We present an approximate analytic solution of the Klein-Gordon equation in the presence of equal scalar and vector generalized deformed hyperbolic potential functions by means of parameteric generalization of the Nikiforov-Uvarov method. We obtain the approximate bound state rotational-vibrational (ro-vibrational) energy levels and the corresponding normalized wave functions expressed in terms of the Jacobi polynomial for a spin-zero particle in a closed form. Special cases are studied including the non-relativistic solutions obtained by appropriate choice of parameters and also the s-wave solutions.
Analytical solutions of the Klein-Gordon equation are obtained by reducing the radial part of the wave equation to a standard form of a second order differential equation. Differential equations of this standard form are solvable in terms of hypergeometric functions and we give an algebraic formulation for the bound state wave functions and for the energy eigenvalues. This formulation is applied for the solutions of the Klein-Gordon equation with some diatomic potentials.
In this paper we present a closed-form expression of the vibrational partition function for the one-dimensional q-deformed Morse potential energy model. Through this function the related thermodynamic functions are derived and studied in terms of the parameters of the model. Specially, we plotted the q-deformed vibrational partition function, and vibrational specific heat for some diatomic molecule systems such as H2, HCl, LiH and CO. The idea of a critical temperature T_{C} is introduced in relation to the specific heat.
We study the quantum dynamics of diatomic molecule driven by a circularly polarized resonant electric field. We look for a quantum effect due to classical chaos appearing due to the overlapping of nonlinear resonances associated to the vibrational and rotational motion. We solve the Schrödinger equation associated with the wave function expanded in term of proper stationary states, $|n>\otimes|lm>$ (vibrational$\otimes$angular momentum states). Looking for quantum-classic correspondence, we consider the Liouville dynamics in the two dimensional phase space defined by the coherent -like state o…
This work presents the bound-state spectra of Morse oscillator, which remains one of the oldest important model potentials for molecules. Accurate ro-vibrational energies are obtained by means of a generalized pseudospectral method that offers an optimal, non-uniform discretization of the radial grid. Both s-wave ($\ell=0$) and rotational ($\ell \neq 0$) states for low and high quantum numbers are calculated for four representative diatomic molecules, namely H$_2$, LiH, HCl and CO. First nine states belonging to a maximum of $n, \ell =2$ are computed with good accuracy, along with nine other h…
The Schrodinger equation for the rotational-vibrational (ro-vibrational) motion of a diatomic molecule with empirical potential functions is solved approximately by means of the Nikiforov-Uvarov method. The approximate ro-vibratinal energy spectra and the corresponding normalized total wavefunctions are calculated in closed form and expressed in terms of the hypergeometric functions or Jacobi polynomials P_{n}^{(μ,ν)}(x), where μ>-1, ν>-1 and x included in [-1,+1]. The s-waves analytic solution is obtained. The numerical energy eigenvalues for selected H_{2} and Ar_{2} molecules are also calcu…
The vibrational motion equations of both homo and hetero-nuclei diatomic molecules are here derived for the first time. A diatomic molecule is first considered as a one dimensional quantum mechanics oscillator. The second and third-order Hamiltonian operators are then formed by substituting the number operator for the quantum number in the corresponding vibrational energy eigenvalues. The expectation values of relative position and linear momentum operators of two oscillating atoms are calculated by solving Heisenbergs equations of motion. Subsequently, the expectation values of potential and …
We study a general problem of the translational/rotational/vibrational/electronic dynamics of a diatomic molecule exposed to an interaction with an arbitrary external electromagnetic field. The theory developed in this paper is relevant to a variety of specific applications. Such as, alignment or orientation of molecules by lasers, trapping of ultracold molecules in optical traps, molecular optics and interferometry, rovibrational spectroscopy of molecules in the presence of intense laser light, or generation of high order harmonics from molecules. Starting from the first quantum mechanical pr…
The Tietz-Hua (TH) potential is one of the very best analytical model potentials for the vibrational energy of diatomic molecules. By using the Nikiforov-Uvarov (NU) method, we have obtained the exact analytical s-wave solutions of the radial Schrödinger equation (SE) for the TH potential. The energy eigenvalues and the corresponding eigenfunctions are calculated in closed forms. Some numerical results for diatomic molecules are also presented.
