Poschl-Teller Research Articles

Collective excitation of quantum droplet with different ranges of the interaction of Pöschl-Teller potential

In this article, we studied quantum droplet with the Pöschl-Teller (PT) interaction potential between the Bose atoms. The Gross–Pitaevskii (GP) equation governs the system. The range and strength of the PT interaction can be adjusted. First, we studied the quantum droplet’s density variation for various PT interaction parameters by the imaginary-time split-step Crank-Nicolson (CN) method. We then used the Bogoliubov theory to examine the collective excitation spectra. We observed that sharp roton forms and phonon modes are missing during long-range interactions. There is a gap at the zero momentum zone due to the long-range PT interaction, which increases with the range and strength of the interaction.

On the solutions of the Schrödinger equation with 2nd Pöschl–Teller potentials

We delve into the intricacies of the hyperbolic Pöschl–Teller potential, often referred to as the 2nd Pöschl–Teller potential (PTP), and explore its implications on the Schrödinger equation for arbitrary l state case. This exploration encompasses a comprehensive review of various methodologies employed in studying the exact solutions of the S wave scenario, along with approximations concerning the centrifugal term within the radial equation for the arbitrary l setup. Our investigation yields exact solutions for the quasi-exact arbitrary l state model. We achieve this by employing a novel approach rooted in Fuchsian differential equations, thereby presenting a potential solution for the S wave case as well. This innovative method holds promise, particularly in seeking solutions for the Schrödinger equation involving non-trivial potentials where exact solutions remain elusive.

Uniform Asymptotic Approximation Method with Pöschl–Teller Potential

In this paper, we study analytical approximate solutions for second-order homogeneous differential equations with the existence of only two turning points (but without poles) by using the uniform asymptotic approximation (UAA) method. To be more concrete, we consider the Pöschl–Teller (PT) potential, for which analytical solutions are known. Depending on the values of the parameters involved in the PT potential, we find that the upper bounds of the errors of the approximate solutions in general are ≲0.15∼10% for the first-order approximation of the UAA method. The approximations can be easily extended to high orders, for which the errors are expected to be much smaller. Such obtained analytical solutions can be used to study cosmological perturbations in the framework of quantum cosmology as well as quasi-normal modes of black holes.

Quantum Otto heat engine with Pöschl–Teller potential in contact with coherent thermal bath

Work and efficiency of quantum Otto heat engines (QOHEs) can increase by using non-thermal baths or by inhomogeneous scaling of energy levels of the working substance. Given these points, at first, we construct the coherent thermal state for a trigonometric Pöschl–Teller (PT) potential. Then using a particle in this potential, which has unequally spaced energy levels, as a working substance, we investigate the work extraction and the efficiency of QOHEs that operates between cold and hot coherent thermal baths. The results show that changing the PT potential parameters in the adiabatic processes of QOHE, which causes an inhomogeneous shift in energy levels or/and make use of the hot coherent thermal bath, improve work extraction and efficiency of QOHE relative to the classical counterpart.

Thermodynamic properties and superstatistics analysis of particle in the non-relativistic system under the influence of trigonometric scarf potential using Dong proper quantization

There has been an intense study on the eigenvalue and eigenfunction of the nonrelativistic quantum system. The Schrodinger equation is important in the nonrelativistic quantum system that is used to analyze the quantum information of the system, so it attached the interest of the researchers to conduct the exploration on it. In most of the studies, the researchers expand the study of the solution of the Schrodinger equation to the physical properties of the system. The thermodynamical property is one of the physical properties and key to analyzing the characteristic of materials. It needs more investigation since it has the potential to form a new design of the material. In this study, the Schrodinger equation with trigonometric Scarf potential was solved using the Supersymmetry WKB quantization condition scheme, where the variable of trigonometric Scarf potential was transformed into the Poschl-Teller potential. The energy spectra were obtained by comparing the constant parameters between Trigonometric Scarf potential and Poschl-Teller potential and by using Dong proper quantization condition of transformed Supersymmetry WKB for trigonometric Scarf potential. The result shows that the energy spectra are influenced by the radial quantum number and the potential width, and the energy increase by the increase of the radial quantum number and the potential width. The thermodynamic properties of the system were obtained approximately using the energy spectra equation and were expressed in the erf function. The superstatistics mechanics was approximately obtained by using modified Delta Dirac function distribution for the Boltzmann factor. These properties are important to analyze the characteristic of the designed solids.

