Kratzer Potential Research Articles

Relativistic treatment of a spin-zero particle subject to a Kratzer-type potential

The Kratzer potential is commonly used for obtaining bound state solutions and describing the vibrational motion of a single particle. The Klein-Gordon equation for Kratzer-type potentials is solved systematically by using the Nikiforov-Uvarov method. The exact bound state energy eigenvalues and the corresponding wave functions of the Klein-Gordon equation are obtained with equal scalar and vector potentials.

Algebraic approach to the Kratzer potential

In this paper, the energy eigenvalues and the corresponding eigenfunctions are calculated for the Kratzer potential. Then we obtain the ladder operators for the one-dimensional (1D) and 3D Kratzer potential. Finally, we show that these operators satisfy the SU(2) commutation relation.

Exact analytical solutions to the Kratzer potential by the asymptotic iteration method

AbstractFor any n and l values, we present a simple exact analytical solution of the radial Schrödinger equation for the Kratzer potential within the framework of the asymptotic iteration method (AIM). The exact bound‐state energy eigenvalues (Enl) and corresponding eigenfunctions (Rnl) are calculated for various values of n and l quantum numbers for CO, NO, O2, and I2 diatomic molecules. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

D-dimensional generalization of the point canonical transformation for a quantum particle with position-dependent mass

The d-dimensional generalization of the point canonical transformation for a quantum particle endowed with a position-dependent mass in Schrodinger equation is described. Illustrative examples including; the harmonic oscillator, Coulomb, spiked harmonic, Kratzer, Morse oscillator, Poschl-Teller and Hulthen potentials are used as reference potentials to obtain exact energy eigenvalues and eigenfunctions for target potentials at different position-dependent mass settings.

On some classical and quantum effects due to gravitational fields

We consider the gravitational fields generated by a cosmic string, a global monopole and a tubular matter with interior magnetic field (Safko-Witten space-time), and examine some classical and quantum effects due to these fields. We investigate the Aharonov-Bohm effect in the space-time of a cosmic string, using the loop variables. In the space-time of a global monopole, we calculate the total energy radiated by a uniformly moving charged scalar particle, for small solid angle deficit. We show that the radiated energy is proportional to the cube of the velocity of the particle and to the cube of the Lorenz factor, in the non-relativistic and ultra-relativistic cases, respectively. In the Safko-Witten space-time, we investigate the existence of an electrostatic self-force on a charged particle. We also consider a hydrogen atom in the background space-time generated by a cosmic string and we find the solutions of the corresponding Dirac equation and we determine the energy levels of the atom. We investigate how the topological features of this space-time lead to shifts in the energy levels as compared with the flat Minkowski space-time. We study the behavior of non-relativistic quantum particles interacting with a Kratzer potential in the space-time generated by a global monopole and we find the energy spectrum in the presence of this topological defect. In the Safko-Witten space-time, an investigation is also made concerning the interaction of an harmonic oscillator with this background gravitational field.

Bound state solutions of the Schrödinger equation for modified Kratzer’s molecular potential

We present the arbitrary angular momentum solutions of the Schrödinger equation for a diatomic molecule with the modified Kratzer potential D e[( r − r e)/ r] 2. The bound state energy eigenvalues and the corresponding eigenfunctions are obtained by the use of the Nikiforov–Uvarov (NU) method which is related to the solutions in terms of Jacobi polynomials. The bound states are calculated for various values of ℓ with n ⩽ 5 and applied to several diatomic molecules. All of these calculations present an effective and methodological procedure under the NU method to investigate the Kratzer’s molecular systems.

Exact Solutions to Three-Dimensional Schrödinger Equation with anExponentially Position-Dependent Mass

For an exponentially position-dependent mass, we obtain the exactsolutions of the three-dimensional Schrödinger equation byusing coordinate transformation method for the reference problemswith Coulomb potential, Kratzer potential, and spherically squarepotential well of infinite depth, respectively. The explicitexpressions for the energy eigenvalues and the correspondingeigenfunctions of the three systems are presented.

Bounded solutions of fermions in the background of mixed vector–scalar inversely linear potentials

The problem of a fermion subject to a general mixing of vector and scalar potentials in a two-dimensional world is mapped into a Sturm–Liouville problem. Isolated bounded solutions are also searched. For the specific case of an inversely linear potential, which gives rise to an effective Kratzer potential in the Sturm–Liouville problem, exact bounded solutions are found in closed form. The case of a pure scalar potential with their isolated zero-energy solutions, already analyzed in a previous work, is obtained as a particular case. The behavior of the upper and lower components of the Dirac spinor is discussed in detail and some unusual results are revealed. The nonrelativistic limit of our results adds a new support to the conclusion that even-parity solutions to the nonrelativistic one-dimensional hydrogen atom do not exist.

