Exponential-type Potential Research Articles

Shannon entropy and thermodynamic properties of an inversely quadratic Yukawa potential model

An inversely quadratic Yukawa potential is an exponential-type potential that has received little attention only in the bound state to the best of our understanding. This study obtained the solution of the Schrödinger equation for an inversely quadratic Yukawa potential via parametric Nikiforov-Uvarov method and supersymmetric approach. Some thermodynamic properties (vibrational enthalpy, vibrational Gibbs free energy and vibrational entropy) for the Inversely quadratic Yukawa potential are examined. The study is also applied to shannon entropy as the theoretic quantity. The results obtained showed that the energy eigenvalue for the potential goes in the opposite direction with the quantum state and the screening parameter. The Shannon entropy obtained as a function of the potential strength only obeyed the Heinsenberg principle at the ground state and excited states. In both cases, the also satisfied Bialynick-Birula, Mycielski inequality. It is noted that the thermal properties studied as a function of the temperature, even though the studied potential is not a molecular potential, the results followed the trend of the molecular potential.

On novel distributed fixed-time formation tracking of multiple hypersonic flight vehicles with collision avoidance

This paper aims to achieve collision-free formation tracking for multiple hypersonic flight vehicles without introducing conventional approximators such as fuzzy logic systems and neural networks. By relying on the hierarchical control theory, a globally fixed-time distributed formation control protocol with guaranteed tracking behaviors is proposed. Firstly, for the purpose of deriving the virtual control commands in the three-dimensional translational dynamics, a fixed-time consensus control input is presented. Then, by designing a comprehensive performance-preserving feedback control law, the actual control commands are derived, so that the virtual control inputs can be tracked timely and accurately. Meanwhile, with resort to a novel exponential-type artificial potential function, a collision-free control input is proposed to avoid close proximity between adjacent hypersonic flight vehicles. Summarily, the outstanding advantage of the proposed protocol is that it simultaneously realize the collision-free formation tracking, fixed-time stability and performance-preserving control for multiple hypersonic flight vehicles with uncertain dynamics. Lyapunov stability criterion verifies the closed-loop stability, showing that the error signals can attenuate to a tiny region around the origin within a user-specified time. Finally, comparative numerical simulations are performed to further demonstrate the theoretical results.

Diatomic molecular energy spectra and thermodynamic properties of exponential inversely quadratic plus improved deformed exponential-type potential

By employing the Pekeris-type approximation to deal with the centrifugal term, we study the thermodynamic properties and approximate analytical solutions of the Schrödinger equation with the exponential inversely quadratic plus improved deformed exponential-type potential using parametric Nikiforov–Uvarov method and hypergeometric functional analysis method. The complex eigen energy equation was presented in a closed and compact form and extended to study partition function and other thermodynamic properties. The total normalized wave functions were also obtained and expressed in terms of hypergeometric Jacobi polynomial. The mathematical accuracy of analytical calculation can be verified consistently by comparing numerical data and results of different arbitrarily l state. The potential also reduces to improved deformed exponential-type potential as a special case. We calculated energy spectra for some selected diatomic molecules namely: Silicon Flouride (SiF+ (X1∑+), Oxygen molecule O2+ (X2∏g), Nitrogen molecule, N2+ (X1∑g+), Sulphur chloride SCl and Rubidium hydride RbH using standard molecular spectroscopic constants as applied to the two methods. The results show that the energy eigenvalues for these selected diatomic molecules are in excellent agreement using the two methods and also increases with an increase in quantum state.