Study of thermomagnetic properties of the diatomic particle using hyperbolic function position dependent mass under the external hyperbolic magnetic and AB force

In this present work, we study the approximate solution of the Schrodinger equation with hyperbolic function position-dependent mass for a symmetrical Modified Poschl-Teller potential. We consider the system influenced by the external hyperbolic magnetic and Aharonov-Bohm (AB) forces. By using the Laplace transform method, the second-order differential equation of the Schrodinger equation is reduced to a first-order differential equation and so the eigenfunction and energy eigenvalues are obtained. The behavior of the bound state energy levels was demonstrated and analyzed using the computational method for various values of the potential parameter, mass parameter, external magnetic force, and AB force. Moreover, the thermomagnetic properties of the system were analyzed for H2, LiH, and HCl diatomic particles, involving the vibrational mean energy, vibrational free energy, vibrational entropy, specific heat capacity, magnetization, magnetic susceptibility, and persistent current.

Riccati generalized resonant states of higher-order nonlinear Schrödinger equation with Pöschl–Teller potential

We present resonant state solutions of the higher-order nonlinear Schrödinger model, with Pöschl–Teller (PT) potential, under certain parametric conditions. The localized solutions are obtained in terms of the hypergeometric functions F(a,b,c;z). The dynamics of these resonant states and their control, for PT potential, is discussed employing isospectral Hamiltonian technique, which is analytically tractable.

Dynamics and stability of matter‐wave solitons in cigar‐shaped Bose–Einstein condensates dragged by Pöschl–Teller potential

AbstractIn this paper, we theoretically investigate the dynamics and stability of matter‐wave solitons (MWS) in a cigar‐shaped Bose–Einstein condensate (BEC) consisting of atoms with attractive interatomic interactions trapped by Pöschl–Teller (PT) potential. In this scenario, we propose the idea of the BEC being dragged by PT potential at a constant speed. The BEC was modeled by the quasi‐one‐dimensional Gross–Piteaevskii Equation (1D‐GPE). In addition, an analytical (time‐dependent variational method) and numerical (split‐step Crank–Nicolson method) approach has been proposed to investigate the dynamic properties of solitons during the temporal evolution of this system. Both analytical and numerical results demonstrated that MWSs dragged by PT potential can be stable. From a general point of view, the results obtained in this paper can motivate both theoretical and experimental investigations on the dynamics and stability of solitons in dilute and ultracold unitary Bose gases trapped by PT potential, nonlinear optical properties in PT quantum wells, and propagation of solitons in nonlinear media.

Time-dependent rationally extended Pöschl–Teller potential and some of its properties

We examine time-dependent Schrodinger equation with oscillating boundary condition. More specifically, we use separation of variable technique to construct time-dependent rationally extended Poschl–Teller potential (whose solutions are given by in terms of $$X_1$$ Jacobi exceptional orthogonal polynomials) and its supersymmetric partner, namely the Poschl–Teller potential. We have obtained exact solutions of the Schrodinger equation with the above-mentioned potentials subjected to some boundary conditions of the oscillating type. A number of physical quantities like the average energy, probability density, expectation values, etc., have also been computed for both the systems and compared with each other.

Energies Spectra and Thermodynamic Properties of Hyperbolic Pöschl–Teller Potential (HPTP) Model

We solved the Dirac equation using modified factorization method with hyperbolic Poschl–Teller potential model. Relativistic and non-relativistic ro-vibrational energy spectra were obtained and numerical solutions for four diatomic molecules $$\left( {{\text{HCl}},\;{\text{H}}_{2} ,\;{\text{CO}}\;{\text{and}}\;{\text{LiH}}} \right)$$ obtained also. The eigenfunction for this potential has been obtained in terms of hypergeometric function. The energy variations with different potential parameters and quantum numbers of the hyperbolic Poschl–Teller potential are also plotted. In addition, we evaluate the vibrational partition function and other thermodynamic properties such as vibrational internal energy, vibrational free energy, vibrational entropy and vibrational specific heat capacity, in terms of temperature and upper bound vibration quantum number. Our results are consistent with those found in the available literature.