We investigate the nonrelativistic magnetic effect on the energy spectra, expectation values of some quantum mechanical observables and diamagnetic susceptibility for some diatomic molecules bounded by the Isotropic oscillator plus inverse quadratic potential. The energy eigenvalues and normalized wavefunctions are obtained via parametric Nikiforov-Uvarov method. The expectation values square of position r2, square of momentum p2, kinetic energy T and potential energy V are obtained by applying Hellmann-Feynman theorem and an expression for the diamagnetic susceptibility X is also derived. Usi…
We investigate the dynamics of a rotating Morse wave packet, appropriate for a ro-vibrating diatomic molecule. The coupling between vibrational and rotational degrees of freedom is explicated in real position space as well as in phase space Wigner distribution of a SU(2) coherent state at various dynamically evolved times. We choose the well studied $I_{2}$ molecule with the parameter values in good agrement with experiments. A quantitative measure of the angles of rotations for different angular momenta is also given.
Within the framework of non-relativistic quantum mechanics, the ro-vibrational energy spectra of the improved deformed exponential-type potential model are obtained using the Greene-Aldrich approximation scheme and an appropriate coordinate transformation. With the help of the energy spectra, analytical expressions of the vibrational partition function and other thermodynamic functions are derived by employing the Poisson summation formula. These thermodynamic functions are studied for the electronic states of hydrogen dimer, carbon monoxide, nitrogen dimer and lithium hydride diatomic molecul…
An algebraic model taking into account the influence of the molecular rotation on the wave functions of vibrational-rotational states of the diatomic molecule using the formalism of the ladder operators and an expansion in a small parameter characterizing the vibrational-rotational interaction have been proposed for the potentials whose the creation and annihilation operators can be constructed. Expressions for the expansion of the wave function of the vibrational-rotational states in a set of wave functions of the rotationless vibrational states have been obtained of the second order in the p…
Explicit algebraic expressions for the expansion of the vibrational matrix elements in series of matrix elements on the wave functions of the ground vibrational state have been obtained for arbitrary sufficiently differentiable functions of the internuclear distance, arbitrary values v and v', and the potential curves whose ladder operators can be constructed. A diagram technique have been developed for it that consists in: 1) the numeration of the matrix elements by points of the 2D diagram with coordinates (l, k), 2) the drawing arrows between points of this diagram corresponding to the acti…
Exact solutions for vibrational levels of diatomic molecules via the Morse potential are obtained by means of the asymptotic iteration method. It is shown that, the numerical results for the energy eigenvalues of $^{7}Li_{2}$ are all in excellent agreement with the ones obtained before. Without any loss of generality, other states and molecules could be treated in a similar way.
In this study, we apply the parametric Nikiforov-Uvarov method to obtain the bound state solution of Schrodinger wave equation in the presence of Kratzer plus generalized Morse potential (KPGM). The energy eigen equation and the corresponding normalised wave function were obtained in closed form. The resulting energy eigen equation were use to study partition function and other thermodynamic properties such as vibrational mean energy, vibrational specific heat capacity, vibrational mean free energy and vibrational entropy for the proposed potential as applied to lithium hydride diatomic molecu…
In this study, we obtain the approximate analytical solutions of the radial Schrodinger equation for the New Generalized Morse-Like Potential in arbitrary dimensions by using the Nikiforov Uvarov Method. Energy eigenvalues and corresponding eigenfunction are obtain analytically. The rotational-vibrational energy eigenvalues for some diatomic molecules are computed with the aid of some spectroscopic parameters. The energy equation for some potentials such as Attarctive Radial and Deng-Fan potentials have also been obtained by varying some potential parameters. Our results excellently agree with…
We obtain the bound-state solutions of the radial Schrödinger equation (SE) with the shifted Deng-Fan (sDF) oscillator potential in the frame of the Nikiforov-Uvarov (NU) method and employing Pekeris-type approximation to deal with the centrifugal term. The analytical expressions for the energy eigenvalues and the corresponding wave functions are obtained in closed form for arbitrary -state. The ro-vibrational energy levels for a few diatomic molecules are also calculated. They are found to be in good agreement with those ones previously obtained by the Morse potential.
