A new approximation method to obtain vibration eigenfunctions suitable for a new oscillator model

A new oscillator model has been proposed by introducing some modifications in the Morse potential function. Its efficacy is tested by taking a number of electronic states of diatomic molecules. For comparison the Hulbert-Hirschfelder model potential is also used. A new approximation method to find the vibrational eigenfunctions suitable for the new oscillator model has been reported. Langer’s method has been used to determine the wavefunctions. Franck-Condon factors andr-centroids are reported for the observed bands ofD ¹Π —X ¹Σ system of SnO molecule.     

An optimization analysis on the molecular vibrational energy eigenvalues of a diatomic gasdynamic laser

We study briefly the eigenvalue spectrum corresponding to the vibrational-energy levels of the molecules as anharmonic oscillators in a diatomic gasdynamic laser by calculating the maximum eigenvalue when all the possible diatomic configurations are considered within the context of gasdynamic lasers. In this context, a parameter which characterizes the anharmonicity of a given diatomic molecule as a one-dimensional oscillator is used in our calculations.     

Dynamical aspects of two-molecule collision process in a diatomic gasdynamic laser

A dynamical analysis on the basic collision process between two molecules is presented within the context of diatomic gasdynamic lasers. In particular, the kinetic energy of the colliding molecules resulting from the process above is considered for quite highly excited bound states from a quasi-classical point of view. In this analysis, the magnitude of molecule velocity is calculated and discussed in terms of the anharmonicity of a given molecule as a quantum anharmonic oscillator.     

Conditional stability of diatomic molecule driven by a weak laser field

Using a direct perturbation method, we investigate the stability of a diatomic molecule modelled by a weakly laser-driven Morse oscillator. It is shown that stationary state solution of the system is stable in the sense of Lyapunov and the periodical one possesses conditional stability, namely its stability depends on the initial conditions and system parameters. The corresponding sufficient and necessary conditions are established that indicate the stable states associated with some discrete energies. The results reveal how a diatomic molecule can be stabilized or dissociated with a weak laser, and demonstrate that the mathematical conditional stability works in the considered physical system.     

分子多光子共振吸收的理论研究

首先利用Floquet理论得到共振态的本征函数,再通过求解束缚态与连续态的耦合矩阵而得到束缚态的解离几率,最后得到任一束缚态通过耦合而解离的解离几率及衰变宽度等,利用我们得到的公式,对不转动的双原子分子及双原子分子模型进行了计算,在外场不是特别强的情况下,得到了一些很好的结果。     

Bound State Solutions of Schrödinger Equation for GeneralizedMorse Potential with Position-Dependent Mass

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 constant mass case. Energy eigenvalues are computed numerically for some diatomic molecules. They are in agreement with the ones obtained before.     

One Dimensional Models with a Singular Potential of the Type −<Emphasis Type="Italic">αδ</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)+<Emphasis Type="Italic">βδ</Emphasis>′(<Emphasis Type="Italic">x</Emphasis>)

We study a one-dimensional singular potential plus two types of regular interactions: constant electric field and harmonic oscillator. In order to search for the bound state energies, we shall use the Lippman-Schwinger Green function technique. Another direct method will be mentioned for the harmonic oscillator. In the electric field case the unique bound state coincides with that found in an earlier study as the field is switched off. For non-zero field the ground state is shifted and positive energy “quasibound states” appear. The harmonic oscillator demonstrates the general result that for a symmetric potential the odd states are not altered whereas the even states energies are lowered or raised accordingly as the delta perturbation is attractive or repulsive. No states are created or annihilated.     

An exactly solvable anharmonic Bohr hamiltonian and its equivalent boson hamiltonian

The eigenenergies and eigenfunctions of the bound states of an anharmonic quadrupole oscillator are derived in closed form. An interacting boson model hamiltonian is found which has the same eigenspectrum.     