Triaxial nuclei and analytical solutions of the conformable fractional Bohr Hamiltonian with some exponential-type potentials

New approximate analytical solutions have been obtained for the conformable fractional collective Bohr Hamiltonian suitable for triaxial nuclei, with the harmonic oscillator in γ-part of the collective potential and different exponential-type potentials (namely Morse, Tietz-Hua, and multi-parameter exponential-type potential) in β-part. The conformable fractional Nikiforov-Uvarov approach is used to derive closed analytical formulas for energy spectra and wave functions. The relation between the conformable fractional spectra of the three potentials and the Z(5) spectrum has been studied. The evolution of the energy spectra as a function of the potential parameters has been investigated for the three potentials. The normalized B(E2) transition rates and spectra have been calculated and compared with the experimental data and theoretical predictions of Kratzer potential. The predictions can describe well the experimental results for 114,116Pd, 126,128Xe, and 192,194Pt isotopes.

Potential parameters, ro-vibrational energy spectra, and expectation values of the improved deformed exponential-type potential

In this contribution, potential parameters were derived for the improved deformed exponential-type potential (IDEP) using Varshni conditions for a diatomic molecule potential. Expression for bound state ro-vibrational energies was obtained by solving the Schrödinger equation in the presence of the IDEP. In modeling the centrifugal term of the Schrödinger equation, a Pekeris-like approximation scheme was employed. Expectation values of potential and kinetic energies were obtained for the IDEP within the framework of Hellmann–Feynman theorem. The IDEP was used to model the spectroscopic data of four diatomic molecules, viz., ICl (X 1Σg+), NaK (c 3Σ+), O2+ (X 2Πg), and NaLi (A 1Σ+). The average absolute deviation from the dissociation energy obtained was 0.1235%, 0.9227%, 1.2275%, and 1.2651%, respectively. The results demonstrated that the IDEP was an ideal model for the diatomic molecules. Data obtained for expectation values of kinetic and potential energies showed that the total energy of the system was conserved and that at upper bound vibration quantum number, the energy of the molecules was entirely kinetic.

Application of Morse potential and improved deformed exponential-type potential (IDEP) model to predict thermodynamics properties of diatomic molecules

The thermal properties of three diatomic molecules such as hydrogen (H2), carbon monoxide (CO) and iodine (I2) are theoretically predicted. To this end, two different potential models are considered, Morse potential and improved deformed exponential-type potential (IDEP). The Schrödinger equation is analytically solved to obtain energy levels. For the Morse and IDEP potential models, the Laplace transformation and the Greene–Aldrich approximation and appropriate coordinate transformation are employed, respectively. In the former potential model, we have obtained energy eigenvalues by considering position-dependent effective mass. Using the energy levels of two models, the partition function is first determined. Second, thermal functions of the molecules such as heat capacity, enthalpy and entropy are obtained and compared with the available results. The calculated enthalpy using the two models for CO and I2 molecules is in excellent agreement in comparison with the available data in a wide temperature range. But, the obtained enthalpy of H2 molecule using the Morse potential at high temperatures does not agree excellently with the experimental data. The entropy of H2, CO and I2 using the two models is tin good agreement in comparison with the available data in a wide temperature range whereas the entropy of H2 at low temperatures does not agree excellently with the experimental data. Our theoretical results using the two potential models do not show a suitable prediction about the heat capacity at a constant pressure of H2, CO and I2 molecules, But the IDEP shows more agreement compared to the Morse potential.

Thermal properties of diatomic molecules with multi parameter exponential-type potential

In this work, we have calculated the thermal properties of H2 and O2 diatomic molecules with multi parameter exponential type potential within the framework of the statistical mechanics. In this regard, using the improved energy spectrum, we have determined the vibrational partition function obtained recently via the path integral formalism. Based on obtained partition function, we find thermodynamic properties of diatomic molecules such as Gibbs free energy, enthalpy and specific heat in constant pressure by the Poisson summation formalism. Also, we have validated our results with experimental data and our results show that there is a good agreement between them. Finally, the average absolute deviations of the calculated data from the experimental data are obtained.