We propose a method for achieving highly efficient transfer between the vibrational states in a diatomic molecule. The process is mediated by strong laser pulses and can be understood in terms of light-induced potentials. In addition to describing a specific molecular system, our results show how, in general, one can manipulate the populations of the different quantum states in double well systems.
In this study, approximate analytical solution of Schrödinger, Klein-Gordon and Dirac equations under the Tietz-Wei (TW) diatomic molecular potential are represented by using an approximation for the centrifugal term. We have applied three types of eigensolution techniques; the functional analysis approach (FAA), supersymmetry quantum mechanics (SUSYQM) and asymptotic iteration method (AIM) to solve Klein-Gordon Dirac and Schrödinger equations, respectively. The energy eigenvalues and the corresponding eigenfunctions for these three wave equations are obtained and some numerical results and fi…
A beam of diatomic molecules scattered off a standing wave laser mode splits according to the rovibrational quantum state of the molecules. Our numerical calculation shows that single state resolution can be achieved by properly tuned, monochromatic light. The proposed scheme allows for selecting non-vibrating and non-rotating molecules from a thermal beam, implementing a laser Maxwell's demon to prepare a rovibrationally cold molecular ensemble.
The energy levels of the Schrödinger equation under the Eckart-Hellmann potential (EHP) energy function are studied by the Nikiforov-Uvarov-Functional Analysis (NUFA) method. We obtained the analytic solution of the energy spectra and the wave function in closed form with the help of Greene-Aldrich approximation. The numerical bound states energy for various screening parameters at different quantum states and vibrational energies of EHP for CuLi, TiH, VH, and TiC diatomic molecules were computed. Four particular cases of this potential were achieved. To test the accuracy of our results, we co…
The analytic solutions of the spatially-dependent mass Schrodinger equation of diatomic molecules with the centrifugal term l(l+1)/r2 for the generalized q-deformed Morse potential are obtained approximately by means of a parametric generalization of the Nikiforov-Uvarov (NU) method combined with the Pekeris approximation scheme. The energy eigenvalues and the corresponding normalized radial wave functions are calculated in closed form with a physically motivated choice of a reciprocal Morse-like mass function, m(r)=m0/(1-deltae^{-a(r-r_{e})})2, 0<delta<1, where a and r_{e} are the range of th…
In this study, we obtain the approximate analytical solutions of the radial Schrödinger equation for the Deng-Fan diatomic molecular potential by using exact quantization rule approach. The wave functions have been expressed by hypergeometric functions via the functional analysis approach. An extension to rotational-vibrational energy eigenvalues of some diatomic molecules are also presented. It is shown that the calculated energy levels are in good agreement with the ones obtained previously $E_{n\ell}-D$ (shifted Deng-Fan).
The study of diatomic molecules plays a central role in the understanding of the chemical bond. For their simplicity, they serve as a model for the study of more complex molecular systems. In this article, we solve the rovibrational Schrödinger equation for diatomic molecules using the Kratzer oscillator, by means of so(2,1) Lie algebra. The energies and bound states for this simple model are obtained through a canonical transformation of the molecular Hamiltonian. The main contribution of the Lie-algebraic approach is that this allows us to reduce the degree of Schrödinger equation, obtaining…
Quantum sensitivity is an important issue in the field of quantum metrology where sub-Planck scale structures play crucial role in the Heisenberg limited measurement. We investigate the mesoscopic superposition structures, particularly for well-known cat-like and compass-like states, in the rotating Morse system where sub-Planck scale structures originate in the dynamics of a suitably constructed SU(2) coherent state. A detail study of the sensitivity analysis reveals that rotational coupling in the vibrational wave packet can be used as a probe to enhance the sensitivity limit in a diatomic m…
