WKB Treatment Research Articles

Multiple scattering of matter waves: An analytic model of the refractive index for atomic and molecular gases

We present an analytic model of the refractive index for matter waves propagating through atomic or molecular gases. The model, which combines a WKB treatment of the long range attraction with the Fraunhofer model treatment of the short range repulsion, furnishes a refractive index in compelling agreement with recent experiments of Jacquey et al. [Phys. Rev. Lett. 98, 240405 (2007)] on Li atom matter waves passing through dilute noble gases. We show that the diffractive contribution, which arises from scattering by a two dimensional "hard core" of the potential, is essential for obtaining a correct imaginary part of the refractive index.

Stokes geometry for the quantum Hénon map

The Stokes geometry for the propagator of the quantum Hénon map is studied. Virtual turning points and new Stokes curves, which have been proposed as entirely new notions appearing only in higher-order differential equations, are defined through deriving a set of differential equations acting on the propagator. A concrete recipe to construct Stokes geometry is presented and its validity is examined using three independent methods. The tree-pruning rule, a phenomenological scheme to cope with the complex WKB description of chaotic systems, is also examined in the light of the exact WKB treatment of the Stokes phenomenon.

The Herman–Kluk approximation: Derivation and semiclassical corrections

The Herman–Kluk (HK) approximation for the propagator is derived semiclassically for a multidimensional system as an asymptotic solution of the Schrödinger equation. The propagator is obtained in the form of an expansion in ℏ, in which the lowest-order term is the HK formula. Thus, the result extends the HK approximation to higher orders in ℏ. Examination of the various terms shows that the expansion is a uniform asymptotic series and establishes the HK formula as a uniform semiclassical approximation. Successive terms in the series should allow one to improve the accuracy of the HK approximation for small ℏ in a systematic and purely semiclassical manner, analogous to a higher-order WKB treatment of time-independent wave functions.

Threshold properties of attractive and repulsive1/r2potentials

We study the near-threshold $(\stackrel{\ensuremath{\rightarrow}}{E}0)$ behavior of quantum systems described by an attractive or repulsive ${1/r}^{2}$ potential in conjunction with a shorter-ranged ${1/r}^{m}$ $(m>2)$ term in the potential tail. For an attractive ${1/r}^{2}$ potential supporting an infinite dipole series of bound states, we derive an explicit expression for the threshold value of the pre-exponential factor determining the absolute positions of the bound-state energies. For potentials consisting entirely of the attractive ${1/r}^{2}$ term and a repulsive ${1/r}^{m}$ term, the exact expression for this prefactor is given analytically. For a potential barrier formed by a repulsive ${1/r}^{2}$ term (e.g., the centrifugal potential) and an attractive ${1/r}^{m}$ term, we derive the leading near-threshold behavior of the transmission probability through the barrier analytically. The conventional treatment based on the WKB formula for the tunneling probability and the Langer modification of the potential yields the right energy dependence, but the absolute values of the near-threshold transmission probabilities are overestimated by a factor which depends on the strength of the ${1/r}^{2}$ term (i.e., on the angular momentum quantum number $l)$ and on the power m of the shorter ranged ${1/r}^{m}$ term. We derive a lower bound for this factor. It approaches unity for large l, but it can become arbitrarily large for fixed l and large values of m. For the realistic example $l=1$ and $m=6$, the conventional WKB treatment overestimates the exact near-threshold transmission probabilities by at least 38%.

Discrete spectrum of the graviton in the AdS5 black hole background

The discrete spectrum of fluctuations of the metric about an AdS5 black hole background are found. These modes are the strong coupling limit of so-called glueball states in a dual 3-d Yang–Mills theory with quantum numbers JPC=2++,1−+,0++. For the ground state modes, we find the mass relation m(0++)<m(2++)<m(1−+). Contrary to expectation, the mass of our new 0++ state (m2=5.4573) associated with the graviton is smaller than the mass of the 0++ state (m2=11.588) from the dilaton. In fact the dilatonic excitations are exactly degenerate with our tensor 2++ states. We find that variational methods gives remarkably accurate mass estimates for all three low-lying levels while a WKB treatment describes the higher modes well.

Dispersion studies of surface-wave-produced discharges with a view to diagnostics

Dispersion of surface-wave-produced plasmas and field behaviour outside the discharges are studied under the influence of collisions and radial and axial density inhomogeneities, the latter density, both in WKB and eigenfunction treatment, is carried through for a cylindrical discharge configuration. The trends of joint influences can be approximated by an extrapolation from WKB treatment, avoiding the large numerical effort of an accurate two-dimensional code. Some consequences for radiophysics diagnostics are pointed out. Whereas standard methods of phase measurements are only weakly affected, amplitude measurements are more subject to various influences. Therefore, potential extensions of methods using test waves of various frequencies can yield additional and improved information on the collision frequency and on the value of the radial inhomogeneity index. A procedure is suggested and exemplified with measurements at a helium discharge.

