSTRACTA new general molecular potential is introduced to model diatomic molecular structures. Eigenfunctions and rotational-vibrational energy eigenvalue equation of the Schrödinger particle in the presence of centrifugal term are obtained for the general molecular potential. By using this rotational-vibrational energy eigenvalue equation, we calculate vibrational energy eigenvalues of 7Li2(6 1Πu), SiC(X 3Π), , ScI(X 1Σ+), and Na2(5 1Δg) molecules and the results are compared with Rydberg–Klein–Rees (RKR) data. We show that the general molecular potential is very convenient for diatomic molecules in fitting RKR data. Also, we show that the general molecular potential can be reduced to Rosen–Morse and Trigonometric Pöschl–Teller potentials which are used for some vibrations of some polyatomic molecules such as NH3 and SO2.

Scattering and delay time for 1D asymmetric potentials: The step-linear and the step-exponential case

We analyze the quantum-mechanical behavior of a system described by a one-dimensional asymmetric potential constituted by a step plus (i) a linear barrier or (ii) an exponential barrier. We solve the energy eigenvalue equation by means of the integral representation method, classifying the independent solutions as equivalence classes of homotopic paths in the complex plane. We discuss the structure of the bound states as function of the height U 0 of the step and we study the propagation of a sharp-peaked wave packet reflected by the barrier. For both the linear and the exponential barrier we provide an explicit formula for the delay time τ ( E ) as a function of the peak energy E. We display the resonant behavior of τ ( E ) at energies close to U 0. By analyzing the asymptotic behavior for large energies of the eigenfunctions of the continuous spectrum we also show that, as expected, τ ( E ) approaches the classical value for E → ∞, thus diverging for the step-linear case and vanishing for the step-exponential one.

Application of Electrodynamic Theory on Quantum Hall Effect

The quantum electrodynamic (QED) behaviour is studied for quantum Hall effect (QHE). Quantum theory with conjecture of fractional charge quantization (quantum dipole moment), eigenfunctions for fractional charge quantization at the surface of a twisted and twigged electron quanta and above its surface, fractional Fourier transform and Hermite function for fractional charge quantization is developed. With energy eigen value equation for QHE and with energy operator on an eigenfunction of a twisted and twigged electron quanta, the corresponding eigenfunctions are normalized with Schrodinger’s quantum wave mechanical equation for electric scalar and magnetic potentials, respectively (QED behavior). The fractional electric and magnetic fields with their corresponding potentials for the quantized fractional states in semiconducting hereto structures are theoretically calculated. Such mathematical expressions are in good agreement with experimental results of Nobel Prize winning scientists Klitzing, Haroche, Peter and Gruebber. Our results can also explain the hybridized states of orbits with emphasis on sigma and pi bonding and their corresponding antibonding orbitals as a manifestation of electrophilic and nucleophilic chemical reactions.