Bound State Solutions Research Articles

Bound State Solution Schrödinger Equation for Extended Cornell Potential at Finite Temperature

In this paper, we study the finite temperature-dependent Schrödinger equation by using the Nikiforov-Uvarov method. We consider the sum of the Cornell, inverse quadratic, and harmonic-type potentials as the potential part of the radial Schrödinger equation. Analytical expressions for the energy eigenvalues and the radial wave function are presented. Application of the results for the heavy quarkonia and B c meson masses are in good agreement with the current experimental data except for zero angular momentum quantum numbers. Numerical results for the temperature dependence indicates a different behaviour for different quantum numbers. Temperature-dependent results are in agreement with some QCD sum rule results from the ground states.

The bound-state soliton solutions of a higher-order nonlinear Schrödinger equation for inhomogeneous Heisenberg ferromagnetic system

The inverse scattering of a higher-order nonlinear Schrödinger equation for inhomogeneous Heisenberg ferromagnetic system with zero boundary condition is calculated by an appropriate Riemann–Hilbert (RH) problem. The RH problem of reflection coefficient with multiple high-order poles is obtained. Meanwhile, the calculation formulas of bound-state (BS) solitons and multiple BS solitons corresponding to one high-order pole and multiple high-order poles are also calculated, respectively. Finally, the corresponding soliton solution model is calculated according to the corresponding formula. Simultaneously, we also obtain the BS soliton, the multiple BS soliton and the interaction between multiple BS solitons and multiple BS solitons.

Rogue Waves With Rational Profiles in Unstable Condensate and Its Solitonic Model

In this brief report we study numerically the spontaneous emergence of rogue waves in 1) modulationally unstable plane wave at its long-time statistically stationary state and 2) bound-state multi-soliton solutions representing the solitonic model of this state. Focusing our analysis on the cohort of the largest rogue waves, we find their practically identical dynamical and statistical properties for both systems, that strongly suggests that the main mechanism of rogue wave formation for the modulational instability case is multi-soliton interaction. Additionally, we demonstrate that most of the largest rogue waves are very well approximated–simultaneously in space and in time–by the amplitude-scaled rational breather solution of the second order.

Bound state solutions, Fisher information measures, expectation values, and transmission coefficient of the Varshni potential

The eigensolutions of the Schrödinger equation under the Varshni potential function are studied with two eigensolution techniques such as the Nikiforov–Uvarov and the semi-classical WKB approximation methods. We extended the work to investigate the analytical and numerical Fisher information measure of complexities and also the expectations values using the Hellmann–Feynman theorem. We determined the transmission coefficient using the WKB method. The WKB energy levels and mean values fluctuate with the potential range compared with the NU derived values. Our results for the Fisher information measure obey the uncertainty relation and the Cramer–Rao inequality for position space (). The mean values conform to the ones reported in existing literature.

Discrete symmetry approach to exact bound-state solutions for a regular hexagon Dirac billiard

We consider solving the stationary Dirac equation for a spin-1/2 fermion confined in a two-dimensional quantum billiard with a regular hexagon boundary, using symmetry transformations of the point group C 6v . Closed-form bound-state solutions for this problem are obtained and the non-relativistic limit of our results are clearly discussed. Due to an adequate choice of confining boundary conditions the upper components of the planar Dirac-spinor eigenfunctions are shown to satisfy the corresponding hexagonal Schrödinger billiard, and the Dirac positive energy eigenvalues are proven to reduce directly to their Schrödinger counterparts in the non-relativistic limit. An illustrative application of our group theoretic method to the well-known square billiard problem has been explicitly provided. The success of our approach in solving equilateral-triangle, square and regular hexagon quantum billiards may well imply a possible applicability to other regular polygonal billiards. A quick look on nodal domains of the Schrödinger eigenfunctions for the hexagon billiard is also considered. Moreover, we have determined a number of distinct non-congruent polygonal billiards that have the same eigenvalue spectrum as that of the regular hexagon.

Bound state solution of the Schrodinger equation for the Woods–Saxon potential plus coulomb interaction by Nikiforov–Uvarov and supersymmetric quantum mechanics methods

The generalized Woods–Saxon potential plus repulsive Coulomb interaction is considered in this work. The supersymmetry quantum mechanics method is used to get the energy spectrum of Schrodinger equation and also the Nikiforov–Uvarov approach is employed to solve analytically the Schrodinger equation in the framework of quantum mechanics. The potentials with centrifugal term include both exponential and radial terms, hence, the Pekeris approximation is considered to approximate the radial terms. By using the step-by-step Nikiforov–Uvarov method, the energy eigenvalue and wave function are obtained analytically. After that, the spectrum of energy is obtained by the supersymmetry quantum mechanics method. The energy eigenvalues obtained from each method are the same. Then in special cases, the results are compared with former result and a full agreement is observed. In the [Formula: see text]-state, the standard Woods–Saxon potential has no bound state, but with Coulomb repulsive interaction, it may have bound state for zero angular momentum.