Ro-vibrational energies of CO molecule via improved generalized Pöschl–Teller potential and Pekeris-type approximation

The improved form of the generalized Poschl–Teller potential (GPT) has been introduced to discuss diatomic molecules. Schrodinger equation has been solved for the improved generalized Poschl–Teller (IGPT) potential by applying a new Pekeris-type approximation to the centrifugal term, and then the rotational–vibrational energy eigenvalue equation has been obtained. By using this ro-vibrational energy eigenvalue equation, firstly, we have calculated the vibrational energies of the $$X^{1}\Sigma ^{+}$$ state of CO molecule and compared them with the experimental Rydberg–Klein–Rees (RKR) data and theoretical results obtained by using Morse, improved Tietz and improved Poschl–Teller potentials. It has been showed that the IGPT potential is one of the best potential energy function in fitting experimental RKR data for $$\hbox {CO}(X^{1}\Sigma ^{+})$$ molecule. Secondly, we have obtained the maximum possible ro-vibrational energies of $$\hbox {CO}(X^{1}\Sigma ^{+})$$ molecule by using the IGPT potential. Also, it has been proved that the Pekeris-type approximation should be used for the centrifugal term instead of Greene–Aldrich approximation in order to get most probable (or more accurate) ro-vibrational energies for the carbon monoxide molecule.

Schrödinger’s Ants: A Continuous Description of Kirman’s Recruitment Model

We show how the approach to equilibrium in Kirman’s ants model can be fully characterized in terms of the spectrum of a Schrodinger equation with a Poschl-Teller (tan2) potential. Among other interesting properties, we have found that in the bimodal phase where ants visit mostly one food site at a time, the switch time between the two sources only depends on the “spontaneous conversion” rate and not on the recruitment rate. More complicated correlation functions can be computed exactly, and involve higher and higher eigenvalues and eigenfunctions of the Schrodinger operator, which can be expressed in terms of hypergeometric functions.

New superintegrable models on spaces of constant curvature

A general class of superintegrable systems on 2D spaces of constant curvature is known for which the potential is not spherically symmetric but allows separation of variables in (geodesic) polar coordinates. The radial parts of these potentials correspond either to an isotropic harmonic oscillator or a generalized Kepler potential. Unlike the radial parts, the angular ones are given implicitly. In the present paper new two-parameter families of angular potentials are constructed in terms of elementary functions. It is shown that for appropriate choice of parameters a family corresponding to the oscillator or Kepler type radial potential reduces to the Poschl–Teller potential. This allows considering Hamiltonian systems defined by this family as generalizations of Tremblay–Turbiner–Winternitz (TTW) or Post–Winternitz (PW) models, both on the plane and on curved spaces of constant curvature.

Study of Bohr–Mottelson with minimal length effect for Q-deformed modified Eckart potential and Bohr–Mottelson with Q-deformed quantum for three-dimensional harmonic oscillator potential

Bohr–Mottelson Hamiltonian on the γ-rigid regime for Q-deformed modified Eckart and three-dimensional harmonic oscillator potentials in the β-collective shape variable was investigated in the presence of minimal length formalism and Q-deformed of the radial momentum part. By introducing new wave function and using the Q-deformed potential concept in Bohr–Mottelson Hamiltonian in the minimal length formalism, the un-normalized wave function and energy spectra equation were obtained by using the hypergeometric method. Meanwhile, the Bohr–Mottelson Hamiltonian in the presence of the quadratic spatial deformation to the momentum in collective shape variable was investigated using transformation of a new variable such as the Schrodinger-like equation with shape invariant potential. The energy equation and un-normalized wave function were obtained using the hypergeometric method. The results showed that the Bohr–Mottelson equations with different energy potentials and different deformation forms in the radial momentum reduced to the similar Schrodinger-like equation with the modified Poschl–Teller potential.