Airy function - exact WKB results for potentials of odd degree

An exact WKB treatment of 1-d homogeneous Schr\"odinger operators (with the confining potentials $q^N$, $N$ even) is extended to odd degrees $N$. The resulting formalism is first illustrated theoretically and numerically upon the spectrum of the cubic oscillator (potential $|q|^3$). Concerning the linear potential (N=1), the theory exhibits a duality in which the Airy functions Ai, Ai' become paired with the spectral determinants of the quartic oscillator (N=4). Classic identities for the Airy function, as well as some less familiar ones, appear in this new perspective as special cases in a general setting.

Correlation energies for two interacting electrons in a harmonic quantum dot

The problem of correlated electrons in a quantum dot is interesting and exciting. Their confinement within a harmonic potential provides not only a needed environment but also a unique signature to the physics of the dynamic correlation. The energy level structure of the relative motion of two electrons in such a quantum dot is studied in detail. Simple but accurate analytic expressions for the correlation energy are found by a double-parabola approximation in a WKB treatment, whose results are shown to be in excellent agreement with those of exact numerical solutions. Based on the analytical study, however, a clear physical picture emerges. It is found that the internal energy levels of a given angular momentum $m$ can be likened to a ladder of nearly equally spaced energy steps placed on a pedestal. The height of the pedestal is given by ${V}_{\mathrm{min}}(m,\ensuremath{\lambda}),$ the minimum of the effective interaction potential associated with a given $m$ and the Coulomb coupling strength \ensuremath{\lambda} relative to the confinement. Superimposing ladders of all possible values of $m$ then yields the entire level structure for the correlated relative motion. Owing to the unique nature of harmonic confinement, correlation thus manifests itself mostly through the \ensuremath{\lambda} dependence of the classical-like ${V}_{\mathrm{min}}(m,\ensuremath{\lambda}),$ in which the wave nature of the electrons plays no role except for the discrete values of $m.$

A Method of Analysis of Vibration–Rotational and Rotational Spectra of Light Diatomic Molecules

The potential model of G. Thompsonet al.[J. Mol. Spectrosc.124,130–138 (1987)] for an analysis of the vibrational–rotational and the rotational spectra of diatomic molecules is modified by an algebraic WKB treatment of the Schrödinger equation given by Watson's effective Hamiltonian. A compact expression is given for the contributions from the adiabatic and nonadiabatic correction terms in the Hamiltonian to the Dunham'sYijcoefficients. The present model is applied to the reported vibrational–rotational and rotational spectra of LiH as a typical light diatomic molecule where effects of the diabatic and nonadiabatic corrections are significant, and all spectral lines of all isotopomers are fitted to a single set of molecular constants simultaneously well within experimental errors. The analytical expressions for Δa,bωand Δa,bBobtained in the present study provide the basis for the validity of the potential model of Thompsonet al.for molecules composed of heavier atoms.

Tight binding electrons on a disordered square lattice in a magnetic field

A semiclassical WKB treatment of the density of states spectrum of tight-binding electrons moving in a disordered two dimensional lattice in the presence of a transverse magnetic field is presented. The disorder is accounted for in the coherent potential approximation and analytical results are derived. For both ordered and disordered systems the line position of magnetic subbands as well as the cluster lineshape of the density of states agree quite well with exact numerical results.

Influence of external potentials on optical Ramsey fringes in atomic interferometry

The interaction of an atomic beam with four traveling-wave laser beams leads to optically induced Ramsey fringes in the atomic beam. This experimental setup has been used as an optical frequency standard as well as a matter-wave interferometer. We calculate for two-level atoms the shift of the fringes due to the influence of fairly general external forces in lowest order of a perturbation approach. A comparison with the phase shift found in a corresponding WKB treatment of the interaction geometry is made. The consequences of beam reversal and cooling of the atomic beam are studied. As an application, we treat the linear acceleration and rotation of the interferometer and discuss the influence of space-time curvature.

A multiple‐transition‐point WKB investigation of complex energy resonances

AbstractApproximate conditions determining the complex energy resonances of a one‐dimensional model potential, originally discussed by Moiseyev et al., are derived within a multiple‐transition‐point WKB treatment. It is shown that an erroneous energy spectrum is generated if too few transition points are considered. The numerical results of a correct semiclassical treatment of the problem, where several transition points are included, agree well with the, presumably exact, results of Rittby, Elander, and Brändas. © 1993 John Wiley &amp; Sons, Inc.

