Jeffreys Wentzel Kramers Brillouin Research Articles

Semiclassical theory and the Koopman-van Hove equation * *Contribution to the Special Collection ‘Koopman Methods in Classical and Quantum-Classical Mechanics’.

The phase space Koopman-van Hove (KvH) equation can be derived from the asymptotic semiclassical analysis of partial differential equations. Semiclassical theory yields the Hamilton–Jacobi equation for the complex phase factor and the transport equation for the amplitude. These two equations can be combined to form a nonlinear semiclassical version of the KvH equation in configuration space. There is a natural injection of configuration space solutions into phase space and a natural projection of phase space solutions onto configuration space. Hence, every solution of the configuration space KvH equation satisfies both the semiclassical phase space KvH equation and the Hamilton–Jacobi constraint. For configuration space solutions, this constraint resolves the paradox that there are two different conserved densities in phase space. For integrable systems, the KvH spectrum is the Cartesian product of a classical and a semiclassical spectrum. If the classical spectrum is eliminated, then, with the correct choice of Jeffreys–Wentzel–Kramers–Brillouin (JWKB) matching conditions, the semiclassical spectrum satisfies the Einstein–Brillouin–Keller quantization conditions which include the correction due to the Maslov index. However, semiclassical analysis uses different choices for boundary conditions, continuity requirements, and the domain of definition. For example, use of the complex JWKB method allows for the treatment of tunneling through the complexification of phase space. Finally, although KvH wavefunctions include the possibility of interference effects, interference is not observable when all observables are approximated as local operators on phase space. Observing interference effects requires consideration of nonlocal operations, e.g. through higher orders in the asymptotic theory.

Near-core Acoustic Glitches Are Not Oscillatory: Consequences for Asteroseismic Probes of Convective Boundary Mixing

Asteroseismology has been used extensively in recent years to study the interior structure and physical processes of main-sequence stars. We consider prospects for using pressure modes (p-modes) near the frequency of maximum oscillation power to probe the structure of the near-core layers of main-sequence stars with convective cores by constructing stellar model tracks. Within our mass range of interest, the inner turning point of p-modes as determined by the Jeffreys–Wentzel–Kramers–Brillouin (JWKB) approximation evolves in two distinct phases during the main sequence, implying a sudden loss of near-core sensitivity during the discontinuous transition between the two phases. However, we also employ non-JWKB asymptotic analysis to derive a contrasting set of expressions for the effects that these structural properties will have on the mode frequencies, which do not encode any such transition. We show analytically that a sufficiently near-core perturbation to the stellar structure results in nonoscillatory, degree-dependent perturbations to the star’s oscillation mode frequencies, contrasting with the case of an outer glitch. We also demonstrate numerically that these near-core acoustic glitches exhibit strong angular degree dependence, even at low degree, agreeing with the non-JWKB analysis, rather than the degree-independent oscillations that emerge from JWKB analyses. These properties have important implications for using p-modes to study near-core mixing processes for intermediate-mass stars on the main sequence, as well as for the interpretation of near-center acoustic glitches in other astrophysical configurations, such as red giants.

Semianalytic Expressions for the Isolation and Coupling of Mixed Modes

Abstract In the oscillation spectra of giant stars, nonradial modes may be seen to undergo avoided crossings, which produce a characteristic “mode bumping” of the otherwise uniform asymptotic p- and g-mode patterns in their respective echelle diagrams. Avoided crossings evolve very quickly relative to typical observational errors and are therefore extremely useful in determining precise ages of stars, particularly in subgiants. This phenomenon is caused by coupling between modes in the p- and g-mode cavities that are near resonance with each other. Most theoretical analyses of the coupling between these mode cavities rely on the Jeffreys–Wentzel–Kramers–Brillouin approach, which is strictly speaking inapplicable to the low-order g modes observed in subgiants or the low-order p modes seen in very evolved red giants. We present both a nonasymptotic prescription for isolating the two mode cavities, as well as a perturbative (and also nonasymptotic) description of the coupling between them, which we show to hold good for the low-order g and p modes in these physical situations. Finally, we discuss how these results may be applied to modeling subgiant stars and determining their global properties from oscillation frequencies. We also make our code for all of these computations publicly available.

