Anharmonic Oscillator Potential Research Articles

Quartic oscillator potential in the γ-rigid regime of the collective geometrical model

A prolate γ-rigid version of the Bohr-Mottelson Hamiltonian with a quartic anharmonic oscillator potential in β collective shape variable is used to describe the spectra for a variety of vibrational-like nuclei. Speculating the exact separation between the two Euler angles and the β variable, one arrives at a differential Schrödinger equation with a quartic anharmonic oscillator potential and a centrifugal-like barrier. The corresponding eigenvalue is approximated by an analytical formula depending only on a single parameter up to an overall scaling factor. The applicability of the model is discussed in connection to the existence interval of the free parameter, which is limited by the accuracy of the approximation, and by comparison with the predictions of the related X(3) and X(3)-β 2 models. The model is applied to qualitatively describe the spectra for nine nuclei which exhibit near-vibrational features.

Construction of the Barut–Girardello quasi coherent states for the Morse potential

The Morse oscillator (MO) potential occupies a privileged place among the anharmonic oscillator potentials due to its applications in quantum mechanics to diatomic or polyatomic molecules, spectroscopy and so on. For this potential some kinds of coherent states (especially of the Klauder–Perelomov and Gazeau–Klauder kinds) have been constructed previously. In this paper we construct the coherent states of the Barut–Girardello kind (BG-CSs) for the MO potential, which have received less attention in the scientific literature. We obtain these CSs and demonstrate that they fulfil all conditions required by the coherent state. The Mandel parameter for the pure BG-CSs and Husimi’s and P-quasi distribution functions (for the mixed-thermal states) are also presented. Finally, we show that all obtained results for the BG-CSs of MO tend, in the harmonic limit, to the corresponding results for the coherent states of the one dimensional harmonic oscillator (CSs for the HO-1D).

Solution of the Dirac equation in the tridiagonal representation with pseudospin symmetry for an anharmonic oscillator and electric dipole ring-shaped potential

An anharmonic oscillatory potential is proposed in which a noncentral electric dipole is included. The pseudospin symmetry for this potential is investigated by working in a complete square integrable basis that supports a tridiagonal matrix representation of the wave operator. The resulting three-term recursion relation for the expansion coefficients of the wavefunctions (both angular and radial) are presented. The angular/radial wavefunction is written in terms of Jacobi/Laguerre polynomials. The discrete spectrum of the bound states is obtained by the diagonalization of the radial recursion relation. The algebraic properties of the energy equation are also discussed, showing the exact pseudospin symmetry.

Solution of Dirac equation with spin and pseudospin symmetry for an anharmonic oscillator

We present the exact solutions of Dirac equation with anharmonic oscillator potential using the Nikiforov–Uvarov method. Taking into account potentials of vector field V(r) and scalar field S(r) in Dirac Hamiltonian, the bound state energy eigenvalues and the corresponding upper and lower two-component spinors of fermion are obtained. These solutions are considered in the framework of the spin and pseudospin symmetry concept.

Energy eigenvalues from an analytical transfer matrix method

A detailed procedure based on an analytical transfer matrix method is presented to solve bound-state problems. The derivation is strict and complete. The energy eigenvalues for an arbitrary one-dimensional potential can be obtained by the method. The anharmonic oscillator potential and the rational potential are two important examples. Checked by numerical techniques, the results for the two potentials by the present method are proven to be exact and reliable.

Bound state solution of the Dirac equation for a new anharmonic oscillator potential**Project supported by the National Natural Science Foundation of China (grant nos 10347003 and 60666001), planned training excellent Scientific and Technological Youth Foundation of Guizhou province, Peoples Republic of China (grant nos 2002 and 2013), the Science Foundation of Guizhou province, Peoples Republic of China, and the creativity

It is shown that the Dirac equation for a new equal scalar and vector anharmonic oscillator potentials could be separated into a solvable angular equation and a radial equation. Corresponding exact solutions of bound states for the Dirac equation have been obtained. The angular solutions are the associated-Legendre polynomial and the radial solutions are expressed in terms of the confluent hypergeometric functions. Finally, the energy equation is found from the boundary condition satisfied by the radial wavefunction.

