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 the potential and the equilibrium position of the nuclei. The constant mass case when delta=0 is also studied. The energy states for H2, LiH, HCl and CO diatomic molecules are calculated and compared favourably well with those obtained by using other approximation methods for arbitrary vibrational n and rotational l quantum numbers.
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.
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.
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…
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 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 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 so…
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…
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…
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.
We present modified $\ell$-states of diatomic molecules by solving the radial and angle-dependent parts of the Schrödinger equation for central potentials, such as Morse and Kratzer, plus an exactly solvable angle-dependent potential $V_θ(θ)/r^2$ within the framework of the Nikiforov-Uvarov (NU) method. We emphasize that the contribution which comes from the solution of the Schrödinger equation for the angle-dependent potential modifies the usual angular momentum quantum number $\ell$. We calculate explicitly bound state energies of a number of neutral diatomic molecules composed of a first-ro…
We present arguments demonstrating that the application of the Nikiforov-Uvarov polynomial method to solve the Schrödinger equation with the Tietz-Hua potential is valid only when $e^{-b_{h}r_{e}}\leq c_{h}<1$ and $r_{0}<r<+\infty $. In particular, it is pointed out that the numerical results with $c_{h}\neq 0$ for the diatomic molecules $\mathrm{HF}$, $\mathrm{N}_{2}$, $\mathrm{I}_{2}$, $\mathrm{H}_{2}$, $\mathrm{O}_{2}$ and $\mathrm{O}_{2}^{+}$ given in Tables 3-5 by Hamzavi and co-workers are wrong. When $-1\leq c_{h}<0$ or $0<c_{h}<e^{-b_{h}r_{e}}$, this approach is not suitable. In both c…
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…
We propose a theoretical scheme for coupling a nanomechanical resonator to a single diatomic molecule via microwave cavity mode of a driven LC resonator. We describe the diatomic molecule by a Morse potential and find the corresponding equations of motion of the hybrid system by using Fokker-Planck formalism. Analytical expressions for the effective frequency and the effective damping of the nanomechanical resonator are obtained. We analyze the ground state cooling of the nanomechanical resonator in presence of the diatomic molecule. The results confirm that presence of the molecule improves t…
We compute the quantum work distribution for a driven Morse oscillator. To this end, we solve the time-dependent dynamics for a scale-invariant process, from which the exact expressions for the transition probabilities are found. Special emphasis is put on the contributions to the work distribution from discrete (bound) and continuous (scattering) parts of the spectrum. The analysis is concluded by comparing the work distribution for the exact Morse potential and the one resulting from a harmonic approximation.
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 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 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…
We show that a direct connection can be drawn, based on fundamental quantum principles, between the Morse potential, extensively used as an empirical description for the atomic interaction in diatomic molecules, and the harmonic potential. This is conceptually achieved here through a non-additive translation operator, whose action leads to a perfect equivalence between the quantum harmonic oscillator in deformed space and the quantum Morse oscillator in regular space. In this way, our theoretical approach provides a distinctive first principle rationale for anharmonicity, therefore revealing a…
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…
Using a version of Witten's supersymmetric quantum mechanics proposed by Caticha, we relate Montroll's kink to a traveling, asymmetric Morse double-well potential suggesting in this way a connection between kink modes and vibrational degrees of freedom along microtubules
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 argue that the textbook method for solving eigenvalue equations is simpler, more elegant and efficient than the Asymptotic Iteration Method (AIM) applied in J. Phys. A {\bf 44} 155205. We show that the Kratzer potential is not a realistic model for the vibration--rotation spectrua of diatomic molecules because it predicts the position of the absorption infrared bands too far from the experimental ones (at least for the $HCl$ and $H_2$ molecules chosen as illustrative examples in that paper).
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…
In this paper we discuss the Morse potential on a quantum computer. The Morse potential is useful to describe diatomic molecules and has a finite number of bound states which can be measured through spectroscopy. It is also a example of an exactly soluble potential using supersymmetric quantum mechanics. Using the the supersymmetric quantum mechanics formalism one can derive a heirachy of Hamiltonians such that the ground state of the next rung on the heirarchy yeids the first excited state of the hamiltonian below it. Using this method one can determine all the states of the Morse potential b…
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…
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…
We solve the parametric generalized effective Schrödinger equation with a specific choice of posi-tion-dependent mass function and Morse oscillator potential by means of the Nikiforov-Uvarov (NU) method combined with the Pekeris approximation scheme. All bound-state energies are found explicitly and all corresponding radial wave functions are built analytically. We choose the Weyl or Li and Kuhn ordering for the ambiguity parameters in our numerical work to calculate the energy spectrum for a few and diatomic molecules with arbitrary vibration and rotation quantum numbers and different positio…
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…
Approximate analytical bound state solutions of the radial Schrödinger equation are studied for a two-term diatomic molecular potential in terms of the hypergeometric functions for the cases where $q\geq1$ and $q=0$. The energy eigenvalues and the corresponding normalized wave functions of the Manning-Rosen potential, the 'standard' Hulthén potential and the generalized Morse potential are briefly studied as special cases. It is observed that our analytical results are the same with the ones obtained before.