Approximate solution of bound state problems through continued fractions

A method to solve ordinary linear differential equations through continued fractions is applied to several physical systems. In particular, results for the Schrödinger equation give a good accuracy for the eigenvalues of bound states in theS-wave Yukawa potential, and the lowest order approximations provide exact-values for the harmonic oscillator and Coulomb potential eigenvalues and eigenfunctions.     

Rovibrational Energies of Triatomic Molecules by Means of the Rayleigh-Schrödinger Perturbation Theory

A Rayleigh-Schr?dinger perturbation theory approach based on the adiabatic (Born-Oppenheimer) separation of vibrational motions was previously developed and used to evaluate for a system of coupled oscillators the adiabatic energy levels and their nonadiabatic corrections. This method is applied here to calculate rotation-vibration energies of the triatomic molecular ions HeH(+)(2) and ArNO(+) consisting of a strongly bound diatomic fragment and a relatively loosely bound rare gas atom. In these systems the high-frequency stretching motion of the diatomic fragment can be separated from the other two low-frequency motions without substantial loss of accuracy. Treating the diatomic fragment as a rigid rotor, the low-frequency stretching motion is decoupled from the bending motion in analogy to the concept of the adiabatic (Born-Oppenheimer) separation of motions and the strong nonadiabatic couplings between these two motions are accounted for perturbationally. Although the resulting perturbation series may show poor convergence, they turn out to be accurately summable by applying standard techniques for the summation of divergent series. Comparison with the results obtained from full-dimensional calculations for the two ions shows that the approach is capable of providing accurate energies for quite a few of the bound rotation-vibration states and that in the case of the HeH(+)(2) ion it is even able to predict the positions and widths of some low-lying resonance states with good accuracy. The perturbation approach yields zeroth-order energies and corrections in terms of the relevant quantum numbers. It thus allows a direct assignment of the energy levels without any reference to the corresponding eigenfunctions. The weak couplings between the high- and low-frequency motions can easily be treated by the same perturbative approach and numerically exact energies can finally be obtained. Copyright 2000 Academic Press.     

Bound States Energies of a Harmonic Oscillator Perturbed by Point Interactions

We determine explicitly the exact transcendental bound states energies equation for a one-dimensional harmonic oscillator perturbed by a single and a double point interactions via Green's function techniques using both momentum and position space representations. The even and odd solutions of the problem are discussed. The corresponding limiting cases are recovered. For the harmonic oscillator with a point interaction in more than one dimension,divergent series appear. We use to remove this divergence an exponential regulator and we obtain a transcendental equation for the energy bound states. The results obtained here are consistent with other investigations using different methods.     

Charge-monopole molecule and vanishing of the Schwinger string

As is well known [1, 2], the wave functions of charge scattering states in the magnetic monopole field are expanded in the eigenfunctions of symmetric quantum top rotation. The established direct relationship of the total momentum operators of these systems causes the Schwinger string to vanish and demonstrates that the charge and monopole system has the property of a diatomic molecule.     

Quantum motion over a finite one-dimensional domain: With and without dissipation

Quantum motion of a single particle over a finite one-dimensional spatial domain is considered for the generalized four parameter infinity of boundary conditions (GBC) of Carreauet al [1]. The boundary conditions permit complex eigenfunctions with nonzero current for discrete states. Explicit expressions are obtained for the eigenvalues and eigenfunctions. It is shown that these states go over to plane waves in the limit of the spatial domain becoming very large. Dissipation is introduced through Schrödinger-Langevin (SL) equation. The space and time parts of the SL equation are separated and the time part is solved exactly. The space part is converted to nonlinear ordinary differential equation. This is solved perturbatively consistent with the GBC. Various special cases are considered for illustrative purposes.     

A new analytic approximation method for the non-zero angular momentum states of the Hulthén potential

We propose a new approximation scheme to obtain analytic expressions for the bond-state energies and eigenfunctions for any arbitrary bound nl-state of the Hulthén potential. The predicted energies E_nl are in excellent agreement with the perturbative results of Lai and Lin. The scope for an extension of the method to the continuum states is also discussed.     