THE ASYMPTOTIC ITERATION METHOD FOR THE EIGENENERGIES OF THE ASYMMETRICAL QUANTUM ANHARMONIC OSCILLATOR POTENTIALS $V(x)=\sum_{j=2}^{2\alpha}A_{j}x^{j}$

The asymptotic iteration method is used to calculate the eigenenergies for the asymmetrical quantum anharmonic oscillator potentials [Formula: see text], with (α = 2) for quartic, and (α = 3) for sextic asymmetrical quantum anharmonic oscillators. An adjustable parameter β is introduced in the method to improve its rate of convergence. Comparing the present results with the exact numerical values, and with the numerical results of the earlier works, it is found that asymptotically, this method gives accurate results over the full range of parameter values Aj.

Exact polynomial eigensolutions of the Schrödinger equation for the pseudoharmonic potential

The polynomial solution of the Schrödinger equation for the Pseudoharmonic potential is found for any arbitrary angular momentum l. The exact bound-state energy eigenvalues and the corresponding eigenfunctions are analytically calculated. The energy states for several diatomic molecular systems are calculated numerically for various principal and angular quantum numbers. By a proper transformation, this problem is also solved very simply by using the known eigensolutions of anharmonic oscillator potential.

SOLUTION FOR THE EIGENENERGIES OF SEXTIC ANHARMONIC OSCILLATOR POTENTIAL V(x)=A6x6+A4x4+A2x2

In this paper a study of the sextic anharmonic oscillator potential V(x)=A6x6+A4x4+A2x2 (A6≠0) using the asymptotic iteration method is presented. We calculate the eigenenergies for different excited states. The used method works very well for this potential and in fact one is able to obtain high accuracy with the asymptotic iteration method. A comparison between our results with other methods found in literature is presented.

Sextic anharmonic oscillators and orthogonal polynomials

Under certain constraints on the parameters a, b and c, it is known that Schroedinger's equation -y"(x)+(ax^6+bx^4+cx^2)y(x) = E y(x), a > 0, with the sextic anharmonic oscillator potential is exactly solvable. In this article we show that the exact wave function y is the generating function for a set of orthogonal polynomials P_n^{(t)}(x) in the energy variable E. Some of the properties of these polynomials are discussed in detail and our analysis reveals scaling and factorization properties that are central to quasi-exact solvability. We also prove that this set of orthogonal polynomials can be reduced,by means of a simple scaling transformation, to a remarkable class of orthogonal polynomials, P_n(E)=P_n^{(0)}(E) recently discovered by Bender and Dunne.

Unified Treatment of Screening Coulomb and AnharmonicOscillator Potentials in Arbitrary Dimensions

A mapping is obtained relating radial screened Coulomb systemswith low screening parameters to radial anharmonic oscillatorsin N-dimensional space. Using the formalism ofsupersymmetric quantum mechanics, it isshown that exact solutions of these potentials exist when theparameters satisfy certain constraints.

Exact Solutions of the Klein–Gordon Equation with a NewAnharmonic Oscillator Potential

We solve the Klein–Gordon equation with a new anharmonicoscillator potential and present the exact solutions. It is shown thatunder the condition of equal scalar and vector potentials, theKlein–Gordon equation could be separated into an angular equation anda radial equation. The angular solutions are the associated-Legendrepolynomial and the radial solutions are expressed in terms of theconfluent hypergeometric functions. Finally, the energy equation isobtained from the boundary condition satisfied by the radial wavefunctions.

The asymptotic iteration method for the eigenenergies of the anharmonic oscillator potential [formula omitted

The asymptotic iteration method is used to calculate the eigenenergies for the anharmonic oscillator potentials V ( x ) = A x 2 α + B x 2 ( A > 0 , B < 0 ), with ( α = 2 ) for quartic, and ( α = 3 ) for sextic anharmonic oscillators. An adjustable parameter β is introduced in the method to improve its rate of convergence. Comparing the present results with the numerical values calculated by earlier workers, it is found that, asymptotically this method gives exact results over the full range of parameter values, A and B.

