WKB Eigenvalues Research Articles

Improving the accuracy of WKB and Langer eigenvalues by higher-order variational method

Higher-order variational method is adapted for improving the accuracy of WKB and Langer eigenvalues by using the WKB and Langer eigenfunctions as trial functions. The results are compared with the first-order variational method using WKB and Langer functions as trial functions and with other approximate methods (finite element method, Langer, first-, third- and fifth-order WKB, first- and third-order WKB variational, first-order perturbation correction).

WKB Level Spectra in Two Inhomogeneous Electron Liquids: Na Clusters and C Cages

After a brief summary of known magic numbers and eigenvalue sums for (a) the hydrogenic potential—Ze 2/r and (b) the three-dimensional isotropic harmonic oscillator potential, approximate semi-classical scaling laws are presented relating to WKB eigenvalues for two more complex central potentials V(r) The first is the so-called Woods-Saxon potential, used in early work to calculate electronic magic numbers in clusters of Na atoms. The second potential V(r) chosen arises from a simple surface charge model of almost-spherical carbon cages such as C60. For these last two potentials, semi-classical theory is shown to lead to qualitative insight, without very lengthy mathematical calculations.

WKB solution of the wave equation for a plane angular sector

The two angular Lamedifferential equations that satisfy boundary conditions on a plane angular sector ~PAS! are solved by the Wentzel-Kramers-Brillouin ~WKB! method. The WKB phase constants are derived by matching the WKB solution with the asymptotic solution of the Weber equation. The WKB eigenvalues and eigenfunctions show excellent agreement with the exact eigenvalues and eigenfunctions. It is shown that those WKB eigenvalues and eigenfunctions that contribute substantially to the scattering amplitude from a PAS can be computed in a rather simple way. An approximate formula for the WKB normalization constant, which is consistent with the WKB assumptions, is derived and compared with the exact normalization constant. @S1063-651X~98!00507-8# PACS number~s!: 42.25.2p I. INTRODUCTION In a previous paper @1#, formulas for the Wentzel- Kramers-Brillouin ~WKB! eigenvalues satisfying Dirichlet or Neumann boundary condition on a plane angular sector ~PAS! were reported and the WKB eigenvalues, $n,m%, were compared with the exact eigenvalues for PAS's of different corner angles ~60°, 90°, and 120° !. A historical review of the solution of the wave equation for a PAS was also given in the above paper. It suffices to say that, to our knowledge, no approximate solution of this problem has been reported in the literature. In this paper, the two coupled Lameequations are solved by the WKB method. The WKB analysis in this paper is valid for large values of n. Depending on the sign of m, one of the two Lameequations has turning points. When umu is small the turning points occur where the angles are small. In this region it proves more accurate to obtain the WKB phase constants by matching it with the asymptotic solution of the Weber equation. For large values of umu, it is shown that this phase constant reduces to the phase constant obtained when the solution is matched with the asymptotic solution of the Airy's equation as is commonly done in quantum mechanics. This paper is organized in the following way: In the sec- ond section the WKB solution is formulated and the WKB phase constants are derived. In the third section formulas for the WKB eigenvalues for Dirichlet and Neumann boundary conditions are derived and a comparison between the WKB and exact eigenfunctions is presented. The fourth section contains a derivation of approximate WKB solutions which are valid for small values of m/n. Finally, an approximate formula for the WKB normalization constant consistent with WKB assumptions is derived in section five.

The second-order WKB approach to electron thermalization in gases

The second-order WKB quantization rule is applied to the determination of the eigenvalues associated with electron thermalization in gases. It is shown that a few percent error in the WKB eigenvalues can be reduced by the second-order correction to less than 0.1% in the case of the electron thermalization in hard-sphere and helium gases with zero electric field. A large error of −116% is drastically reduced to 5% for the lowest eigenvalue associated with electron attachment to SF 6 in He. The usefulness of the second-order WKB correction is thus confirmed.

Semiclassical theory of spin-orbit coupling.

Newly developed methods for finding the semiclassical or WKB eigenvalues and eigenfunctions for vector wave fields are applied to spinning particles moving in central potentials subject to spin-orbit forces. The new methods are like the familiar Bohr-Sommerfeld or Einstein-Brillouin-Keller quantization methods for scalar wave fields, but involve additional issues relating to Berry's phase, gauge structures, and monopolelike singularities. All these extra issues occur in the asymptotics of spin-orbit coupling. The role of angular momentum and conservation laws is particularly interesting, due to the role played by gauge fields in the classical-quantum correspondence. The work presented is a contribution to ongoing semiclassical studies of shell structure in nuclei and other fermion systems such as metal clusters, as well as an example of the general methods of semiclassical vector wave quantization.

