Nonrelativistic Wave Equation Research Articles

Bottomonia in quark-antiquark confining potential

This paper comprehensively explores bottomonia mass spectra and their decay properties by solving the non-relativistic Schrodinger wave equation numerically with an approximate quark-antiquark potential form. We also incorporate spin-dependent terms—spin-spin, spin-orbit, and tensor terms to remove mass degeneracy and to obtain excited states (nS, nP, nD, nF, n = 1, 2, 3, 4, 5) mass spectra. Using the Van Royen—Weisskopf formula, we investigate leptonic decay constants, di-leptonic, di-gamma, tri-gamma, di-gluon decay widths and incorporate first-order radiative corrections. We also computed radiative transition widths, which provide better insight into the non-perturbative aspects of QCD. The states ϒ(10750), ϒ(10860), ϒ(11020) are analyzed as pure states 33 D 1, 53 S 1 and 43 D 1, respectively. The present mass spectroscopy and decay properties results align with available experimental values and other theoretical predictions. Our results may provide better insight into upcoming experimental information in the near future.

Topological defects on solutions of the non-relativistic equation for extended double ring-shaped potential

In this study, we focus into the non-relativistic wave equation described by the Schrödinger equation, specifically considering angular-dependent potentials within the context of a topological defect background generated by a cosmic string. Our primary goal is to explore quasi-exactly solvable problems by introducing an extended ring-shaped potential. We utilize the Bethe ansatz method to determine the angular solutions, while the radial solutions are obtained using special functions. Our findings demonstrate that the eigenvalue solutions of quantum particles are intricately influenced by the presence of the topological defect of the cosmic string, resulting in significant modifications compared to those in a flat space background. The existence of the topological defect induces alterations in the energy spectra, disrupting degeneracy. Afterwards, we extend our analysis to study the same problem in the presence of a ring-shaped potential against the background of another topological defect geometry known as a point-like global monopole. Following a similar procedure, we obtain the eigenvalue solutions and analyze the results. Remarkably, we observe that the presence of a global monopole leads to a decrease in the energy levels compared to the flat space results. In both cases, we conduct a thorough numerical analysis to validate our findings.

Non-relativistic energy equations for diatomic molecules constrained in a deformed hyperbolic potential function.

In this paper, the approximate analytical energy equations for the deformed hyperbolic potential have been obtained for arbitrary parameters of the potential. The potential function was transformed to a molecular potential by subjecting it to the Varshni conditions which allows for the determination of the energy levels of diatomic molecules. The molecular vibrational energy spectra for [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] diatomic molecules were obtained and found to match with the results obtained with another analytical approach, potential functions, and experimental data. The noticeable slight differences in the approximate energy spectra obtained in this work and existing literature may be ascribed to the analytical method, computational approach, and the accuracy of the molecular potential functions. The obtained energy equations were used to determine the energy of a particle for arbitrary parameters of the potential function. The obtained energy is bounded and increases with the increase in the quantum numbers. The results conformed to the ones obtained via the path integral approach and numerical solutions obtained via the MATHEMATICA program. The energy spectra equations were obtained via the Nikiforov-Uvarov approach and semi-classical WKB approximation. The Pekeris approximation has been applied to resolve the difficulty in solving the complete energy spectrum of the non-relativistic wave equation for the potential function. The numerical data of the energy spectra was obtained using the MATHEMATICA program.

Information theory and thermodynamic properties of diatomic molecules using molecular potential.

Owing to the devise applications of molecules in industries, the bound state solution of the non-relativistic wave equation with a molecular potential function has been obtained in a closed-form using the Nikiforov-Uvarov method. The solutions of the bound state are then applied to study the information-theoretic measures such as the one-dimensional Shannon and Renyi entropic densities. The expectation values for the position and momentum spaces were obtained to verify the Heisenberg's uncertainty principle. Utilizing the energy spectrum equation, the thermodynamic vibrational partition function is obtained via the Poisson summation. Other thermodynamic function variations with absolute temperature have been obtained numerically for four diatomic molecules (H2, N2, O2, and HF) using Maple 18 software. The Shannon global entropic sum inequality has also been verified. The Renyi sum for constrained index parameters satisfies the global entropic inequality. The thermodynamic properties of the four molecules are similar and conform to works reported in the existing literature. The obtained vibrational energies are in fair agreement with the ones obtained using other forms of potential energy. The result further indicates that the lowest bounds for the Shannon, Renyi, and Heisenberg inequalities are ground states phenomena.