The bound-state solutions of the one-dimensional hydrogen atom

The one-dimensional hydrogen atom is an intriguing quantum mechanics problem that exhibits several properties which have been continually debated. In particular, there has been variance as to whether or not even-parity solutions exist, and specifically whether or not the ground state is an even-parity state with infinite negative energy. We study a “regularized” version of this system, where the potential is a constant in the vicinity of the origin, and we discuss the even- and odd-parity solutions for this regularized one-dimensional hydrogen atom. We show how the even-parity states, with the exception of the ground state, converge to the same functional form and become degenerate for x > 0 with the odd-parity solutions as the cutoff approaches zero. This differs with conclusions derived from analysis of the singular (i.e., without regularization) one-dimensional Coulomb potential, where even-parity solutions are absent from the spectrum.

Attractive inverse-square–type interaction induced by Lorentz symmetry violation by fixed vector background

Based on a model that goes beyond the Standard Model, we study quantum effects of the violation of the Lorentz symmetry at a low-energy regime. From a background of the Lorentz symmetry violation determined by a space-like vector field and electric fields, we obtain an attractive inverse-square–type interaction. In addition, we obtain the bound-states solutions to the Schrödinger equation.

The Relativistic and Nonrelativistic Solutions for the Modified Unequal Mixture of Scalar and Time-Like Vector Cornell Potentials in the Symmetries of Noncommutative Quantum Mechanics

Abstract: In this work, we have obtained analytically the bound state solution for both the relativistic modified Klein-Gordon equation MKG and non-relativistic modified Schrödinger equation for the modified unequal mixture of scalar and time-like vector Cornell (MUSVC) potentials in the relativistic noncommutative three-dimensional real space (RNC: 3D-RS) symmetries. The unequal mixture of scalar and time-like vector Cornell potentials is extended by including new radial terms. Also, MUSVC potentials are proposed as a quark-antiquark interaction potential for studying the masses of heavy and heavy-light mesons in (RNC: 3D-RSP) symmetries. The ordinary Bopp’s shift method and perturbation theory are surveyed to get generalized excited states’ energy as a function of shift energy and the energy of USVC potentials in the relativistic quantum mechanics RQM and NRQM. Furthermore, the obtained preservative solutions of discrete spectrum depended on the parabolic cylinder function, the gamma function, the ordinary discrete atomic quantum numbers, as well as the potential parameters and the two infinitesimal parameters (θ and σ) which are generated with the effect of (space-space) noncommutativity properties. We have also applied our obtained results for bosonic particles, like the charmonium cc ¯ and bottomonium bb ¯ mesons (that have quark and antiquark flavour) and cs ¯ mesons with spin-(0 and 1) and shown that MKG equation under MUSVC potentials becomes similar to the Duffin–Kemmer equation. We have shown that the degeneracy of the initial spectral under USVC potentials in RQM is changed radically and replaced by the newly triplet degeneracy of energy levels under the MUSVC potentials; this gives more precision in measurement and better results compared to the results of ordinary RQM under USVC potentials. Keywords: Klein-Gordon equation, Schrödinger equation, Unequal mixture of scalar and time-like vector Cornell potentials, Noncommutative quantum mechanics, Star product, Bopp’s shift method, Heavy–light mesons. PACS Nos.: 03.65.Ta; 03.65.Ca; 03.65.Ge.

Effects of Cornell-type scalar potential on a generalized KG-oscillator field in Kaluza–Klein theory

We study a generalized KG-oscillator in the five-dimensional cosmic string geometry background with a magnetic field and quantum flux using Kaluza–Klein theory under the effects of a Cornell-type scalar potential, and observe the gravitational analogue of the Aharonov–Bohm effect. We see that the scalar potential allows the formation of bound states solution, and the energy eigenvalue depends on the global parameter characterizing the space–time. We also see that the magnetic field depends on quantum numbers of the relativistic system which shows a quantum effect.

Attractive inverse-square interaction from Lorentz symmetry breaking effects at a low energy scenario

We analyze nonrelativistic quantum effects on a neutral particle due to the presence of an attractive inverse-square potential that stems from the effects of the Lorentz symmetry violation determined by the parity-even sector of the tensor [Formula: see text]. We show that bound states solutions to the Schrödinger equation can be achieved. We go further by considering a repulsive inverse-square potential yielded by Lorentz symmetry breaking effects, which are also determined the parity-even sector of the tensor [Formula: see text]. Then, we analyze the influence of this repulsive inverse-square potential on a neutral particle confined to two cylindrical surfaces and a cylindrical surface.

