Energy Eigenvalue Equation Research Articles

Bound State and Ro-Vibrational Energies Eigenvalues of Selected Diatomic Molecules with a Class of Inversely Quadratic Yukawa Plus Hulthén Potential Model

The Nikiforov-Uvarov approach is used in this study to solve the Schrödinger equation utilizing a class of inversely quadratic Yukawa plus Hulthén potential model with an approximation to the centrifugal term. The normalized wave function and energy eigenvalue equation were obtained. The numerical bound state for a few diatomic molecules (N2, O2, NO, and CO) for various rotational and vibrational quantum numbers was calculated using the energy equation and the related spectroscopic data. Our results show that, with no divergence between the s-wave and l-wave, the energy eigenvalues are very sensitive to the potential and diatomic molecule properties, suggesting that the approximation approach is appropriate for this set of potentials. The results are consistent with earlier studies in the literature, and we also found four special cases of this potential.

Study of energies spectra and thermodynamic properties of the relativistic Dirac equation using Feynman path integral method

In the current paper, the contribution of a Coulomb-like potential tensor interaction on the solution of the Dirac equation with a new generalized Morse-like potential is investigated using the Feynman path integral method. Relativistic and non-relativistic energy spectra were obtained. It has been established that the Coulomb-like potential eliminates the degeneracy of all pairs of spin doublets. Using the resulting non-relativistic energy eigenvalues equation, three diatomic molecules (H2, LiH, and HCl) were investigated and their thermodynamic properties, including mean energy, free energy, entropy, and specific heat capacity were shown. A comparison with the available literature shows that the thermodynamic plots obtained are consistent with previous work.

Solutions of the Dirac equation with position-dependent mass for q-deformed Pöschl–Teller and Coulomb-like tensor potentials

We consider the Dirac equation with position-dependent mass for a q-deformed Pöschl–Teller potential plus a Coulomb-like tensor interaction in the limits of spin and pseudospin symmetries. Under the condition of spin symmetry, the bound state energy eigenvalue equation and the two radial components of the Dirac wave function in terms of the Jacobi polynomials are obtained approximately for an arbitrary shifted spin-orbit quantum number λκ = κ + H and for any value of the deformation parameter q ≥ 1. In the case of the pseudospin symmetry limit, there are no bound states. The Dirac equation describes a free physical system and the two components of the wave function are expressed in terms of the spherical Bessel functions.

Recovery of a generic local Hamiltonian from a steady state

With the development of the quantum many-body simulator, Hamiltonian tomography has become an increasingly important technique for verification of quantum devices. Here we investigate recovering the generic Hamiltonians of two spin chains with two-local interactions and three-local interactions by measuring local observables. For these two models, we show that, when the chain length reaches a certain critical number, we can recover the generic local Hamiltonian from its one steady state by solving the homogeneous operator equation (HOE) developed by Bairey, Arad, and Lindner [Phys. Rev. Lett. 122, 020504 (2019)]. To explain the existence of such a critical chain length, we develop an alternative method to recover the Hamiltonian by solving the energy eigenvalue equations (EEE). By using the EEE method, we completely recovered the numerical results from the HOE method. Then we theoretically prove the equivalence between the HOE method and the EEE method. In particular, we obtain the analytical expression of the rank of the constraint matrix in the HOE method by using the EEE method, which can be used to determine the correct critical chain length in all the cases.

Improved q-deformed Scarf II oscillator

In this work, the improved q-deformed Scarf II oscillator (IQSO) is formulated through the Varshni conditions for diatomic molecule potential. Equation for bound state energy eigenvalues of the IQSO is obtained by the SUSYQM method, a Pekeris-like approximation scheme is used to model the centrifugal term of the Schrödinger equation. Expressions for molar entropy and Gibbs free energy are deduced for the IQSO from equations of energy eigenvalues and partition function within the context of Poisson summation formula. The IQSO is used to model experimental Rydberg-Klein-Rees (RKR) data of six diatomic molecules including: KRb (B 1Π), CO+ (X 2Σ+), Cl2 (X 1Σg +), ICl (X 1Σg +), NbO (X 1Σ+), and K2 (X 1Σg +). The average absolute deviation and mean absolute percentage deviation are used as accuracy indicators. Computed data revealed that the IQSO is an excellent model in fitting the RKR data of the diatomic molecules examined in this work. Numerical values of molar entropy and Gibbs free energy calculated for ICl (X 1Σg +) molecule are almost indistinguishable from existing experimental data for gaseous ICl (X 1Σg +) molecules. This suggests that the IQSO is a near perfect model for representing the internal vibrations of the ICl (X 1Σg +) molecule.