The generator coordinate representation in an natural state formalism

The generator coordinate method for bound states is reformulated using the classical Fredholm theory of linear integral equations. A natural state formalism is developed to investigate the properties of the variational Hill-Wheeler equations. A generator coordinate representation of the one-dimensional harmonic oscillator is worked out analytically.     

Analytical solution of the inverse spectroscopic problem: Determination of electron moments of vibrational transitions of diatomic molecules by their probabilities

The inverse problem of spectroscopy of vibrational transitions of diatomic molecules was analytically solved on the basis of wave functions of the Morse oscillator. An example of the determination of the transition dipole-moment function by intensities of several lines of molecule vibrational-rotational spectra measured experimentally was considered. A similar approach based on simple expressions for matrix elements can be applied to consider optimal lasing conditions of the CO electronic transition laser.     

Computationally efficient recurrence relations for one-dimensional Franck–Condon overlap integrals

Calculations based on analytical expressions for the harmonic oscillator Franck–Condon factors often yield numerically unstable and erroneous results for large values of the oscillator quantum numbers. This instability arises from inherent machine precision limits and large number round-off associated with the products and ratios of factorial and gamma functions in these expressions; the analytical expressions themselves are exact. This paper presents, first, efficient, exact recurrence relations to evaluate Franck–Condon factors for the harmonic oscillator model. The recurrence relations, which are similar to those originally found by Manneback, Wagner and Ansbacher avoid the direct use of the factorial and gamma functions. Second, a variational strategy for the evaluation of Franck–Condon factors for the Morse oscillator is proposed. The Schrödinger equation for the Morse model is solved variationally with a large enough basis set of one-dimensional harmonic oscillator functions to get good agreement with the analytic eigenvalues of the Morse potential itself. The eigenvectors of this analysis are then used together with the associated harmonic oscillator Franck–Condon overlap matrix elements to evaluate the overlap for the Morse potential. This approach allows one, in principle, to estimate Franck–Condon overlap up to states near to the dissociation limit of the Morse oscillator.     

Dynamically based fitting law for cross-sections of vibrational energy transfer

A three-dimensional semiclassical analytical model for cross-sections of vibrational energy transfer in collisions between an atom and a diatomic molecule has been developed. The model is based on the Bessel uniform approximation for transition probabilities valid for highly excited states of a molecule represented by the Morse oscillator. Three fitting parameters of the model are expressed in terms of the features characterizing the anisotropic intermolecular potential. The accuracy and validity of this law are tested by comparison with large Δn transitions and isotope effects in the crossed beam inelastic scattering of I₂ from He, H₂ and D₂.     

Topological modes in radiofrequency resonator arrays

Topological properties of solid states have sparked considerable recent interest due to their importance in the physics of lattices with a non-trivial basis and their potential in the design of novel materials. Here we describe an experimental and accompanying numerical toolbox to create and analyze topological states in radiofrequency resonator arrays including non-local coupling. These arrays are very easily constructed, offer a variety of geometric configurations, and their eigenfunctions and eigenvalues are amenable to detailed analysis. They offer well defined analogs to coupled oscillator systems in general in that they are characterized by resonances whose frequency spectra depend on the individual resonators, their interactions, and boundary conditions. A comparison of a small one-dimensional experimental system with theory by means of easy to measure S-parameters shows excellent agreement. The numerical toolbox allows for simulations of arbitrarily large systems, revealing an astonishing richness of band structures under systematic parameter variation.     

Studies on the 3D confined potentials using generalized pseudospectral approach

In presence of the spherically confined three-dimensional potentials with impenetrable boundaries, the generalized pseudospectral method is shown to provide accurate eigenvalues, eigenfunctions, and radial expectation values for (a) the isotropic harmonic oscillator, (b) the H atom and (c) the Davidson oscillator. Several novel degeneracy conditions are obtained for (a) when the radius of confinement is suitably chosen at the radial nodes corresponding to the free states.