Radial solution of Schrödinger equation with Hulthen-Yukawa-Inverse quadratic potential in a Point-Like defect under AB-flux field

In this paper, we determine the approximate eigenvalue solution of the non-relativistic wave equation in the presence of the Aharonov-Bohm flux field with Hulthen-Yukawa-Inverse Quadratic potential in a topological defect via point-like global monopole (PGM) geometry. We use the Greene-Aldrich improved approximation scheme into the centrifugal and reciprocal terms appear in the radial Schrödinger equation. We then solve this radial equation using the parametric Nikiforov-Uvarov method and analyze the effects on the eigenvalue solution. We see that the energy levels and the radial wave functions get modified by the topological defect of a point-like global monopole and the magnetic flux field that shows an analogue of the Aharonov-Bohm effect for the bound state. Finally, we utilize the eigenvalue solution to some potential models, such as Hulthen potential, Hulthen plus Yukawa potential, and Hulthen plus inverse quadratic potential and discuss the results.

Fisher information of a modified trigonometric inversely quadratic potential

• A new form of potential was proposed • The solution of the Schrödinger equation with the potential was obtained. • Using expectation values, the Fisher information was calculated. • The Cramer-Rao inequality was verified • The results of some special cases of the potential were obtained. The solution of the non-relativistic wave equation for modified trigonometric inversely quadratic potential (MTIAP) is obtained. The energy equation obtained, and the Hamiltonian which comprises the inverse squared term and the potential function were used to obtain the various radial expectation values. These expectation values were used to estimate the Fisher information for position configuration and momentum space. Numerical results were computed for Fisher information-based uncertainty relation with m = 0 and m = 1 . Some useful potentials like the isotropic harmonic oscillator and hydrogen atom were deduced from the modified trigonometric inversely quadratic potential and the Fisher information for the two potentials was also studied.

Energies of some halogen molecules for the improved Tietz potential model

A solution of the radial Schrödinger equation for an improved Tietz potential function was obtained using the concept of the supersymmetric approach. The energy equation obtained for the non-relativistic wave equation was used to generate numerical values for chlorine dimer, iodine dimer and bromine dimer with the aid of their respective spectroscopic constants. The calculated results were compared with RKR values for each of the molecules. The calculated results were found to be in excellent agreement with the RKR values.

A new method for theoretical calculation of atomic hyperfine structure

Schrödinger equation is a nonrelativistic wave equation, which does not have Lorentz invariance. Therefore, this equation has a large theoretical error in the precise calculation of hydrogen-like system. So the commonly used method is Dirac-Hartree-Fock approximation in the calculation of atomic system. However, we have found a new eigen equation, whose eigenvalue of the hydrogen-like system approximates the calculation of quantum electrodynamics. Hence, we propose a new calculation scheme for the atomic hyperfine structure based on the eigen equation and the basic principle of Hartree-Fock variational method, and come to our conclusion through the correlation calculation of excited single states of hydrogen atom, U 91+ ion, helium atom and lithium atom as well as the comparison with NIST, that is, our method is a better improved model of the stationary Schrödinger equation. Meanwhile, we list the correlation algorithms of energy functional, two-electron coupling integral and radial generalized integral in the appendix.

The Spinning Electron

The notion introduced by Ohanian that spin is a wave property is implemented, both in Dirac and in Schrödinger quantum mechanics. We find that half-integer spin is the consequence of azimuthal dependence in two of the four spinor components, relativistically and non-relativistically. In both cases the spinor components are free particle wavepackets; the total wavefunction is an eigenstate of the total angular momentum in the direction of net particle motion. In the non-relativistic case we make use of the Lévy-Leblond result that four coupled non-relativistic wave equations, equivalent to the Pauli-Schrödinger equation, represent particles of half-integer spin, with g-factor 2. An example of an exact Gaussian solution of the non-relativistic equations is illustrated.