Seismic characterization of red giants going through the helium-core flash

Context. First-ascent red giants in the approximate mass range 0.7 ≲ M/M⊙ ≲ 2 ignite helium in their degenerate core as a flash. Stellar evolution codes predict that the He flash consists of a series of consecutive subflashes. Observational evidence of the existence of the He flash and subflashes is lacking. The detection of mixed modes in red giants from space missions CoRoT and Kepler has opened new opportunities to search for such evidence. Aims. During a subflash, the He-burning shell is convective, which splits the cavity of gravity modes in two. We here investigate how this additional cavity modifies the oscillation spectrum of the star. We also address the question of the detectability of the modes, to determine whether they could be used to seismically identify red giants passing through the He flash. Methods. We calculate the asymptotic mode frequencies of stellar models going through a He subflash using the Jeffreys-Wentzel-Kramers-Brillouin (JWKB) approximation. To predict the detectability of the modes, we estimate their expected heights, taking into account the effects of radiative damping in the core. Our results are then compared to the oscillation spectra obtained by numerically calculating the mode frequencies during a He subflash. Results. We show that during a He subflash, the detectable oscillation spectrum mainly consists of modes trapped in the acoustic cavity and in the outer g-mode cavity. The spectrum should thus at first sight resemble that of a core-helium-burning giant. However, we find a list of clear, detectable features that could enable us to identify red giants passing through a He subflash. In particular, during a He subflash, several modes that are trapped in the innermost g-mode cavity are expected to be detectable. We show that these modes could be identified by their frequencies or by their rotational splittings. Other features, such as the measured period spacing of gravity modes or the location of the H-burning shell within the g-mode cavity could also be used to identify stars going through a He subflash. Conclusions. The features derived in this study can now be searched for in the large datasets provided by the CoRoT and Kepler missions.

Calculations with experimental validations of cross-sections and transport coefficients of Xe+ colliding with Xe

Mobility and diffusion coefficients of ions in xenon gas and related momentum transfer cross-sections (MTCSs) for /Xe collisions are calculated using two different sets of diatomic interaction potentials and various theoretical approaches. MTCSs are obtained using four different methods: quantum method where both electrons and atomic nuclei are treated quantum mechanically, semi-classical method taking advantage of the Jeffreys–Wentzel–Kramers–Brillouin approximation applied to atomic nuclei, hybrid method where electrons are treated quantum mechanically while classical treatment is used for atomic nuclei, and a semi-empirical inverse method within which the cross-sections are extracted from available measurements on the mobility in xenon gas. Mobility and diffusion coefficients are calculated at ambient pressure and temperature for a broad range of the reduced electric field ( Td) using a Monte Carlo approach. A thorough comparison with available experimental data is provided.

On the asymptotic analysis of the JWKB method via change of dependent variable in the first-order Bessel’s equation

The first-order Jeffreys–Wentzel–Kramers–Brillouin method (called (JWKB)1) is a conventional semi-classical approximation method used in quantum mechanical systems for accurate solutions. It is known to give accurate energy and wave-function in the classically accessible region of the related quantum mechanical system defined by Schroedinger’s equation whereas the solutions in the classically inaccessible region require special treatment, conventionally known as the asymptotic matching rules. In this work, (JWKB)1 solution of the Bessel differential equation of the first order (called (BDE)1), chosen as a mathematical model, is studied by being transformed into the normal form via the change of dependent variable. General JWKB solution of the initial value problem where appropriately chosen initial values are applied is studied in both normal and standard form representations to be analyzed by the generalized JWKB asymptotic matching rules regarding the Sij matrix elements defined in the literature. Consequently, regions requiring first-order and zeroth-order JWKB approximations are determined successfully.

Tight-binding tunneling amplitude of an optical lattice

The particle in a periodic potential is an important topic in an undergraduate quantum mechanics curriculum and a stepping stone on the way to more advanced topics, such as courses on interacting electrons in crystalline solids, and graduate-level research in solid-state and condensed matter physics. The interacting many-body phenomena are usually described in terms of the second quantized lattice Hamiltonians which treat single-particle physics on the level of tight-binding approximation and add interactions on top of it. The aim of this paper is to show how the tight-binding tunneling amplitude can be related to the strength of the periodic potential for the case of a cosine potential used in the burgeoning field of ultracold atoms. We show how to approach the problem of computing the tunneling amplitude of a deep lattice using the JWKB (Jeffreys–Wentzel–Kramers–Brillouin, also known as semiclassical) approximation. We also point out that care should be taken when applying the method of the linear combination of atomic orbitals (LCAO) in an optical lattice context. A summary of the exact solution in terms of Mathieu functions is also given.