Phonon-assisted cyclotron resonance in Pöschl-Teller quantum well

The phonon-assisted cyclotron-resonance (PACR) effect due to the one-photon (1P) and two-photon (2P) absorption processes in a quantum well (QW) with the Pöschl-Teller (PT) potential is studied. The magneto-optical absorption coefficient as functions of photon energy for different QW-parameters, magnetic fields, and temperatures is presented. The calculations are done for four different possible cases: both 1P- and 2P-processes as well as both phonon emission and absorption processes. The numerical results for a GaAs, InAs, InSb, and GaSb PT-QW show that the PACR peaks are blueshifted with κ and magnetic field, redshifted with QW-length, but remained unchanged with temperature. Peak intensities are found to increase with κ and magnetic field and decrease with QW-length but have a complex dependency on the temperature. The full-width at half-maximum is found to increase with κ, magnetic field, and temperature but found to decrease with QW-length. The phonon emission process is always dominant in comparison with absorption. The 2P-process is comparable with the 1P-process and should be taken into account when studying the PACR in QWs.

Quasi-exactly solvable extended trigonometric Pöschl-Teller potentials with position-dependent mass

Infinite families of quasi-exactly solvable position-dependent mass Schrodinger equations with known ground and first excited states are constructed in a deformed supersymmetric background. The starting points consist in one- and two-parameter trigonometric Poschl-Teller potentials endowed with a deformed shape invariance property and, therefore, exactly solvable. Some extensions of them are considered with the same position-dependent mass and dealt with by a generating function method. The latter enables to construct the first two superpotentials of a deformed supersymmetric hierarchy, as well as the first two partner potentials and the first two eigenstates of the first potential from some generating function W+(x) (and its accompanying function W-(x) . The generalized trigonometric Poschl-Teller potentials so obtained are thought to have interesting applications in molecular and solid state physics.

On the spin-1/2 Aharonov-Bohm problem for modified Pöschl-Teller potential, physical regularization and self-adjoint extensions

We study the bound-state energies for a spin-1/2 particle subjected to the modified Poschl-Teller potential and an Aharonov-Bohm field in the nonrelativistic limit. The problem of the $ \delta$ -function singularity which describes the interaction between spin and magnetic flux is solved using two different approaches already known in the literature, the first is a physical regularization based on a redefinition of the Aharonov-Bohm field as a limit of another one which is more physical and less singular. The second is the self-adjoint extensions method, we obtain an expression for the self-adjoint extension parameter in terms of the physics of the problem.

Nonclassical and nonlocal properties of atom-field systems in the framework of Pöschl-Teller potentials

Abstract We study the nonclassical and nonlocal properties for atom-field systems in the context of Poschl-Teller potential. Using the coherent states for Poschl-Teller potentials, we propose a Jaynes Cummings model that characterizes the interaction among two-level atoms and fields with this kind of potential. After solving the motion equation, we examine the dynamical change of atomic inversion, entanglement, fidelity, and squeezing phenomenon with respect to initial parameter values. We show how the quantities can be controlled through the proper choice of the potential parameters. Finally, we give the connection among the quantum quantifiers for both whole system and subsystems during the evolution.

On Some Families of New Constructed Polynomials

The purpose of this paper is to introduce some new polynomials obtained from second- and third-order algebraic equations by using a simple iterative method. One-variable polynomials obtained in this study deal with special form of Poschl–Teller potential with constant energy, and the two-variable polynomials are related to time-dependent wave equations. We present some recurrence relations, Binet formula and get various families of linear, multilinear and multilateral generating functions for these polynomials. In addition, we derive some special cases. At the end of the paper we also give an extension to the multidimensional case of our results.

A Simplified Pöschl–Teller Potential: An Instructive Exercise for Introductory Quantum Mechanics

The quantum mechanics of a simplified Poschl–Teller potential in one dimension is analyzed at an elementary level. The number of bound states of this potential is determined by a parameter l characterizing the depth of the potential energy. The resulting collection of potentials with varying depth, as well as their ground state energies, is found by requiring the ground state wave function to be the hyperbolic secant function raised to power l. A step-by-step operator-based strategy for the construction of the eigenfunctions corresponding to all other bound states and their eigenenergies is discussed. We also show how all these results can be alternatively obtained by means of the Legendre differential equation, familiar from the rigid-rotor and hydrogen atom problems. The connection between all these problems is discussed while highlighting the use of this model for teaching introductory quantum mechanics.