Scattering of electrons and positrons from argon and krypton in the GUP framework

We study the scattering of electrons and positrons by argon and krypton atoms considering a generalized uncertainty principle (GUP). To proceed, we first prove the modified non-relativistic wave equation due to the minimal length in the GUP framework and then determine the wave function of incident particles applying a scattering potential including the static and the energy-dependent correlation–polarization contributions. Finally, we compute the transmission and reflection probabilities from potential barrier. It will be shown that some exotic phenomena occur in this framework which cannot be justified by the ordinary quantum mechanics.

Eigensolutions and theoretic quantities under the nonrelativistic wave equation

The radial Schrödinger equation was solved with the combination of three important potentials with [Formula: see text] as deformed parameter via the parametric Nikiforov–Uvarov method and the energy equation as well as the corresponding normalized radial wave function were obtained in close and compact form. The energy equation obtained was used to study eight molecules. The effect of the deformed parameter on energy eigenvalues was also studied numerically. The subset of the combined potential was also studied numerically and the results were found to be in agreement with the previous results. To extend the application of our work, the wave function obtained was used to calculate some theoretic quantities such as the Tsallis entropy, Rényi entropy and information energy. By putting the Tsallis index to 2, we deduced the information energy from Tsallis entropy. Finally, the effect of the deformed parameter and screening parameter, respectively, on the theoretic quantities were also studied.

Study of the radial Schrödinger equation with external magnetic and AB flux fields by the extended Nikiforov–Uvarov method

Applicability of the extended Nikiforov–Uvarov method in order to obtain eigenstate solutions of the radial Schrodinger equation in the presence of external magnetic field and Aharonov–Bohm (AB) flux fields is investigated. This study includes two analytical solutions of two different physical problems: solution of the non-relativistic wave equation for radial scalar power potential and solution for a charged particle with position-dependent mass interacted via Morse plus Coulomb potential under the influence of external magnetic field and AB flux fields. Wave function solutions for each case are achieved in terms of biconfluent Heun polynomials.

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.

Space-time L 2 estimates, regularity and almost global existence for elastic waves

Abstract In this paper, we first establish a kind of weighted space-time L 2 {L^{2}} estimate, which belongs to Keel–Smith–Sogge-type estimates, for perturbed linear elastic wave equations. This estimate refines the corresponding one established by the second author [D. Zha, Space-time L 2 L^{2} estimates for elastic waves and applications, J. Differential Equations 263 2017, 4, 1947–1965] and is proved by combining the methods in the former paper, the first author, Wang and Yokoyama’s paper [K. Hidano, C. Wang and K. Yokoyama, On almost global existence and local well posedness for some 3-D quasi-linear wave equations, Adv. Differential Equations 17 2012, 3–4, 267–306] and some new ingredients. Then, together with some weighted Sobolev inequalities, this estimate is used to show a refined version of almost global existence of classical solutions for nonlinear elastic waves with small initial data. Compared with former almost global existence results for nonlinear elastic waves due to John [F. John, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm. Pure Appl. Math. 41 1988, 5, 615–666] and Klainerman and Sideris [S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D, Comm. Pure Appl. Math. 49 1996, 307–321], the main innovation of our result is that it considerably improves the amount of regularity of initial data, i.e., the Sobolev regularity of initial data is assumed to be the smallest among all the admissible Sobolev spaces of integer order in the standard local existence theory. Finally, in the radially symmetric case, we establish the almost global existence of a low regularity solution for every small initial data in H 3 × H 2 {H^{3}\times H^{2}} .

Introducing a new family of short-range potentials and their numerical solutions using the asymptotic iteration method

The goal of this work is to derive a new class of short-range potentials that could have a wide range of physical applications, specially in molecular physics. The tridiagonal representation approach has been developed beyond its limitations to produce new potentials by requiring the representation of the Schrodinger wave operator to be multidiagonal and symmetric. This produces a family of Hulthen potentials that has a specific structure, as mentioned in the introduction. As an example, we have solved the nonrelativistic wave equation for the new four-parameter short-range screening potential numerically using the asymptotic iteration method, where we tabulated the eigenvalues for both s -wave and arbitrary l -wave cases in tables.