A solution of the fractional N-dimensional radial Schrodinger equation with the Deng-Fan potential is investigated by the generalized fractional NU method. The analytical formulas of energy eigenvalues and corresponding eigen functions for the Deng-Fan potential are generated. Furthermore, the current results are applied to several diatomic molecules for the Deng-Fan potential as well as the shifted Deng Fan potential. For both the Deng-Fan potential and its shifted potential, the effect of the fractional parameter on the energy levels of various diatomic molecules is examined numerically and …
Two physically important potentials (Manning-Rosen and Pöschl-Teller) are considered for the ro-vibrational energy in diatomic molecules. An improved new approximation is invoked for the centrifugal term, which is then used for their solution within the Nikiforov-Uvarov framework. This employs a recently proposed scheme, which combines the two widely used Greene-Aldrich and Pekeris-type approximations. Thus, approximate analytical expressions are derived for eigenvalues and eigenfunctions. The energies are examined with respect to two approximation parameters, $λ$ and $ν$. The original approxi…
We demonstrate a scheme for optical cycling in the polar, diatomic molecule strontium monofluoride (SrF) using the $X ^2Σ^+\toA^2Π_{1/2}$ electronic transition. SrF's highly diagonal Franck-Condon factors suppress vibrational branching. We eliminate rotational branching by employing a quasi-cycling $N=1\to N^\prime=0$ type transition in conjunction with magnetic field remixing of dark Zeeman sublevels. We observe cycling fluorescence and deflection through radiative force of an SrF molecular beam using this scheme. With straightforward improvements our scheme promises to allow more than $10^5$…
We solve the relativistic equations(Klein-Gordon and Dirac equation) via the conventional Nikiforov-Uvarov method. In order to overcome the centrifugal barrier, we employed the well-known Greene and Aldrich approximation scheme. The corresponding normalized eigenfunctions was also obtained in each case. It was shown that in the non-relativistic limits, both energy equations obtained by solving Klein-Gordon and Dirac equations, and wavefunctions reduced to the non-relativisitc energy Equation. The bound state energy eigenvalues for N_2, CO, NO, CH and HCl diatomic molecules were computed for va…
In this paper we use the Nikiforv-Uvarov method to obtain the approximate solutions of the Klein-Gordon equation with deformed five parameter exponential type potential (DFPEP) model. We also obtain the solutions of the Schrödinger equation in the presence of the DFPEP in the non-relativistic limits. In addition, we calculate in the nonrelativistic limits the thermodynamics properties such as vibrational mean energy U, free energy F and the specific heat capacity C . Special cases of the potential are also discussed.
We explore the electronic structure and rovibrational properties of an ultralong-range triatomic Rydberg molecule formed by a Rydberg atom and a ground state heteronuclear diatomic molecule. We focus here on interaction of Rb($27s$) Rydberg atom with KRb($N=0$) diatomic polar molecule. There's significant electronic hybridization of Rb($n=24$, $l\ge 3$) degenerate manifold. The polar diatomic molecule is allowed to rotate in the electric fields generated by the Rydberg electron and core as well as an external field. We investigate the metamorphosis of the Born-Oppenheimer potential curves, ess…
For Klein-Gordon equation a consistent physical interpretation of wave functions is reviewed as based on a proper modification of the scalar product in Hilbert space. Bound states are then studied in a deep-square-well model where spectrum is roughly equidistant and where a fine-tuning of the levels is mediated by PT-symmetric interactions composed of imaginary delta functions which mimic creation/annihilation processes.
In the present article, we introduce a model to investigate the energy spectrum of a relativistic rotor by considering the Klein-Gordon Hamiltonian. Rotational spectral lines are a signature of homonuclear and heteronuclear systems and play a key role in understanding diatomic molecules. We show that the energy-correction term arising due to unequal masses influences the line separation. Determining the rotational constant enables one to calculate the moment of inertia and bond length of the molecule.