On the position-dependent mass Schrödinger equation for Mie-type potentials

The exactly solvable Position Dependent Mass Schrödinger Equation (PDMSE) for Mie-type potentials is presented. To that, by means of a point canonical transformation the exactly solvable constant mass Schrödinger equation is transformed into a PDMSE. The mapping between both Schrödinger equations lets obtain the energy spectra and wave functions for the potential under study. This happens for any selection of the O von Roos ambiguity parameters involved in the kinetic energy operator. The exactly solvable multiparameter exponential-type potential for the constant mass Schrödinger equation constitutes the reference problem allowing to solve the PDMSE for Mie potentials and mass functions of the form given by m(x) = skx s-1/(xs + 1))2. Thereby, as a useful application of our proposal, the particular Lennard-Jones potential is presented as an example of Mie potential by considering the mass distribution m(x) = 6kx 5/(x 6 + 1))2. The proposed method is general and can be straightforwardly applied to the solution of the PDMSE for other potential models and/or with different position-dependent mass distributions.

Molecular energies of a modified and deformed exponential-type potential model

The solutions of a one-dimensional relativistic Klein-Gordon equation for a modified and deformed exponential-type potential are obtained using supersymmetric approach and parametric Nikiforov-Uvarov method respectively. The numerical values for lithium molecule were obtained using the energy equations from the two methods. The non-relativistic limit of the relativistic Klein-Gordn equation was obtained for each of the energy equation using a popular transformation invoked on the solution of the Klein-Gordon equation. Numerical values for lithium and sodium molecules were also generated based on the two methods. The generated values for both relativistic and non-relativistic were compared with the experimental values. These results agreed perfectly with the experimental values of the studied molecules.

Effect of the deformation parameter on the nonrelativistic energy spectra of the q-deformed Hulthen-quadratic exponential-type potential

In this study, an approximate solution of the Schr�dinger equation for the q-deformed Hulthen-quadratic exponential-type potential model within the framework of the Nikiforov�Uvarov method was obtained. The bound state energy equation and the corresponding eigenfunction was obtained. The energy spectrum is applied to study H2, HCl, CO and LiH diatomic molecules. The effect of the deformation parameters and other potential parameters on the energy spectra of the system were graphically and numerically analyzed in detail. Special cases were considered when the potential parameters were altered, resulting in deformed Hulthen potential, Hulthen potential, deformed quadratic exponential-type potential and quadratic exponential-type potential. The energy eigenvalues expressions agreed with what obtained in literature. Finally, the results can find many applications in quantum chemistry, atomic and molecular physics.

Neutron-Proton Scattering Phase Shifts in S-Channel using Phase Function Method for Various Two Term Potentials

The scattering phase shifts for n-p scattering have been modeled using various two term exponential type potentials such as Malfliet-Tjon, Manning-Rosen and Morse to study the phase shifts in the S-channels. As a first step, the model arameters for each of the potentials are determined by obtaining binding energy of the deuteron using matrix methods vis-a-vis Variational Monte-Carlo (VMC) technique to minimize the percentage error w.r.t. the experimental value. Then, the first order ODE as given by phase function method (PFM), is numerically solved using 5th order Runge-Kutta (RK-5) technique, by substituting the obtained potentials for calculating phase shifts for the bound 3S1 channel. Finally, the potential parameters are varied in least squares sense using VMC technique to obtain the scattering phase-shifts for each of the potentials in the 1S0 channel. The numerically obtained values are seen to be matching with those obtained using other analytical techniques and a comparative analysis with the experimental values up to 300 MeV is presented.

Eigensolutions and thermodynamic properties of deformed and modified Morse potential model

A four parameter exponential-type potential known as the deformed and modified Morse potential model was proposed. The solution of the one-dimensional radial Schrӧdinger equation was obtained in the presence of the proposed potential for an arbitrary ℓ-state using a parametric Nikiforov–Uvarov method. The energy equation was used to obtain the eigenvalues for the cesium molecule and tin carbide for various states. The effects of the deformed parameter on the eigenvalues were studied. The result for the Morse potential was obtained by fixing the deformed parameter to zero. Finally, the thermal properties of the proposed potential and the Morse potential were studied in detail. The result obtained revealed that the behavior of the energy eigenvalue for the deformed and modified Morse potential with the deformed parameter (b) equals one, is same as the behavior of the energy eigenvalue for the actual Morse potential.