Tunneling of Relativistic Bosons Induced by Magnetic Fields in the Magnetar's Crust

The present work is devoted to the study of bosons evolving in the frozen magnetar's crust endowed with an ultra-strong magnetic field orthogonal to an electric field, both described by periodic functions. We discuss the quantum tunneling process through the one-dimensional potential barrier along Oz. The solutions to the Klein–Gordon equation are expressed in terms of Mathieu's functions which, for computable particle's energy range, are turning from oscillatory to exponentially growing modes along Oz. Within the Jeffreys–Wentzel–Kramers–Brillouin framework, the transmission coefficient is computed for the particle momentum in the middle of the instability range.

Gate Tunnelling Current Model for Nanoscale MOSFETs with Varying Surface Potential

ABSTRACTAt nanoscale, the channel of metal oxide semiconductor field effect transistors (MOSFETs) acts like a potential well within which electron energy levels are quantized and a nonzero wavefunction is obtained at the oxide–semiconductor interface. As a result the gate tunnelling current has emerged a significant constraint in respect of scaling of ultrathin gate oxides. We have developed an analytical model considering varying surface potential with applied voltage for evaluating gate tunnelling current through thin dielectrics in nanoscale MOSFETs. The electron wavefunction has been calculated by treating the band profile in the channel as a triangular potential well. The tunnelling probability through the gate oxide has been evaluated using Jeffreys–Wentzel–Kramers–Brillouin approximation method. The tunnelling current density is estimated from the evaluated interface wavefunction along with the tunnelling probability. The results from the present model compare well with the published Mondal–Dutta model and the experimental data. The novelty of the present model lies in its simplicity and its analyticity requiring much less computational efforts for its implementation.

Coaxial propagation of Laguerre–Gaussian (LG) and Gaussian beams in a plasma

AbstractThis paper investigates the non-linear coaxial (or coupled mode) propagation of Laguerre–Gaussian (LG) (in particular L01 mode) and Gaussian electromagnetic (em) beams in a homogeneous plasma characterized by ponderomotive and relativistic non-linearities. The formulation is based on numerical solution of non-linear Schrödinger wave equation under Jeffreys–Wentzel–Kramers–Brillouin approximation, followed by paraxial approach applicable in the vicinity of intensity maximum of the beams. A set of coupled differential equations for spot size (beam width) and phase evolution with space corresponding to coupled mode has been derived and numerically solved to determine the propagation dynamics. Using focusing equation a critical condition describing the self-trapped (i.e., spatial soliton) mode of laser beam propagation in the plasma has been discussed; as a consequence oscillatory focusing/defocusing of the beams in coupled mode propagation have been analyzed and presented graphically. As an important outcome, significant enhancement in the intensity of LG beam is noticed when it is coupled with the Gaussian mode.

Basic data of ions in He-air mixtures for fluid modeling of low temperature plasma jets

The basic ion data such as interaction potential parameters, elastic and inelastic collision cross sections, transport coefficients (reduced mobility and diffusion coefficients) and reaction coefficients have been analysed and determined for the case of He+, N2+, and O2+ in He-dry air mixtures. The ion transport and reaction coefficients have been determined from an optimized Monte Carlo simulation using calculated elastic and experimentally fitted inelastic collision cross sections. The elastic momentum transfer cross sections have been calculated from a semi-classical JWKB (Jeffreys Wentzel Kramers Brillouin) approximation based on a (6-4) rigid core interaction potential model. The inelastic cross sections have been fitted using the measured reaction coefficients, such as, for instance, the non resonant charge transfer coefficients. The cross section sets involving elastic and inelastic processes were then validated using either the measured reduced mobility whenever available in the literature or the zero-field mobility calculated from Satoh’s relation, and potential parameters available in the literature. From the sets of elastic and inelastic collision cross sections thus obtained for the first time for He+/N2, He+/O2, N2+/He, and O2+/He systems, the ion transport and reaction coefficients were calculated in the pure gases over a wide range of the density reduced electric field E/N. Then, from the present cross section and other literature sets, the ion mobility and the longitudinal and transverse diffusion coefficients were calculated for different concentrations of air in He in the case of He+, N2+, O2+, and also O− ions.