We apply the asymptotic iteration method (AIM) to obtain the solutions of Schrodinger equation in the presence of Poschl-Teller (PT) potential. We also obtain the solutions of Dirac equation for the same potential under the condition of spin and pseudospin (p-spin) symmetries. We show that in the nonrelativistic limits, the solution of Dirac system converges to that of Schrodinger system. Rotational-Vibrational energy eigenvalues of some diatomic molecules are calculated. Some special cases of interest are studied such as s-wave case, reflectionless-type potential and symmetric hyperbolic PT p…
The Deng-Fan-Eckart (DFE) potential is as good as the Morse potential in studying atomic interaction in diatomic molecules. By using the improved Pekeris-type approximation, to deal with the centrifugal term, we obtain the bound-state solutions of the radial Schrödinger equation with this adopted molecular model via the Factorization Method. With the energy equation obtained, the thermodynamic properties of some selected diatomic molecules(H2 , CO , and ScN ) were obtained using Poisson summation method.. The unnormalized wave function is also derived. The energy spectrum for a set of diatomic…
We have calculated the isotropic $C\_6$ coefficients characterizing the long-range van der Waals interaction between two identical heteronuclear alkali-metal diatomic molecules in the same arbitrary vibrational level of their ground electronic state $X^1Σ^+$. We consider the ten species made up of $^7$Li, $^{23}$Na, $^{39}$K, $^{87}$Rb and $^{133}$Cs. Following our previous work [M.~Lepers \textit{et.~al.}, Phys.~Rev.~A \textbf{88}, 032709 (2013)] we use the sum-over-state formula inherent to the second-order perturbation theory, composed of the contributions from the transitions within the gr…
By making use of the quantization rule of Raab and Friedrich [P. Raab and H. Friedrich, Phys. Rev. A 78, 022707 (2008)], we derive simple and accurate formulae for the number of rotational states supported by a weakly-bound vibrational level of a diatomic molecule and the rotational constants of any such levels up to the threshold, and provide a criterion for determining whether a given weakly-bound vibrational level is rotationless. The results depend solely on the long-range part of the molecular potential and are applicable to halo molecules.
Atomic lattice clocks have spurred numerous ideas for tests of fundamental physics, detection of general relativistic effects, and studies of interacting many-body systems. On the other hand, molecular structure and dynamics offer rich energy scales that are at the heart of new protocols in precision measurement and quantum information science. Here we demonstrate a fundamentally distinct type of lattice clock that is based on vibrations in diatomic molecules, and present coherent Rabi oscillations between weakly and deeply bound molecules that persist for 10's of milliseconds. This control is…
The laser control of photodissociation branching in a diatomic molecule is demonstrated to be effectively achieved with use of the complete reflection phenomenon. The phenomenon and the control condition can be nicely formulated by the semiclassical (Zhu-Nakamura) theory. The method is applied to the branching between I($^2 P_{3/2}$) (HI $\to$ H + I) and I$^*(^2 P_{1/2})$ (HI $\to$ H + I$^*$) formation, and nearly complete control is shown to be possible by appropriately choosing an initial vibrational state and laser frequency in spite of the fact that there are three electronically excited s…
Relativistic ab initio study on the spectroscopic and radiative properties of the lowest states and modeling of the optical cycles for the LiFr molecule
Arbitrary $\ell$-state solutions of the Klein-Gordon equation with the Eckart plus a class of Yukawa potential and its non-relativistic thermal properties
We consider an ensemble of diatomic molecules resonantly coupled to an optical cavity under strong coupling conditions at normal incidence. Photodissociation dynamics is examined via direct numerical integration of the coupled Maxwell-Schrodinger equations with molecular ro-vibrational degrees of freedom explicitly taken into account. It is shown that the dissociation is significantly affected (slowed down) when the system is driven at its polaritonic frequencies. The observed effect is demonstrated to be of transient nature and has no classical analog. An intuitive explanation of the dissocia…
According to the Schiff theorem, the atomic electrons completely screen the atomic nucleus from an external static electric field. However, this is not the case if the field is time-dependent. Electronic orbitals in atoms either shield the nucleus from an oscillating electric field when the frequency of the field is off the atomic resonances or enhance this field when its frequency approaches an atomic transition energy. In molecules, not only electronic, but also rotational and vibrational states are responsible for the screening of oscillating electric fields. As will be shown in this paper,…
Analytic potentials are built for all four $^{2S+1}Π_{u/g}$ states of Li$_{2}$ dissociating to Li$(2S)$ + Li$(3P)$: $3b(3^{3}Π_{u})$, $3B(3^{1}Π_{u})$, $3C(3^{1}Π_{g}),$ and $3d(3^{3}Π_{g})$. These potentials include the effect of spin-orbit coupling for large internuclear distances, and include state of the art long-range constants. This is the first successful demonstration of fully analytic diatomic potentials that capture features that are usually considered too difficult to capture without a point-wise potential, such as multiple minima, and shelves. Vibrational energies for each potentia…