Enthalpy, Gibbs free energy and specific heat in constant pressure for diatomic molecules using improved deformed exponential-type potential (IDEP)

In this work, we have studied such thermal function of diatomic molecules like hydrogen dimer, carbon monoxide, nitrogen dimer and lithium hydride using improved deformed exponential-type potential (IDEP). To this end, the energy spectra of the IDEP are obtained applying Greene-Aldrich approximation and appropriate coordinate transformation within the framework of non-relativistic quantum mechanics. With calculated energy eigenstates, we have deduced the partition function and such thermodynamic functions like specific heat in constant pressure, enthalpy and Gibbs free energy by employing the Poisson summation formula. We have compared our results with experimental data, and there is a good agreement between them.

Specific heat at constant pressure, enthalpy and Gibbs free energy of boron nitride (BN) using [formula omitted]-deformed exponential-type potential

In this paper, we have obtained the thermodynamic properties of boron nitride (BN) using q-deformed exponential-type potential within the framework of the statistical mechanics. To this end, the energy spectra of the potential model are calculated using benefit approximation and appropriate coordinate transformation within the non-relativistic quantum mechanics framework. The partition function of BN has been obtained analytically and was used to find its thermodynamic functions such as specific heat at constant pressure, enthalpy and Gibbs free energy by the Poisson summation formula. The average absolute deviations of the predicted results from the experimental data are 3.9716%, 2.3567% and 8.5868% for specific heat at constant pressure, Gibbs free energy and enthalpy, respectively.

Quintessential inflation for exponential type potentials: scaling and tracker behavior

We will show that for exponential type potentials of the form V(varphi )sim e^{-gamma varphi ^n/M_{pl}^n}, which are used to depict quintessential inflation, the solutions whose initial conditions take place during the slow roll phase in order to describe correctly the inflationary period, do not belong for large values of the parameter n to the basin of attraction of the scaling solution – a solution of the scalar field equation whose energy density scale as the one of the fluid component of the universe during radiation or the matter domination period –, meaning that a late time mechanism to exit this behavior and depict correctly the current cosmic acceleration is not needed. However, in such cases, namely n large enough, these potentials cannot correctly depict the current cosmic acceleration. This is the reason why the potential must be improved introducing another parameter -as the one in the well-known Peebles–Vilenkin quintessential inflation model, which depends on two parameters, one to describe inflation and the other one to correctly depict the present accelerated evolution – able to deal with the late time acceleration of our universe.

Solution of the Dirac equation with exponential-type potential under the GUP

In quantum gravity theories, when the scattering energy is comparable to the Planck energy, the usual Heisenberg uncertainty principle breaks down and is replaced by generalized uncertainty principle (GUP). In this paper, the Dirac equation is studied for a single particle with spin and pseudospin symmetry in the presence of GUP, in [Formula: see text] dimensions. For arbitrary wave [Formula: see text], the Dirac equation with multiparameter exponential-type potential is solved by applying the approximation of the centrifugal term and the Nikiforov–Uvarov method. The corresponding energy spectra and eigenvalue function are obtained in the closed form and depend on the GUP parameter. In addition, several interesting cases have been discussed.