Effects of Coulomb- and Cornell-types potential on a spin-0 scalar particle under a magnetic field and quantum flux in topological defects space-time

We investigate quantum motion of spin-0 scalar particles in the presence of a uniform magnetic field including quantum flux under the effects of Cornell- and Coulomb-type potentials in cosmic string space-time. We evaluate the energy eigenvalues and eigenfunctions and analyze a relativistic analogue of the Aharonov-Bohm effect. We show that the potentials allow the formation of bound states solution, and the dependence of the magnetic field on quantum numbers of the system gives a quantum effect.

Nature of the Λnn(Jπ=1/2+, I=1) and Λ3H*(Jπ=3/2+, I=0) states

The nature of the $\Lambda nn$ and ${\rm ^3_\Lambda H^*} (J^\pi=3/2^+,~I=0)$ states is investigated within a pionless effective field theory at leading order, constrained by the low energy $\Lambda N$ scattering data and hypernuclear 3- and 4-body data. Bound state solutions are obtained using the stochastic variational method, the continuum region is studied by employing two independent methods - the inverse analytic continuation in the coupling constant method and the complex scaling method. Our calculations yield both the $\Lambda nn$ and ${\rm ^3_\Lambda H^*}$ states unbound. We conclude that the excited state ${\rm ^3_\Lambda H^*}$ is a virtual state and the $\Lambda nn$ pole located close to the three-body threshold in a complex energy plane could convert to a true resonance with Re$(E)>0$ for some considered $\Lambda N$ interactions. Finally, the stability of resonance solutions is discussed and limits of the accuracy of performed calculations are assessed.

Analytical Bound State Solutions of the Klein-Fock-Gordon Equation for the Sum of Hulthén and Yukawa Potential within SUSY Quantum Mechanics

The relativistic wave equations determine the dynamics of quantum fields in the context of quantum field theory. One of the conventional tools for dealing with the relativistic bound state problem is the Klein-Fock-Gordon equation. In this work, using a developed scheme, we present how to surmount the centrifugal part and solve the modified Klein-Fock-Gordon equation for the linear combination of Hulthén and Yukawa potentials. In particular, we show that the relativistic energy eigenvalues and corresponding radial wave functions are obtained from supersymmetric quantum mechanics by applying the shape invariance concept. Here, both scalar potential conditions, which are whether equal and nonequal to vector potential, are considered in the calculation. The energy levels and corresponding normalized eigenfunctions are represented as a recursion relation regarding the Jacobi polynomials for arbitrary l states. Beyond that, a closed form of the normalization constant of the wave functions is found. Furthermore, we state that the energy eigenvalues are quite sensitive with potential parameters for the quantum states. The nonrelativistic and relativistic results obtained within SUSY QM overlap entirely with the results obtained by ordinary quantum mechanics, and it displays that the mathematical implementation of SUSY quantum mechanics is quite perfect.

Statistical analysis and information theory of screened Kratzer–Hellmann potential model

In this research, the radial Schrödinger equation for a newly proposed screened Kratzer–Hellmann potential model was studied via the conventional Nikiforov–Uvarov method. The approximate bound state solution of the Schrödinger equation was obtained using the Greene–Aldrich approximation in addition to the normalized eigenfunction for the new potential model, both analytically and numerically. These results were employed to evaluate the rotational–vibrational partition function and other thermodynamic properties for the screened Kratzer–Hellmann potential. The results obtained have been graphically discussed. Also, the normalized eigenfunction has been used to calculate some information-theoretic measures including Shannon entropy and Fisher information for low-lying states in both position and momentum spaces numerically. The Shannon entropy results obtained agreed with the Bialynicki-Birula and Mycielski inequality, while the Fisher information results obtained agreed with the Stam, Cramér–Rao inequality. Also, alternating increasing and decreasing localization across the screening parameter for both eigenstates was observed.