The system of two $Q$-deformed oscillators coupled so that the total Hamiltonian has the su$_Q$(2) symmetry is proved to be equivalent, to lowest order approximation, to a system of two identical Morse oscillators coupled by the cross-anharmonicity usually used empirically in describing vibrational spectra of triatomic molecules. The symmetry also imposes a connection between the self-anharmonicity of the Morse oscillators and the cross-anharmonicity strength, which can be removed by replacing the $Q$-oscillators by deformed anharmonic oscillators. The generalization to $n$ oscillators is stra…
The effective mass one-dimensional Schrödinger equation for the generalized Morse potential is solved by using Nikiforov-Uvarov method. Energy eigenvalues and corresponding eigenfunctions are computed analytically. The results are also reduced to the case of constant mass. Energy eigenvalues are computed numerically for some diatomic molecules. The results are in agreement with the ones obtained before.
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.
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 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…
The shifted Tietz-Wei (sTW) oscillator is as good as traditional Morse potential in simulating the atomic interaction in diatomic molecules. By using the Pekeris-type approximation to deal with the centrifugal term, we obtain the bound-state solutions of the radial Schrödinger equation with this typical molecular model via the exact quantization rule (EQR). The energy spectrum for a set of diatomic molecules ($NO \left(a^4Π_i\right)$, $NO \left(B^2Π_r\right)$, $NO \left(L'^2ϕ\right)$, $NO \left(b^4Σ^{-}\right)$, $ICl\left(X^1Σ_g^{+}\right)$, $ICl\left(A^3Π_1\right)$ and $ICl\left(A'^3Π_2\right…
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.
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).
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 …
We have studied the variation of the position space statistical complexity measure defined by López-Ruiz, Mancini, and Calbet such as the product of exponential of the Shannon information entropy and the disequilibrium by using the 1D-normalized probability densities derived from solutions of the Schrödinger equation corresponding to the q-deformed harmonic oscillator and q-deformed Morse potentials. An analysis of the numerical results in terms of Shannon information entropy, disequilibrium and complexity measure are presented. In q-deformed harmonic oscillator, q-dependence of the complexity…
For non-zero $\ell$ values, we present an analytical solution of the radial Schrödinger equation for the rotating Morse potential using the Pekeris approximation within the framework of the Asymptotic Iteration Method. The bound state energy eigenvalues and corresponding wave functions are obtained for a number of diatomic molecules and the results are compared with the findings of the super-symmetry, the hypervirial perturbation, the Nikiforov-Uvarov, the variational, the shifted 1/N and the modified shifted 1/N expansion methods.
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 present an algebraic method for treating molecular vibrations in the Morse potential perturbed by an external laser field. By the help of a complete and normalizable basis we transform the Schrödinger equation into a system of coupled ordinary differential equations. We apply our method to calculate the dissociation probability of the NO molecule excited by chirped laser pulses. The dependence of the molecular dipole-moment on the interatomic separation is determined by a quantum-chemical method, and the corresponding transition dipole moments are given by approximate analytic expressions. …
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…
The trigonometric Pöschl-Teller (PT) potential describes the diatomic molecular vibration. By using the Nikiforov-Uvarov (NU) method, we have obtained the exact analytical s-wave solutions of the radial Schrödinger equation (SE) for the trigonometric PT potential. The energy eigenvalues and corresponding eigenfunctions are calculated in closed forms. Some numerical results are presented too.
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…
The trigonometric Pöschl-Teller (PT) potential describes the diatomic molecular vibration. We have obtained the approximate solutions of the radial Schrödinger equation (SE) for the rotating trigonometric PT potential using the Nikiforov-Uvarov (NU) method. The energy eigenvalues and their corresponding eigenfunctions are calculated for arbitrary -states in closed form. In the low screening region, when the screening parameter the potential reduces to Kratzer potential. Further, some numerical results are presented for several diatomic molecules.
The pseudoharmonic oscillator potential is studied in non relativistic quantum mechanics with a generalized uncertainty principle characterized by the existence of a minimal length scale. By using a perturbative approach, we analytically compute the correction to the energy spectrum in the first order of the minimal length parameter \b{eta}. We investigate the effect of this fundamental length on the vibration-rotation energy levels of diatomic molecules with this potential function interaction. We explicitly show that the minimal length would have some importance in studying the spectra of di…