On the improvement of q-deformed hyperbolic functions

In the search of quantum potential models used in the theoretical treatment of diatomic molecules, some of them have been constructed by using standard hyperbolic functions as well as from the so-called q-deformed hyperbolic functions (sc q-dhf). To improve the scope of hyperbolic functions, in this work, a class of generalized q-deformed hyperbolic functions is presented. Specifically, such improvement contains the particular case sc q-dhf besides that the can be expressed in terms of standard hyperbolic functions. As a useful application of the proposal, new q-deformed exactly solvable exponential-type potentials to be used in quantum chemical calculations are given as examples. At this regard, we are considering multi-parameter exponential-type potentials which means that the single choice of each set of q-dependent parameters leads to a different particular case of q-deformed exponential potential totally characterized. That is, with the proposed approach it is not necessary to make use of specialized methods to solve the involved differential equation for each specific case of exponential q-deformed potentials.

The unified treatment of the non-relativistic bound state solutions, thermodynamic properties, and expectation values of exponential-type potentials

In this paper, we obtained the analytical N-dimensional bound state solutions of exponential-type potential functions using the semi-classical Wentzel–Kramers–Brillouin (WKB) approximation method with an improved approximation to the orbital centrifugal barrier. Furthermore, we derived the partition function with the associated thermodynamic properties. We calculated the expectation values (〈1/r2〉) via the Hellmann–Feynman theorem. Our results for the energy levels are in excellent agreement with the best available results obtained by other methods in the existing literature. The variations of the thermodynamic functions with the temperature parameter (β) agree very well with the other results obtained under different potential energy functions. The expectation values conformed to the ones obtained by another approach in previous work. This work demonstrates the exactness of the leading-order WKB approximation.

Solution of the Ultra Generalized Exponential–Hyperbolic Potential in Multi-dimensional Space

In this work, we proposed ultra generalized exponential–hyperbolic potential (UGEHP) and derived various well known exponential–hyperbolic type potentials by setting parameters in UGEHP and using approximation suggested by Greene–Aldrich. The bound state solutions of the multi (D)-dimensional Schrodinger equation for UGEHP have been presented using the parametric Nikiforov–Uvarov method. The approximate analytical bound state energy eigenvalues and the corresponding un-normalized eigenfunctions expressed in terms of hypergeometric functions were obtained. We also investigated the rotational vibrational (RV) partition function from the eigenvalue of UGEHP. By the setting parameters, we obtained eigenvalue spectrum and RV partition function for the screened cosine Kratzer potential, screened Kratzer potential, attractive radial potential, quadratic exponential-type potential, Manning Rosen with class of Yukawa potential and Yukawa potential, class of Yukawa potential, mixed class of Yukawa potential, quantum interaction potential or Hulthen–Yukawa inversely quadratic potential, Hulthen plus inversely quadratic exponential Mie-type potential and Hulthen plus exponential Coulombic potential with centrifugal potential barrier. We studied the behavior of energy eigenvalues and RV partition function for the UGEHP. We computed and tabulated numerical results for $$CO, NO, I_2$$ , HCl and LiH diatomic molecules and compared with numerical results available in literature for same diatomic molecules.

Energy Eigenvalues and Eigenfunctions of a Diatomic Molecule in Quadratic Exponential-type Potential

We have employed the exact quantization rule to obtain closed form expression for the bound state energy eigenvalues of a molecule in quadratic exponential-type potential. To deal with the spin-orbit centrifugal term of the effective potential energy function, we have used a Pekeris-type approximation scheme, we have also obtained closed form expression for the normalized radial wave functions by solving the Riccati equation with quadratic exponential-type potential. Using our derived energy eigenvalue formula, we have deduced expressions for the bound state energy eigenvalues of the Hulthén, Eckart and Deng-Fan potentials, considered as special cases of the quadratic exponential-type potential. Our deduced energy eigenvalues are in excellent agreement with those in the literature. We have computed bound states energy eigenvalues for six diatomic molecules viz: HCl, LiH, H2, SeH, VH and TiH. Our results are in total agreement with existing results in the literature for the s-wave and in good agreement for higher quantum states. By solving the Riccati equation, we have obtained normalized radial wave functions of the quadratic exponential-type potential, our results show higher probabilities of finding the molecule in the region 0.1 ≤ y ≤ 0.2