Teller Potential Research Articles

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.

Removal of Lindane from Aqueous Solution Using Aluminum Hydroxide Nanoparticles with Surface Modification by Anionic Surfactant.

In the present study, we investigated the removal of an emerging pesticide lindane from aqueous solution using synthesized aluminum hydroxide Al(OH)3 (bayerite) nanomaterials with surface modification by an anionic surfactant sodium dodecyl sulfate (SDS). The Al(OH)3 nanoparticles were characterized by X-ray diffraction (XRD), Fourier transform infrared spectroscopy (FT-IR), scanning electron microscopy (SEM), Brunauer–Emmett–Teller (BET) and zeta potential. The lindane removal using SDS-modified nano-aluminum hydroxide nanoparticles (SMNAH) achieved removal of up to 93.68%, which was 3.3 times higher than that of nano-aluminum hydroxide nanoparticles. The adsorptive removal conditions were studied and found to have an adsorption time of 60 min, a pH of 6, an adsorbent dosage of 25 mg/mL and an ionic strength of 10 mM NaCl. After reusing four times, the removal efficiency of lindane using SMNAH still reached 75%. Two-step adsorption can fit adsorption isotherms of lindane onto SMNAH at two salt concentrations. On the basis of the change in zeta potential, surface functional groups and adsorption isotherms, we suggest that the formation of a bilayer micelle induced the removal of lindane.

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.

Exact and approximate energy sums in potential wells

Sums of the N lowest energy levels for quantum particles bound by potentials are calculated, emphasising the semiclassical regime N ≫ 1. Euler-Maclaurin summation, together with a regularisation, gives a formula for these energy sums, involving only the levels N + 1, N + 2…. For the harmonic oscillator and the particle in a box, the formula is exact. For wells where the levels are known approximately (e.g. as a WKB series), with the higher levels being more accurate, the formula improves accuracy by avoiding the lower levels. For a linear potential, the formula gives the first Airy zero with an error of order 10−7. For the Pöschl–Teller potential, regularisation is not immediately applicable but the energy sum can be calculated exactly; its semiclassical approximation depends on how N and the well depth are linked. In more dimensions, the Euler–Maclaurin technique is applied to give an analytical formula for the energy sum for a free particle on a torus, using levels determined by the smoothed spectral staircase plus some oscillatory corrections from short periodic orbits.

A renormalized HVT approach for a class of central potential wells

An investigation is carried out to consider a renormalized HVT approach in the context of s-power series expansions for the energy eigenvalues of a particle moving non-relativistically in a central potential well belonging to the class V(r)=−Df(rR), D>0 where f is an appropriate even function of x=r/R and the dimensionless quantity s = (h^2/2μDR)^{1/2} is assumed to be sufficiently small. Previously, the more general class of central potentials of even power series in r is considered and the renormalized recurrence relations from which the expansions of the energy eigenvalues follow, are derived. The s-power series of the renormalized expansion are then given for the initial class of potentials up to third order in s (included) for each energy-level Enl . It is shown that the renormalization parameter Κ enters the coefficients of the renormalized expansion through the state-dependent quantity a_{nl}χ^{1/2} =a_{nl}(1+K ((−d_1D)R^2))^{½}, a_{nl}=(2n+l+32). The question of determining χ is discussed. Our first numerical results are also given and the utility of potentials of the class considered (to which belong the well-known Gaussian and reduced Poschl- Teller potentials) in the study of single–particle states of a Λ in hypernuclei is pointed out.

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.

Bohr Hamiltonian with trigonometric Pöschl-Teller potential in $ \gamma$-unstable and $ \gamma$-stable pictures

In this paper, we present an analytical solution for the Bohr Hamiltonian with the trigonometric P\"oschl Teller (P.T) potential in the cases of {\gamma} unstable nuclei and {\gamma} stable axially symmetric prolate deformed ones with {\gamma} = 0. The energy spectra and corresponding wave functions are derived by means of the asymptotic iteration method. In addition, B(E2) transition rates are calculated and compared with experimental data. Overall good agreement is obtained for inter and intra band transitions within ground state and \b{eta} bands. Our numerical results, particularly for transition rates are much closer to experimental ones in comparison with those obtained by Davidson and Kratzer potentials which are widely used in the literature.