Exact solution for a fermion in the background of a scalar inversely linear potential

The problem of a fermion subject to a general scalar potential in a two-dimensional world is mapped into a Sturm–Liouville problem for nonzero eigenenergies. The searching for possible bounded solutions is done in the circumstance of power-law potentials. The normalizable zero-eigenmode solutions are also searched. For the specific case of an inversely linear potential, which gives rise to an effective Kratzer potential, exact bounded solutions are found in closed form. The behaviour of the upper and lower components of the Dirac spinor is discussed in detail and some unusual results are revealed.

Integrals containing confluent hypergeometric functions with applications to perturbed singular potentials

We show that many integrals containing products of confluent hypergeometric functions follow directly from one single integral that has a very simple formula in terms of Appell's double series F2. We present some techniques for computing such series. Applications requiring the matrix elements of singular potentials and the perturbed Kratzer potential are presented.

Size of two-body weakly bound objects: short versus long range potentials.

The variation of the size of two-body objects is investigated, as the separation energy approaches zero, with both long range potentials and short range potentials having a repulsive core. It is shown that long range potentials can also give rise to very extended systems. Except for the l = 0 state, the asymptotic laws differ according to the range of the potential. For short range potentials defined by two and three parameters, their sensitivity to the shape and length is studied. These ideas as well as the transition from the short to the long range regime for the l = 0 case are illustrated using the Kratzer potential.

Bound states in the Kratzer plus polynomial potentials and the new form of perturbation theory

The Schrodinger equation with potentials of the Kratzer plus polynomial type (say, quartic V(r) = Ar 4 + Br 3 + Cr 2 + Dr + F/r + G/r 2 etc.) is considered and a new method of exact construction of some of its bound states is presented. Our approach is made feasible via a combination of the traditional use of the infinite series ψ(r)(terminated rigorously after N + 1 terms at certain specific couplings and energies) with several new ideas. We proceed in two steps. Firstly, in the strong coupling regime with G → ∞, we find the exact, complete and compact unperturbed solution of our N + 1 coupled and nonlinear algebraic conditions of the termination. Secondly, we adapt the current Rayleigh–Schrodinger perturbation theory to our nonlinear equations and define the general G < ∞ bound states via an innovated, triple perturbation series. In its tests we show how all the corrections appear in integer arithmetics and remain, therefore, exact.

A Closed Rovibrational Energy Formula Based on a Modified Kratzer Potential

The energy eigenvalue expression for a diatomic vib-rotor with the simple Kratzer potentialDe[(r−re)/r]2{xmathmode}approximates poorly most of the anharmonic and rotation–vibration interaction terms, as shown by its expansion in a Dunham-type series and comparison with experimentally determined DunhamYijcoefficients. To correct this behavior, the expression was modified by terms dependent on the vibrational and rotational quantum numbers built directly into the formula. The formula thus modified turns out to be a useful alternative to the Dunham expansion for not-too-long progressions or sequences of rovibrational bands as it provides a precision of fit better or at least comparable to the Dunham series with the same number of adjustable parameters. Application of this modified energy expression to individual bands (such as fundamental bands) is shown to yield accurate estimates for equilibrium rotational constantsBeand equilibrium internuclear distancesrewhich are in many cases better than values obtained from standard power series analyses of single bands. Useful estimates are also obtained for the zero-order vibrational frequencies ωeand for the positions of the lower overtone or combination bands. Such information is otherwise not accessible from a conventional analysis of an individual band.

1/ η expansion for quasistationary states: complex energies and complex angular momenta

The recently developed 1/ η expansion technique, where η is a dimensionless well depth parameter, for the study of bound states is extended to treat resonances. It is shown that the method, with a minimum of numerical computation, permits the accurate calculation of values of complex energies and of complex angular momentum quantum numbers (Regge pole positions) comparable with numerical and semiclassical results. It is shown that this technique can be considered as complementary to the dimensional scaling method and, in several cases studied in this article, allows even greater accuracy. For bound states the method involves minimizing an effective potential function and evaluating the exact energy when η→∞, then computing approximate results for η finite by performing a perturbation expansion in 1/ η about this limit. For resonances we obtain complex energies by allowing the coordinate to be complex. For energy resonances we study several systems which range from one-dimensional (symmetric and nonsymmetric Eckart barriers) to three-dimensional ones: inverted Kratzer potential, Lennard-Jones potential, and the H 2 molecule. For Regge poles we study a Lennard-Jones (12,6) and a Lennard-Jones (6,4) potentials, and a potential which describes the elastic scattering of an electron by Ar. For the Eckart potential barrier this method offers a simple way to evaluate Siegert complex eigenstates which are becoming important in Chemical Physics due to the recent association of transition states with resonances. In all the cases studied, comparison with results of numerical, exact, and semiclassical calculations show that the 1/ η expansion technique is capable of producing accurate values of complex energies and complex angular momentum quantum numbers.