Improving the accuracy of WKB eigenvalues

A simple method is proposed for improving the accuracy of WKB eigenvalues by using the WKB eigenfunctions as trial functions in a variational-principle expression for the eigenvalues. The first-order eigenvalues obtained from this estimate are shown to differ from the exact values by terms of order ε6 (where ε is an appropriately defined small parameter that specifies the accuracy of the WKB approximation). For comparison, the fifth-order WKB eigenvalues also differ from the exact values by terms of order ε6. Higher-order variational eigenvalues are also defined, and the third-order estimate is shown to differ from the exact by terms of order ε10. The accuracy of these variational and WKB estimates are illustrated with a numerical example.

Modified semiclassical quantization conditions for double minimum potentials: Applications to anharmonic oscillators.

WKB-type connection formulas, established by Miller and Good and based on the quadratic mapping of a section of a potential containing a local minimum, are used to obtain ``modified-well'' quantization conditions for general double minimum potentials. Significant improvements in the usual WKB eigenvalues may be achieved by combining these with the corresponding ``modified-barrier'' rules. This is shown to be true in the context of the anharmonic oscillator potentials V(x)=-${\mathrm{kx}}^{2}$+\ensuremath{\lambda}${x}^{4}$ (k,\ensuremath{\lambda}>0) for a wide range of \ensuremath{\lambda}. The characteristic features of the different modified formalisms are also discussed.

Characterization of exit-channel barriers for chemical reactions producing specific vibrational states

Use of the reaction-path Hamiltonian in natural collision coordinates with WKB eigenvalues for stretching vibrations is tested for predicting state-specific vibrationally adiabatic barriers for the fully three-dimensional reaction F+D2→FD (n′=4)+D. A range of potential energy surfaces, including one specifically designed to have no delayed threshold for the highest energetically allowed product vibrational state, are considered, and both approximate quantal scattering calculations and quasiclassical trajectory calculations are performed for comparison to the adiabatic predictions. We find that the adiabatic predictions correlate well with the quantal scattering results. We also study other features of the dynamics on the surface with no delayed vibrational threshold to see how changing the surface to eliminate the delayed threshold affects other reaction attributes. We find that the approximate quantal differential cross sections for this surface show an appreciable probability for forward scattering of the products. This trend is not exhibited by the quasiclassical trajectory calculations for the same surface.

On the asymptotic eigenvalues of an inhomogeneous circular waveguide

An asymptotic formula is presented by means of which WKB's eigenvalues of a circular waveguide can be corrected in analytic form. Maslov's formula derived in 1960 for the determination of Schrödinger's eigenvalues is given for comparison. Both results are shown to be in complete agreement in a specific case.

Upper and Lower Bounds of WKB Eigenvalues

Upper and Lower Bounds of WKB Eigenvalues

Approximate Analytic Eigenvalues of Lennard–Jones Potentials

Approximate analytic expressions of eigenvalues for Lennard–Jones (10,6) and (12,6) potentials are obtained. The eigenvalues obtained from such expressions are compared with the WKB or numerical eigenvalues for the estimation of accuracy.

Effects of Finite Boundaries on a One-Dimensional Harmonic Oscillator

In the study of the effects of finite boundaries on the magnetic properties of a solid, one encounters the problem of finding the energy eigenvalues of a one-dimensional linear harmonic oscillator located in a potential enclosure. The WKB method is employed to solve this problem for an oscillator whose center is fixed at some arbitrary position inside a potential box. Both numerical and analytical approximations of the WKB method are employed to find the energy eigenvalues over a wide range of the parameters of the problem. Numerical methods are also used on the exact series solutions of a bound oscillator to find the exact eigenvalues for the first few quantum states. A comparison of the two methods show that, in general, the WKB eigenvalues are accurate to much better than 1% except when the classical turning points are near the wall. Here the difference is of the order of 10% for the worst possible cases. The eigenvalues of a bound oscillator are shown to reduce to unbound-oscillator energy eigenvalues if the classical turning points are inside the potential enclosure and not near the walls. At the other extreme the eigenvalues are shown to become plane wave box eigenvalues when the separation of the classical turning points is large compared to the size of the enclosure. Also included is the relationship of the WKB solution to the Bohr-Sommerfeld quantization rule for the bound oscillator.

Use of the WKB Method for Obtaining Energy Eigenvalues

A simple, general method is derived for evaluating the second- and third-order WKB energy integrals by rewriting the integrals having nonintegrable singularities in terms of derivatives, with respect to the energy, of integrals having integrable singularities. As an example, it is shown that the higher-order WKB integrals vanish for the one-dimensional linear harmonic oscillator. A calculation of some eigenvalues using this method is made for potentials of the form V(x)=λx2ν and the results are compared to the ``exact'' results obtained from a numerical integration of the Schrödinger equation. It is observed that inclusion of the third-order integral improves the accuracy of WKB eigenvalues.