Two- and three-point functions in the D = 1 matrix model

The critical behavior of the genus-zero two-point function in the D = 1 matrix model is carefully analyzed for arbitrary embedding-space momentum. Kostov's result is recovered for momenta below a certain value P 0 (which is 1/√α′ in the continuum language), with a non-universal form factor which is expressed simply in terms of the critical fermion trajectory. For momenta above P 0, the Kostov scaling term is found to be subdominant. We then extend the large- N WKB treatment to calculate the genus-zero three-point function, and elucidate its critical behavior when all momenta are below P 0. The resulting universal scaling behavior, as well as the non-universal form factor for the three-point function, are related to the two-point functions of the individual external momenta, through the factorization familiar from continuum conformal field theories.

Rossby radii and phase speeds in the Baltic Sea

On the basis of a data set recorded during seasonal cruises of the R.V.s A. v. Humboldt and Prof. A. Penck of the Academy of Sciences, G.D.R., in the Baltic Sea during 1977–1987, mean Brunt—Väisäläfrequency profiles were derived in order to compute vertical eigenvalues and internal, or baroclinic, Rossby radii for different seasons in various parts of the Baltic Sea. Due to regional variations of stratification and the depth of the different basins, these parameters vary considerably. Moreover, the Rossby radii show a seasonal cycle with minimum values during the winter and autumn, and maxima during summer. The largest Rossby radius, 7 km, was found in the Bornholm Basin and the smallest ones, 1.3 km, in the Belt Sea and in the Gulf of Finland during autumn. Approximated Rossby radii, as obtained by a WKB treatment, appear to be too small by about 10–30%.

Dendritic growth rate at arbitrary undercooling.

We study the growth rate of a dendritic crystal in two and three dimensions in the symmetric model of solidification. We clarify the question of the variation of the ``stability constant'' with the P\'eclet number in the regime of moderate to large values. A careful WKB treatment shows that the eigenvalues selected by anisotropy of surface tension are strongly P\'eclet-number dependent in contradiction with some earlier treatments. We also perform a numerical analysis of this problem and our results are compared with recent experimental data.

Polarization and energy evolution of electromagnetic waves in sheared toroidal plasmas

The evolution, through a toroidal magnetoactive plasma of the polarization of an electromagnetic wave and of the energy of the modes into which it splits, is analyzed here under the condition of negligible geometric refraction, also taking into account the possible wave absorption as a result of electron cyclotron resonances. The problem is reduced to the solution of a simple set of first-order differential equations, avoiding both the complexity of the full Maxwell system and the limitations of a WKB treatment. This differential set is solved numerically, and its results are presented and discussed.

Analytic theory of the Saffman-Taylor fingers.

A complete analytic solution is presented for the selection of symmetric Saffman-Taylor fingers, in the small--surface-tension limit. In a previous paper [Combescot et al., Phys. Rev. Lett. 56, 2036 (1986)] we showed that the selection can be understood in terms of a nonlinear eigenvalue problem that describes the region where ordinary perturbation theory fails to converge. This ``inner'' problem is completely solved by systematically studying the asymptotic series of the (formal) solution of the problem. Two methods are proposed to extract the selection mechanism from this asymptotic series. Numerical results agree very well with other existing numerical results, and, somewhat surprisingly, with the results that we obtained by a WKB solution of the inner problem. Last, we complete our WKB treatment by determining explicitly the additive constant appearing in the WKB formula.

Dendritic growth in a channel.

We study the growth of dendritic crystals from a supersaturated solution in a channel geometry. This model provides a continuous interpolation between the Saffman-Taylor problem and the (two-dimensional) free-space dendrite. We derive an integral equation for the shape of steady-state propagating fingers, which we treat by discretizing the curve and solving the resulting set of nonlinear algebraic equations by Newton's method. For general Peclet numbers, in the absence of anisotropy the finger widths are always greater than half the channel width, but with anisotropy all widths are obtained. The numerical results are in rough agreement with an approximate WKB treatment for small anisotropy and zero Peclet number. Finally, we speculate on the emergence of sidebranches in this system.

Multi-dimensional WKB treatment of first passage time

The problem of first passage time is mathematically equivalent to the tunnelling of a particle out of a potential well and hence can be treated by the multi-dimensional WKB technique. In general, the most probable tunnelling path (MPTP) is curved and is difficult to determine. The authors discuss a model potential for which the curved MPTP can be solved analytically. It is shown that the rate of tunnelling calculated along the straight path as usually assumed is unreliable. Numerical results of the mean switching time for the bistable two-mode laser and comparison with experiments and other theories are also discussed.

Adiabatic Viewpoint for the WKB Treatment of Coupled Channel System: Appearance of the Berry Phase and Another Extra Phase Accompanying the Adiabatic Motion

Adiabatic Viewpoint for the WKB Treatment of Coupled Channel System: Appearance of the Berry Phase and Another Extra Phase Accompanying the Adiabatic Motion