Using the AIM for solving the non-relativistic wave equation for a new class of infinite one-dimensional well with non-flat bottom

The main goal of this work is to solve the non-relativistic wave equation for a new potential configuration that describes the quantum states of a particle that lies within a one-dimensional infinite well of width L using the asymptotic iteration method (AIM). This potential was introduced recently by Alhaidari to be added to the class of exactly solvable potentials in the tridiagonal representation approach (TRA). We have obtained the energy eigenvalues for different choices of the potential parameters. A good match between our results and the ones obtained by the TRA together with new results for the energy spectrum have been presented.

Solution of the nonrelativistic wave equation using the tridiagonal representation approach

We choose a complete set of square integrable functions as a basis for the expansion of the wavefunction in configuration space such that the matrix representation of the nonrelativistic time-independent linear wave operator is tridiagonal and symmetric. Consequently, the matrix wave equation becomes a symmetric three-term recursion relation for the expansion coefficients of the wavefunction. The recursion relation is then solved exactly in terms of orthogonal polynomials in the energy. Some of these polynomials are not found in the mathematics literature. The asymptotics of these polynomials give the phase shift for the continuous energy scattering states and the spectrum for the discrete energy bound states. Depending on the space and boundary conditions, the basis functions are written in terms of either the Laguerre or Jacobi polynomials. The tridiagonal requirement limits the number of potential functions that yield exact solutions of the wave equation. Nonetheless, the class of exactly solvable problems in this approach is larger than the conventional class (see, for example, Table XII in the text). We also give very accurate results for cases where the wave operator matrix is not tridiagonal but its elements could be evaluated either exactly or numerically with high precision.

Study of a 2D charged particle confined by a magnetic and AB flux fields under the radial scalar power potential

We present a more general form of the Schrodinger equation in curved space by introducing the magnetic fields. Further, we solve the non-relativistic wave equation with the radial scalar power potential (RSPP) under the influence of magnetic and Aharonov-Bohm (AB) flux fields by using the curvilinear coordinates system in such space. With this requirement, the energy spectrum and the corresponding wave functions have been calculated by means of the series method. Our analytical results are compared with other results and found to be in a good agreement. Furthermore, the main thermodynamic functions, such as the free energy, the mean energy, the entropy, the specific heat, the persistent currents and the magnetization, have been calculated by using the characteristic function. Some plots of the numerical results of the thermodynamic quantities are shown. Finally, we discuss our results.

A Green's function approach to the nonrelativistic radial wave equation of hydrogen atom

We report an efficient and consistent method for the solution of Schrödinger wave equation using the Green's function technique and its successful application to the nonrelativistic radial wave equation of hydrogen atom. For the radial wave equation, the Green's function is worked out analytically by means of Laplace transform method and the wave function is proposed under the boundary conditions. Computationally the product of potential term, the proposed wave function and the Green's function are integrated iteratively to get the nonrelativistic radial wave function. The resultant wave after each iteration is normalized and plotted against the standard nonrelativistic radial wave function. The solution converges to the standard wave with the increasing number of iterations. Results are verified for the first 15 states of hydrogen atom. The method adopted here can be extended to many‐body problem and hope that it can enhance our knowledge about complex systems.

Spectra generated by a non-central generalized Kratzer potential and explicit expressions for expectation values of rs in N-dimensions

Analytical solutions of the N-dimensional non-relativistic wave equation with the non-central generalized Kratzer potential with arbitrary angular momentum have been investigated within the framework of the asymptotic iteration method. In hyperspherical coordinates, the normalized wave functions for rovibrational states are obtained in terms of generalized Laguerre and Gegenbauer polynomials. We have also derived the recurrence formulas and the radial expectation values of the reduced internuclear distances in N-dimensions. Rovibrational expectation values are given for .