Applications of the worldline Monte Carlo formalism in quantum mechanics

In recent years efficient algorithms have been developed for the numerical computation of relativistic single-particle path integrals in quantum field theory. Here, we adapt this “worldline Monte Carlo” approach to the standard problem of the numerical approximation of the non-relativistic path integral, resulting in a formalism whose characteristic feature is the fast, non-recursive generation of an ensemble of trajectories that is independent of the potential, and is thus universally applicable. The numerical implementation discretises the trajectories with respect to their time parametrisation but maintains a continuous spatial domain. In the case of singular potentials, the discretised action gets adapted to the singularity through a “smoothing” procedure. We show for a variety of examples (the harmonic oscillator in various dimensions, the modified Pöschl–Teller potential, delta-function potentials, the Coulomb and Yukawa potentials) that the method allows one to obtain fast and reliable estimates for the Euclidean propagator and use them in a certain time window suitable for extracting the ground state energy. Usually Monte Carlo calculations in quantum mechanics are performed using algorithms that involve a more specific adaptation to the potential, and we briefly compare our method with the most widely used such algorithms, which are of Metropolis type. As an aside, we apply it for studying the classical limit where nearly classical trajectories are expected to dominate in the path integral. We expect the advances made here to be useful also in the relativistic case.

One- and two-photon-induced cyclotron–phonon resonance in modified-Pöschl–Teller quantum well

We study the two-photon-induced cyclotron–phonon resonance (CPR) effect resulting from the interaction between electrons and phonons in a quantum well with the modified-Poschl–Teller (MPT) potential. The CPR is considered via the magneto-optical absorption coefficient (MOAC) and the full-width at half-maximum (FWHM). Numerical results are presented for a typical GaAs MPT quantum well. It is found that the quantum well parameters, such as well-depth and well-width, the magnetic field, and the temperature affect significantly the energy separation, the MOAC, and the FWHM. Obtained results agree well with previous theoretical and experimental results. Especially, we found two new rules of the well-depth and well-width-dependent FWHM, which should be tested by experimental works to examine their validities.

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.

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.

Dielectric Function of a Quantum-Confined Thin Film with a Modified Pöschel–Teller Potential

The spatial and time dispersions of the dielectric permittivity of an electron gas in quasi-two-dimensional quantum nanostructures are studied. The screening of the charge-carrier scattering potential in a quantum-confined film with a modified P¨oschel–Teller potential is considered for the first time. Analytical expressions for the dielectric permittivity are obtained.

Exact results for the infinite supersymmetric extensions of the infinite square well

One-dimensional potentials defined by (for integer S) arise in the repeated supersymmetrization of the infinite square well, here defined over the region (0, a). We review the derivation of this hierarchy of potentials and then use the methods of supersymmetric quantum mechanics, as well as more familiar textbook techniques, to derive compact closed-form expressions for the normalised solutions, , for all V(S)(x) in terms of well-known special functions in a pedagogically accessible manner. We also note how the solutions can be obtained as a special case of a family of shape-invariant potentials, the trigonometric Pöschl–Teller potentials, which can be used to confirm our results. We then suggest additional avenues for research questions related to, and pedagogical applications of, these solutions, including the behaviour of the corresponding momentum-space wave functions for large and general questions about the supersymmetric hierarchies of potentials which include an infinite barrier.

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.