Supersymmetric WKB and WKB phase shifts for the Coulomb potential, the inverse-square potential, and their combinations

In order to assess the accuracy of the supersymmetric WKB (SWKB) approximation for the calculation of phase shifts, the SWKB and WKB (with the Langer modification) phase shifts up to are compared numerically to the exact calculations for the Coulomb potential B/r, the inverse-square potential and their combinations , which include the Kratzer and inverted Kratzer potentials. For the inverse-square potential both the SWKB and the WKB methods yield the exact phase shifts. For the other two potentials considered, the results show that the SWKB phase shifts are in many cases superior; however, it is found that the WKB Coulomb phase shifts are more accurate for with . In addition, the excellent agreement of the SWKB phase shifts up to with the exact ones shows that the SWKB series expansion in powers of is rapidly convergent for the potentials considered.

5538024 Cleaning method and cleaning apparatus: Inada Minoru; Kabuki Kimiak; Imajo Yasutak; Yagi Noriaki; Saitoh Nobuhiro, Yokohama, Japan, assigned to Kabushiki Kaisha Toshiba; Japan Field Company Ltd

We investigate the exact solution of the Dirac equation for the Mie-type potentials under the conditions of pseudospin and spin symmetry limits. The bound state energy equations and the corresponding two-component spinor wave functions of the Dirac particles for the Mie-type potentials with pseudospin and spin symmetry are obtained. We use the asymptotic iteration method in the calculations. Closed forms of the energy eigenvalues are obtained for any spin–orbit coupling term κ. We also investigate the energy eigenvalues of the Dirac particles for the well-known Kratzer–Fues and modified Kratzer potentials which are Mie-type potentials.

On numerical solution of the Schrödinger equation: the shooting method revisited

An alternative formulation of the “shooting” method for a numerical solution of the Schrödinger equation is described for cases of general asymmetric one-dimensional potential (planar geometry), and spherically symmetric potential. The method relies on matching the asymptotic wavefunctions and the potential core region wavefunctions, in course of finding bound states energies. It is demonstrated in the examples of Morse and Kratzer potentials, where a high accuracy of the calculated eigenvalues is found, together with a considerable saving of the computation time.

Improved recursion formulas for the calculation of two-center central potential integrals

A commutator algebra procedure is used to get improved recurrence relations for the calculation of any f(r) off-diagonal two-center matrix elements in the general case of displaced arbitrary central potential wave functions. As expected, the proposed formulas reduce properly to the generalized recursion equations for the calculation of one-center integrals. Besides, when f(r) = rk, one obtains the equivalent of the Kramer rule for two-center multipolar matrix elements for arbitrary potentials, and when f(r) is constant, the Wu formula (for the calculation of Franck–Condon factors) is ameliorated in the sense that the new relation not necessarily considers equal mass. Furthermore, all diagonal matrix elements as well as all off-diagonal integrals between nondisplaced potentials appear, in our treatment, as particular cases in good agreement with already published results. As a useful application, the corresponding recurrence relations for the calculation of one-center hydrogenic matrix elements and two-center Kratzer potential integrals are given as examples. However, our approach is general and can be easily extended to obtain recursion formulae for other potential wave functions. © 1995 John Wiley & Sons, Inc.

Kratzer potential algebraic representation and matrix elements recurrence formulae

Generalized recurrence relations for the calculation of multipole matrix elements for Kratzer potential wave functions are obtained operationally. These formulas have been determined by using a non-analytical procedure based on the algebraic representation of the Kratzer eigenfunctions along with the usual ladder properties and commutation relations. For that, the creation and annihilation operators are adequately derived by means of an alternative approach to the factorization method and the exact expressions for matrix elements are achieved with the aid of a relationship between the ladder operators associated with the bra and theket. The proposed algebraic approach as well as the formulas for the calculation of matrix elements thus derived are quite simple and direct when compared with other alternative expressions already obtained analytically or pseudo-algebraically by means of the hypervirial theorem commutator algebra.

A vibrational Hamiltonian model for triatomic molecules based on the Kratzer and Poschl Teller potentials

AbstractA Hamiltonian model to describe molecular vibrations of triatomic molecules is proposed. The Hamiltonian is based on the use of the Kratzer potential variable for the stretching motions and a perturbed Poschl Teller potential for the bending one. The perturbation and variational treatments to compute the vibrational energies of this Hamiltonian can be developed using a zero‐order system that includes part of the couplings between the stretching and bending motions. All the matrix elements involved in these calclations can be then evaluated in closed form. A numerical application to the HCN molecule is made. © 1994 John Wiley &amp; Sons, Inc.