Exact solutions for the field electron emission achieved from a flat metal using the standard Fowler–Nordheim equation with a correction factor that accounts for the electric field, the work function, and the Fermi energy of the emitter

The author uses a transfer-matrix technique to simulate field electron emission from a flat metal. The author compares in particular the results provided by this numerical scheme with those predicted by the standard Fowler–Nordheim equation. This comparison aims at establishing the influence of different approximations introduced in the standard Fowler–Nordheim theory (in particular the use of the Jeffreys–Wentzel–Kramers–Brillouin approximation for evaluating the transmission coefficient of the surface barrier and the series expansion of this coefficient when integrating over the normal-energy distribution of the incident electrons). In addition to the field and work function considered in previous work, the author explores the dependence of the emission current on the Fermi energy of the emitter. This physical parameter, which is related to the density of free carriers in the emitter, does not appear in the final form of the standard Fowler–Nordheim equation. It is therefore discarded from most analysis of field-emission data. The author shows, however, by a series of arguments that the emission currents are affected by the Fermi energy of the emitter. The author finally establishes a correction factor to be used with the Murphy–Good expression that accounts for the field, for the work function, and for the Fermi energy of the emitter and provides the exact solution for the emission achieved from a flat metal.

Numerical testing of the Fowler–Nordheim equation for the electronic field emission from a flat metal and proposition for an improved equation

The author uses a transfer-matrix technique to simulate field electronic emission from a flat metal. He compares, in particular, the results provided by this numerical scheme with those predicted by the standard Fowler–Nordheim equation. He considers for this study electric fields between 1 and 10 V/nm as well as work functions between 1.5 and 5 eV. The results demonstrate that the Fowler–Nordheim theory and the transfer-matrix calculations are globally in good agreement. With the Fermi energy of 10 eV considered in this work, the results provided by the standard Fowler–Nordheim equation are, however, systematically larger than the quantum-mechanical result, especially for low values of the work function and for high electric fields. This is essentially due to the fact the standard Fowler–Nordheim theory relies on the simple Jeffreys–Wentzel–Kramers–Brillouin approximation for evaluating the electronic transmission through the surface barrier of the emitter. A correction factor is thus established that enables the temperature-dependent version of the standard Fowler–Nordheim equation to match the exact quantum-mechanical result.

Electric field emission of electrons from spherical metallic dust particles

In this paper, an almost exact expression for the tunneling probability of an electron through the potential barrier on account of a negative charge on a spherical metallic particle has been derived and used to obtain the field emission current density from the surface of the particle. Based on these results a parametric analysis of the phenomenon and comparison with the results of the Jeffreys–Wentzel–Kramers–Brillouin (JWKB) approximation (similar to the Fowler–Nordheim theory) has been presented. It is seen that the JWKB theory underestimates the tunneling probability. The investigation is of relevance to dusty plasmas in space and in the laboratory.

The formal derivation of an exact series expansion for the principal Schottky–Nordheim barrier function v, using the Gauss hypergeometric differential equation

The standard theory of Fowler–Nordheim tunnelling and cold field electron emission (CFE) employs a mathematical function v, sometimes called the principal field emission elliptic function, but better called the principal Schottky–Nordheim barrier function. This function arises when the simple-JWKB (Jeffreys–Wentzel–Kramers–Brillouin) method is applied to solve the Schrödinger equation approximately, for the image-rounded tunnelling barrier introduced by Schottky and then used by Nordheim in late 1928. An exact series expansion was recently found for v, as a function of a complementary elliptic variable l′ equal to y2, where y is the Nordheim parameter. The expansion was originally found by using an algebraic manipulation package. It was subsequently discovered that v(l′) is a particular solution of the ordinary differential equation (ODE) l′(1 − l′) d2v/dl′2 = (3/16)v. This ODE is now recognized to be a special case of the Gauss hypergeometric differential equation. This paper uses known special-case solutions of the hypergeometric equation to formally derive the series expansion for v(l′). It notes how to derive the defining ODE, and then uses an 1876 result from Cayley to derive the boundary condition that dv/dl′ satisfies as l′ tends to zero. It then establishes the series expansion for v(l′), by applying this and the boundary condition v(0) = 1. This mathematical derivation underpins earlier results, including good approximate expressions for v(l′). Its outcome proves that terms involving ln l′ are part of a mathematically correct solution, but fractional powers of l′ are not. It also implies that simple Taylor-expansion methods cannot easily generate good approximation formulae valid over the whole range 0 ⩽ l′ ⩽ 1; this may also apply to barriers of other shapes. This derivation should bring closure to the particular line of mathematical analysis of CFE theory initiated by Nordheim in 1928. It is hoped that the derivation might also serve as a model for analysing other tunnelling-barrier problems.