A new class of Pöschl–Teller potentials with inverse square singularity and their spectra using the asymptotic iteration method

The aim of this work is to introduce a new family of potentials with inverse square singularity which we called the Pöschl–Teller family of potentials. We enforced the matrix representation of the wave operator to be symmetric and (2k[Formula: see text]+[Formula: see text]1) band-diagonal with respect to a square integrable basis set. This, in principle, is only satisfied for specific potential functions within the used basis set. The basis functions we used here are written in terms of Jacobi polynomials, which is the same basis used in the Tridiagonal Representation Approach (TRA). This yield a more general form of Pöschl–Teller potential that can have many terms which could be beneficial for modeling different physical systems where this potential applies. As an illustration, we have studied a specific new five-parameter potential that belongs to this new family and calculated the bound states for both s-wave and l-wave cases using the Asymptotic Iteration Method (AIM). Along the way, we have introduced new approximation schemes to deal with the l-wave centrifugal potential within the AIM at different approximation orders.

Shape invariance and equivalence relations for pseudo-Wronskians of Laguerre and Jacobi polynomials

In a previous paper we derived equivalence relations for pseudo-Wronskian determinants of Hermite polynomials. In this paper we obtain the analogous result for Laguerre and Jacobi polynomials. The equivalence formulas are richer in this case since rational Darboux transformations can be defined for four families of seed functions, as opposed to only two families in the Hermite case. The pseudo-Wronskian determinants of Laguerre and Jacobi type will thus depend on two Maya diagrams, while Hermite pseudo-Wronskians depend on just one Maya diagram. We show that these equivalence relations can be interpreted as the general transcription of shape invariance and specific discrete symmetries acting on the parameters of the isotonic oscillator and Darboux–Pöschl–Teller potential.

The asymmetric quantum Rabi model and generalised Pöschl–Teller potentials

Starting with the Gaudin-like Bethe ansatz equations associated with the quasi-exactly solved (QES) exceptional points of the asymmetric quantum Rabi model (AQRM) a spectral equivalence is established with QES hyperbolic Schrödinger potentials on the line. This leads to particular QES Pöschl–Teller potentials. The complete spectral equivalence is then established between the AQRM and generalised Pöschl–Teller potentials. This result extends a previous mapping between the symmetric quantum Rabi model and a QES Pöschl–Teller potential. The complete spectral equivalence between the two systems suggests that the physics of the generalised Pöschl–Teller potentials may also be explored in experimental realisations of the quantum Rabi model.

Qubit-nonlinear-oscillator systems: from the moderate-coupling limit to the ultrastrong-coupling regime

This work aims to provide an alternative approach to modeling a two-state system (qubit) coupled to a nonlinear oscillator. Within a single algebraic scheme based upon the f-deformed oscillator description, hard and soft nonlinearities are proposed to be simulated by making use of fitting algebraic models extracted from the trigonometric and modified Pöschl–Teller potentials, respectively. In the regime where the strength of the coupling is considered to be moderate, this approach allows for an analytic, albeit approximate, diagonalization process of the proposed Hamiltonian through using the Van Vleck perturbation theory and embracing the two types of nonlinear features. In the ultrastrong-coupling regime, the effect of such nonlinearities upon the squeezing and phase space properties of the ground state of the composite system is also explored by numerical means.

WKB approximation for a deformed Schrodinger-like equation and its applications to quasinormal modes of black holes and quantum cosmology

In this paper, we use the WKB approximation method to approximately solve a deformed Schrodinger-like differential equation: [−ħ2∂ξ2g2(−iħα∂ξ)−p2(ξ)]ψ(ξ)=0, which are frequently dealt with in various effective models of quantum gravity, where the parameter α characterizes effects of quantum gravity. For an arbitrary function g(x) satisfying several properties proposed in the paper, we find the WKB solutions, the WKB connection formulas through a turning point, the deformed Bohr–Sommerfeld quantization rule, and the deformed tunneling rate formula through a potential barrier. Several examples of applying the WKB approximation to the deformed quantum mechanics are investigated. In particular, we calculate the bound states of the Pöschl–Teller potential and estimate the effects of quantum gravity on the quasinormal modes of a Schwarzschild black hole. Moreover, the area quantum of the black hole is considered via Bohr's correspondence principle. Finally, the WKB solutions of the deformed Wheeler–DeWitt equation for a closed Friedmann universe with a scalar field are obtained, and the effects of quantum gravity on the probability of sufficient inflation is discussed in the context of the tunneling proposal.