H. Taşeli

Exact solutions for vibrational levels of the Morse potential

The vibrational levels of diatomic molecules via Morse potentials are studied by means of the system confined in a spherical box of radius . It is shown that there exists a critical radius at which the spectrum of the usual unbounded system can be calculated to any desired accuracy. The results are compared with those of Morses classical solution which is based on the assumption that the domain of the internuclear distance r includes the unphysical region . By determining numerically exact lower and upper bounds for the energy eigenvalues of molecule, it is deduced here that Morses approach is perfect and gives very impressive results.

On the exact solution of the Schrödinger equation with a quartic anharmonicity

A new version of solutions in the form of an exponentially weighted power series is constructed for the two-dimensional circularly symmetric quartic oscillators, which reflects successfully the desired properties of the exact wave function. The regular series part is shown to be the solution of a transformed equation. The transformed equation is applicable to the one-dimensional problem as well. Moreover, the exact closed-form eigenfunctions of the harmonic oscillator can be reproduced as a special case of the present wave function. 0 1996 John Wiley & Sons, Inc.

Modified Laguerre Basis for Hydrogen-like Systems

N-dimensional Schrodinger equation with isotropic nonpolynomial ¨ perturbations is studied. A Laguerre basis, which is different from that of the hydrogen atom in nature, has been introduced and applied to screened Coulomb potentials. Certain very useful recurrence relations are developed for the evaluation of matrix elements analytically. Specimen eigenvalue calculations to illustrate the method as well as its extension to other potentials are presented. Q 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 949)959, 1997

A FOURIER-BESSEL EXPANSION FOR SOLVING RADIAL SCHRODINGER EQUATION IN TWO DIMENSIONS

The spectrum of the two-dimensional Schrodinger equation for polynomial oscillators bounded by infinitely high potentials, where the eigenvalue problem is defined on a finite interval r ∈ [0, L), is variationally studied. The wave function is expanded into a Fourier-Bessel series, and matrix elements in terms of integrals involving Bessel functions are evaluated analytically. Numerical results presented accurate to thirty digits show that, by the time L approaches a critical value, the low-lying state energies behave almost as if the potentials were unbounded. The method is applicable to multi-well oscillators as well.

Studies on algebraic methods to solve linear eigenvalue problems: generalised anharmonic oscillators

A detailed presentation of the recently introduced integration-free method, with applications to determine the energy levels of the generalised quantum anharmonic oscillators, are given. Numerical calculations are realised for the quartic and the sextic oscillators. Energy eigenvalues obtained for the ground state as well as for the first few excited states accurate to thirty digits are very impressive and demonstrate the efficiency of the method. Certain remarks about the selection of the basis functions and a convergence discussion on the presented simple approximation scheme are also included in this paper.

An Eigenfunction Expansion for the Schrödinger Equation with Arbitrary Non-Central Potentials

An eigenfunction expansion for the Schrödinger equation for a particle moving in an arbitrary non-central potential in the cylindrical polar coordinates is introduced, which reduces the partial differential equation to a system of coupled differential equations in the radial variable r. It is proved that such an orthogonal expansion of the wavefunction into the complete set of Chebyshev polynomials is uniformly convergent on any domain of (r,θ). As a benchmark application, the bound states calculations of the quartic oscillator show that both analytical and numerical implementations of the present method are quite satisfactory.

Accurate numerical bounds for the spectral points of singular Sturm-Liouville problems over 0 < x < ∞

The eigenvalues of singular Sturm-Liouville problems defined over the semi-infinite positive real axis are examined on a truncated interval 0 < x < l as functions of the boundary point l. As a basic theoretical result, it is shown that the eigenvalues of the truncated interval problems satisfying Dirichlet and Neumann boundary conditions provide, respectively, upper and lower bounds to the eigenvalues of the original problem. Moreover, the unperturbed system in a perturbation problem, where l remains sufficiently small, admits analytical solutions in terms of the Bessel functions of the first kind. Applications to the Schrodinger equations of diatomic molecules and a harmonic oscillator confirm the practical implementation of this approach in calculating highly accurate numerical eigenvalue enclosures. It is worth mentioning that this study is, therefore, a completion of the paper (J. Comput. Appl. Math. 115 (2000) 535) where similar problems on the whole real axis -∞ < x < ∞ were treated along the same lines.

The scaled Hermite-Weber basis still highly competitive

The effectiveness of the usual harmonic oscillator basis is demonstrated on a wide class of Schrödinger Hamiltonians with various spectral properties. More specifically, it is shown numerically that an appropriately scaled Hermite–Weber basis yields extremely accurate results not only for the energy eigenvalues which differ slighly from the harmonic oscillator levels, but also for the states which reflect a purely anharmonic character.

Bessel basis with applications: N-dimensional isotropic polynomial oscillators

The efficient technique of expanding the wave function into a Fourier–Bessel series to solve the radial Schrodinger equation with polynomial potentials, , in two dimensions is extended to N-dimensional space. It is shown that the spectra of two- and three-dimensional oscillators cover the spectra of the corresponding N-dimensional problems for N. Extremely accurate numerical results are presented for illustrative purposes. The connection between the eigenvalues of the general anharmonic oscillators and the confinement potentials of the form is also discussed.

An alternative series solution to the isotropic quartic oscillator inN dimensions

The series solution of theN-dimensional isotropic quartic oscillator weighted by an appropriate function which exhibits the correct asymptotic behavior of the wave function is presented. The numerical performance of the solution in Hills determinant picture is excellent, and yields the energy spectrum of the system to any desired accuracy for the full range of the coupling constant. Furthermore, it converges to the well-known exact solution of the unperturbed harmonic oscillator wave function, when the anharmonic interaction vanishes.