Thermodynamic properties and bound state solutions of Schrodinger equation with Mobius square plus screened-Kratzer potential using Nikiforov-Uvarov method

In this work, we applied the Nikiforov-Uvarov (NU) method to calculate the bound state energies for two diatomic molecules such as Carbon(II)Oxide (CO) and Scandium Flouride (ScF) using a new potential model called Mobius Square Plus Screen Kratzer potential (MSPSKP). The bound state energies obtained numerically are in excellent agreement compared to the results obtained with the semi-classical WKB approximation approach. We extended the work to obtain the partition function and other thermodynamic properties of the diatomic molecules under the study using the resulting energy eigenvalue equation. The thermodynamic properties of these molecules produce excellent trends reported in the existing literature. The novel potential also reduces to Mobius square, Screened-Kratzer and the Kratzer potentials as special cases. The numerical results are in excellent agreement to existing literature.

PT -Symmetric Qubit-System States in the Probability Representation of Quantum Mechanics

PT-symmetric qubit-system states are considered in the probability representation of quantum mechanics. The new energy eigenvalue equation for probability distributions identified with qubit and qutrit states is presented in an explicit form. A possibility to test PT-symmetry and its violation by measuring the probabilities of spin projections for qubits in three perpendicular directions is discussed.

Thermal properties of a two-dimensional Duffin–Kemmer–Petiau oscillator under an external magnetic field in the presence of a minimal length

Generalized uncertainty principle puts forward the existence of the shortest distances and/or maximum momentum at the Planck scale for consideration. In this article, we investigate the solutions of a two-dimensional Duffin–Kemmer–Petiau (DKP) oscillator within an external magnetic field in a minimal length (ML) scale. First, we obtain the eigensolutions in ordinary quantum mechanics. Then, we examine the DKP oscillator in the presence of an ML for the spin-zero and spin-one sectors. We determine an energy eigenvalue equation in both cases with the corresponding eigenfunctions in the non-relativistic limit. We show that in the ordinary quantum mechanic limit, where the ML correction vanishes, the energy eigenvalue equations become identical with the habitual quantum mechanical ones. Finally, we employ the Euler–Mclaurin summation formula and obtain the thermodynamic functions of the DKP oscillator in the high-temperature scale.

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.

Solutions of Schrodinger equation for the modified Mobius square plus Kratzer potential

In this paper, we obtain the approximate solutions of the Schrodinger equation with the modified Mobius square plus Kratzer potential using the Nikiforov–Uvarov method and employing the approximation scheme for the centrifugal term. We obtain the energy eigenvalue equation and corresponding wave functions. Finally, some numerical results and special cases are also reported.

PHENOMENOLOGICAL RELATIVISTIC STUDY OF THE ENERGY OF Α Λ IN ITS GROUND AND EXCITED STATES IN HYPERNUCLEI

The binding energy B_Λ of a Λ-particle in hypernuclei is studied by means of the Dirac equation containing attractive and repulsive potentials of orthogonal shapes. The energy eigenvalue equation in this case is obtained analytically for every bound state. An attempt is also made to investigate the possibility of deriving in particular cases approximate ana­lytic expressions for B_Λ.

Thermodynamic functions for diatomic molecules with modified Kratzer plus screened Coulomb potential

We have solved the Schrodinger equation with the modified Kratzer plus screened Coulomb potential using the modified factorization method. We have also employed both the Greene–Aldrich approximation scheme and a suitable transformation scheme to obtain the energy eigenvalues equation and its corresponding energy eigenfunctions for CO, NO, and N2 diatomic molecules. Numerical results of the energy eigenvalues for the selected diatomic molecules have been computed and discussed. The effects of the combined potential parameters on the energy eigenvalues have been evaluated. We have used the energy eigenvalues of the modified Kratzer plus screened Coulomb potential to obtain the vibrational partition functions and other thermodynamic functions for the selected diatomic molecules. Our results are relevant in other areas of theoretical physics and chemistry, and they agree perfectly with the results obtained in the literature.

Two anyons on the sphere: Nonlinear states and spectrum

We study the energy spectrum of two anyons on the sphere in a constant magnetic field. Making use of rotational invariance we reduce the energy eigenvalue equation to a system of linear differential equations for functions of a single variable, a reduction analogous to separating center of mass and relative coordinates on the plane. We solve these equations by a generalization of the Frobenius method and derive numerical results for the energies of non-analytically derivable states.

Spin-one (1 + 3)-dimensional DKP equation with modified Kratzer potential in the non-commutative space

In this paper, the spin-one Duffin–Kemmer–Petiau equation in (1 + 3) dimensions with a modified Kratzer potential is considered in the non-commutative space framework. The energy eigenvalue equation and the corresponding eigenfunctions are derived analytically. Furthermore, the energy shift due to the space non-commutativity effect is also obtained using the perturbation theory. In particular, it is shown that the degeneracy of the initial spectral line is broken, where the space non-commutativity plays the role of a magnetic field. This behavior is very similar to the Zeeman effect.