Analytical bound state solutions of the Dirac equation with the Hulthén plus a class of Yukawa potential including a Coulomb-like tensor interaction

We examine the bound state solutions of the Dirac equation under the spin and pseudospin symmetries for a new suggested combined potential, Hulthén plus a class of Yukawa potential including a Coulomb-like tensor interaction. An improved scheme is employed to deal with the centrifugal (pseudocentrifugal) term. Using the Nikiforov–Uvarov and SUSYQM methods, we analytically develop the relativistic energy eigenvalues and associated Dirac spinor components of wave functions. We find that both methods give entirely the same results. Modification of our results to fit some particular potentials, which are useful in some other systems, is also discussed. We obtain complete agreement with the findings of previous works. The spin and pseudospin bound state energy spectra for various levels are presented in the absence as well as the presence of tensor coupling. Both energy spectrums are sensitive with regard to the quantum numbers $$\kappa $$ and n, as well as the parameter $$\delta $$ . We also notice that the degeneracies between Dirac spin and pseudospin-doublet eigenstate partners are completely removed by the tensor interaction. Finally, we present the parameter space of allowable bound state regions of potential strength $$V_0$$ with constants $$C_{S}$$ and $$C_{PS}$$ for both symmetry limits.

Generalized tanh-shaped hyperbolic potential: bound state solution of Schrödinger equation

The development of potential theory offers compelling coarse-grained descriptions of fundamental interactions in quantum field theory. In this paper, we propose $$V(r)=V_{1}+V_{2}\tanh (\alpha {r})+V_{3}\tanh ^{2}(\alpha {r})$$ generalized tanh-shaped hyperbolic potential, which in itself contains several important physical potentials. Next, we present the bound state solution of the modified radial Schrodinger equation with this potential by using the Nikiforov–Uvarov method. The obtained energy eigenvalues and corresponding radial wave functions are expressed in terms of the Jacobi polynomials for arbitrary l states. It is also shown that the energy eigenvalues are sensitively associated with potential parameters for quantum states. The generalized tanh-shaped hyperbolic potential and its obtained energy eigenvalues are in excellent overlap with the already reported results in some instances. Altogether, the potential model is predicted to be a possible candidate for prescribing multiple quantum systems simultaneously.

Spin and pseudospin solutions to Dirac equation and its thermodynamic properties using hyperbolic Hulthen plus hyperbolic exponential inversely quadratic potential

In this research article, the modified approximation to the centrifugal barrier term is applied to solve an approximate bound state solutions of Dirac equation for spin and pseudospin symmetries with hyperbolic Hulthen plus hyperbolic exponential inversely quadratic potential using parametric Nikiforov–Uvarov method. The energy eigen equation and the unnormalised wave function were presented in closed and compact form. The nonrelativistic energy equation was obtain by applying nonrelativistic limit to the relativistic spin energy eigen equation. Numerical bound state energies were obtained for both the spin symmetry, pseudospin symmetry and the non relativistic energy. The screen parameter in the potential affects the solutions of the spin symmetry and non-relativistic energy in the same manner but in a revised form for the pseudospin symmetry energy equation. In order to ascertain the accuracy of the work, the numerical results obtained was compared to research work of existing literature and the results were found to be in excellent agreement to the existing literature. The partition function and other thermodynamic properties were obtained using the compact form of the nonrelativistic energy equation. The proposed potential model reduces to Hulthen and exponential inversely quadratic potential as special cases. All numerical computations were carried out using Maple 10.0 version and Matlab 9.0 version softwares respectively.

Аналитическое решение уравнения Даффина - Кеммера - Петьо для суммы потенциала Маннинга - Розена и класс Юкавы

This paper presents an analytical bound-state solution to the Duffin - Kemmer - Petiau equation for the new putative combined Manning - Rosen and Yukawa class potentials. Using the developed scheme to approximate and overcome the difficulties arising in the centrifugal part of the potential, the bound-state solution of the modified Duffin - Kemmer - Petiau equation is found. Analytical expressions of energy eigenvalue and the corresponding radial wave functions are obtained for an arbitrary value of the orbital quantum number l . Also, eigenfunctions are expressed in terms of hypergeometric functions. It is shown that energy levels and eigenfunctions are quite sensitive to the choice of radial and orbital quantum numbers.

Thermodynamic Properties and Bound State Solutions of Schrodinger Equation with Inversely Quadratic Yukawa/Attractive Coulomb Potential Plus a Modified Kratzer Potential

In this paper, the energy spectra of the Inversely Quadratic Yukawa/ Attractive Coulomb Potential plus a Modified Kratzer Potential (IQYCFMP) are obtained using the formula method within the framework of non-relativistic quantum mechanics (Schrodinger Wave Equation). The energy spectra obtained was found to correspond with that obtained using WKB (Werner-Kramer-Brillouin). With the energy spectrum obtained the vibrational partition function is calculated in a closed form and is used to obtain an expression for other thermodynamic functions such as vibrational mean energy U, vibrational free energy F, vibrational entropy S, and vibrational specific capacity C for some diatomic molecules. The study reveals that the Inversely Quadratic Yukawa/ Attractive Coulomb Potential plus a Modified Kratzer Potential (IQYCFMP) can be used to predict the thermodynamic properties of diatomic molecules.