Exact solutions to the Schr?dinger equation for the anharmonic oscillator potential

The radial wave function of Schr?dinger equation for the anharmonic oscillator potential V(r)=ar2+br4+cr6 can be written in the form of a product of an exponential function and a polynomial function .The exact energy and wave function of the potential are obtained by using the relation for the coefficient of the polynomial function. In the bound states, the results show that parameters a,b and c in the model potential have to satisfy relevant restraint conditions.

Generalized infinite-dimensional Fresnel integrals

A generalized infinite dimensional oscillatory integral with a polynomially growing phase function is defined and explicitly computed in terms of an absolutely convergent Gaussian integral. The results are applied to the Feynman path integral representation for the solution of the Schrödinger equation with an anharmonic oscillator potential. To cite this article: S. Albeverio, S. Mazzucchi, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Polarizability and chemical hardness?A combined study of wave function and density functional theory approach

A flexible model for generating the molecular wave function and electron density for diatomic molecules is developed employing the quadratic anharmonic oscillator and Morse potential. The chemical hardness, Fukui function, and polarizability were calculated using the electron density of the molecules, and the values are found to be reasonably good. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001

The density matrix renormalization group applied to single-particle quantum mechanics

A simplified version of White's density matrix renormalization group (DMRG) algorithm has been used to find the ground state of the free particle on a tight-binding lattice. We generalize this algorithm to treat the tight-binding particle in an arbitrary potential and to find excited states. We thereby solve a discretized version of the single-particle Schrödinger equation, which we can then take to the continuum limit. This allows us to obtain very accurate results for the lowest energy levels of the quantum harmonic oscillator, anharmonic oscillator and double-well potential. We compare the DMRG results thus obtained with those achieved by other methods.

Sturmian basis functions for the harmonic oscillator

We define Sturmian basis functions for the harmonic oscillator and investigate whether recent insights into Sturmians for Coulomb-like potentials can be extended to this important potential. We also treat many body problems such as coupling to a bath of harmonic oscillators. Comments on coupled oscillators and time-dependent potentials are also made. It is argued that the Sturmian method amounts to a non-perturbative calculation of the energy levels, but the limitations of the method is also pointed out, and the cause of this limitation is found to be related to the divergence of the potential. Thus the divergent nature of the anharmonic potential leads to the Sturmian method being less acurate than in the Coulomb case. We discuss how modified anharmonic oscillator potentials, which are well behaved at infinity, leads to a rapidly converging Sturmian approximation.

Superrevivals in the quantum dynamics of a particle confined in a finite square-well potential

We examine the revival features in wave packet dynamics of a particle confined in a finite square well potential. The possibility of tunneling modifies the revival pattern as compared to an infinite square well potential. We study the dependence of the revival times on the depth of the square well and predict the existence of superrevivals. The nature of these superrevivals is compared with similar features seen in the dynamics of wavepackets in an anharmonic oscillator potential.

On the equivalence of moment quantization and continuous wavelet transform analysis

The space of polynomials maps onto itself under affine transformations, . This suggests that a moment reformulation of continuous wavelet transform (CWT) theory (the affine convolution, , of a signal, or wavefunction, ) should lead to significant simplifications in its implementation. We present a comprehensive formalism, with numerical examples, that inextricably links moment quantization (MQ) and CWT theory. For rational fraction potential problems and mother wavelets of the form (Q(x) an appropriate polynomial), MQ permits a more efficient and accurate (in a pointwise convergent sense) CWT implementation; whereas, CWT broadens the scope of applicability for MQ methods, and is its natural extension when a more global approximation is desired. Our formalism also gives one justification for the empirical superiority manifested by previous MQ studies, as compared with dyadic wavelet reconstruction methods. We implement our formalism in the context of the quartic, sextic and octic anharmonic oscillator potentials, and demonstrate the flexibility of the method by treating both the Mexican hat wavelet transform, as well as that based on the mother wavelet .