Spectroscopic transitions supporting chemiluminescence in the reaction of calcium with hydrogen chloride

The main aim of this study was to show that chemiluminescence transitions of free radicals and molecules formed during the reaction are important for stability. After detection of certain geometrical structures that are valuable for the reaction, the most likely electrical and vibrational transitions in density functional theory were determined. Another factor that suppresses or promotes electronic transitions, as well as geometric position, is the vibrational energy levels of the molecular system. In experimental studies, vibrational energy distributions are worth examining while studying electron density or population. In these geometric regions, the quantum states where the infrared transitions take place were calculated through the energy eigenvalues equation. Thus, the potential energy surface obtained by density functional theory method was compared with a realistic potential energy surface and experimental values in the literature. The stability of the nascent calcium chloride molecule is determined by the position of the hydrogen atom separated from the chlorine atom rather than the energy of the calcium atom.

A Simple Matrix Approach to Determination of the Helium Atom Energies

Calculation of He atomic energy levels using the first order perturbation theory taught in the Basic Quantum Mechanics course has led to relatively large errors. To improve its accuracy, several methods have been developed but most of them are too complicated to be understood by undergraduate students. The purposes of this study are to apply a simple matrix method in calculating some of the lowest energy levels of He atom (1s2, triplet 1s2s, and singlet 1s2s states) and to reduce errors obtained from calculations using the standard perturbation theory. The convergence of solutions as a function of the number of bases is also examined. The calculation is done analytically for 3 bases and computationally with the number of bases using MATHEMATICA. First, the 2-electron wave function of the Helium atom is written as the multiplication of two He+ ion wave functions, which are then expanded into finite dimension bases. These bases are used to calculate the elements of the Hamiltonian matrix, which are then substituted back to the energy eigenvalue equation to determine the energy values of the system. Based on the calculation results, the error obtained for the He ground state energy using 3 bases is 2.51 %, smaller than the errors of the standard perturbation theory (5.28 %). Despite the fact that the error is still relatively large from the analytical calculations for singlet-triplet 1s2s energy splitting of He atom, this error is successfully reduced significantly as more bases were used in the numerical calculations. In particular, for n = 25, the current calculation error for all states is much smaller than the errors obtained from calculations using standard perturbation theory. In conclusion, the analytical calculations for the energy eigenvalue equation for the 3 lowest states of the Helium atom using 3 bases have been carried out. It was also found in this study that increasing the number of bases in our numerical calculations has significantly reduced the errors obtained from the analytical calculations.

Rotation-vibrational energies for some diatomic molecules with improved Rosen-Morse potential in D-dimensions.

By using the Nikiforov-Uvarov method, we solve the Schrödinger equation for the improved Rosen-Morse potential model in D spatial dimensions. We obtained the rotation-vibrational energies and the wave function, respectively. The ro-vibrational energies spectral of NO(a4πi) and [Formula: see text] in D-dimensions have been computed by using the rotation-vibrational energy eigenvalues equation.

Relativistic spectral bounds for the general molecular potential: application to a diatomic molecule.

We tackle with the Dirac equation in the presence of the general molecular potential (GMP). The spin symmetric solution of the relativistic wave equation is obtained by considering the Pekeris-type approximation scheme to deal with the centrifugal term, and in order to solve second-order differential equation, the asymptotic iteration method (AIM) is used. The closed form of the energy eigenvalue equation is found out for any values of the angular momentum quantum number. We calculate the relativistic vibrational bound state energies of 51Δg state of Na2 molecule and compare them with the Rydberg-Klein-Rees (RKR) data. We show that relativistic vibrational energies of this molecule, which are found in the spin symmetric case, are more convenient with experimental RKR data than non-relativistic vibrational energies. We also obtain normalization constant by considering inductive approach and investigate the radial eigenfunctions and probability density functions corresponding to different eigenvalues graphically for the Na2(51Δg) molecule.

Harmonic oscillator in Ads and dS spaces

In one-dimensional (anti)-de Sitter background, all commutation relations are modified and the Heisenberg uncertainty principle is changed to the so-called extended uncertainty principle. In this scenario, the commutator between position and momentum operators contains a linear power of coordinate, instead of a constant, and the coordinate representation of the momentum operators for this model becomes coordinate dependent. In this paper, we present an exact solution of the one-dimensional harmonic oscillator in Ads and dS spaces. The eigenfunctions are determined for both cases, and the energy eigenvalues equation is obtained. From the obtained spectrum energy, the thermodynamic properties of the harmonic oscillator in AdS space are then analyzed.

Analytical solutions of the Klein-Gordon equation for the deformed generalized Deng-Fan potential plus deformed Eckart potential

In this paper, we study approximate analytical solutions of the Klein-Gordon equation with arbitrary l state for the deformed generalized Deng-Fan potential plus deformed Eckart potential using the Nikiforov-Uvarov method and employing the approximation scheme for the centrifugal term. We obtain the energy eigenvalue equation and corresponding wave functions. Also, we discuss nonrelativistic limit of the energy equation. Finally, some numerical results are presented and show that these results are in good agreement with those obtained previously by other methods.