On the need for a tunneling pre-factor in Fowler–Nordheim tunneling theory

This paper argues that a tunneling prefactor should appear in expressions for the tunneling probability D relevant to cold field electron emission (CFE) and in Fowler–Nordheim (FN) type equations. Except in the case of the “ideally smooth” parabolic barrier, a prefactor is always present for barriers where D can be found by exact solution of the Schrödinger equation. A review of the Jeffreys–Wentzel–Kramers–Brillouin (JWKB) approach to solving the Schrödinger equation shows that tunneling barriers should be classified according to whether they are weak or strong and ideally smooth or not: there are four different JWKB-type formulas, depending on the nature of the barrier. CFE tunneling barriers are not ideally smooth but since the 1950s have usually been analyzed using the JWKB formulas for ideally smooth barriers. These analyses, and the standard Murphy–Good FN-type equation, seem mathematically and physically incomplete. The FN-type equations currently used to describe CFE should be revised to explicitly include a tunneling prefactor. Some implications are explored.

Semiclassical path integral theory of a double‐well potential in an electric field

AbstractA recently published methodology based on semiclassical path integral (SCPI) theory was implemented in the case of a model of a double‐well potential perturbed by a static electric field, with application to the inversion frequency of NH3. This model was chosen as an idealized case for testing of the present approach, as well as for quantum mechanical models that might be applied in the future. The calculations were concerned with the variation of the frequency of inversion as a function of field strength, F, and of distance, xf (from the symmetric point xo = 0), where the field is “felt.” It is found that this variation occurs sharply in very small regions of values of these parameters, and the system switches from internal oscillation to diffusion to the continuum. The fact that the theory is in analytic form allows the extraction of results and conclusions not only at the full SCPI level, but also at the Jeffreys–Wentzel–Kramers–Brillouin (JWKB) level. Comparison shows that the discrepancy sets in as the field strength increases, in accordance with the well‐known limitations of the JWKB method regarding its dependence on the degree of variation of the potential as a function of position. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

On the Bidirectionality of the JWKB Connection Formula at a Linear Turning Point

The bidirectionality of the Jeffreys-Wentzel-Kramers-Brillouin (JWKB) connection formula is demonstrated by a simple numerical example constructed from a linear potential and a box.

On the semiclassical approach to cold atomic collisions

We extend the semiclassical description of two-state atomic collisions to low energies for which the impact parameter treatment fails. The problem reduces to solving a system of first-order differential equations with coefficients whose semiclassical asymptotes experience the Stokes phenomenon in the complex coordinate plane. Primitive semiclassical and uniform Airy approximations are discussed. The theory of collisions between two atomic systems goes back to the early days of quantum mechanics (Mott 1931; Teller 1937; Landau & Teller 1936) and the basic 1930s models of state interaction are detailed in the recent review of Nikitin (1999). Typically, the colliding atoms undergo electronic transitions and one needs to solve a number (in the simplest case, two) of coupled radial Schrodinger equations. It has been noted since the early thirties that the relative motion of the heavy nuclei can be described classically (Mott 1931; Rosen & Zener 1932). Semiclassically, Stueckelberg (1932) first suggested the analytical continuation of the JWKB (Jeffreys-Wentzel- Kramers-Brillouin) wave functions into the complex plane of the internuclear sepa- ration and a proper handling of the Stokes phenomenon. His solution of the avoided crossing time-independent problem (Stueckelberg 1932) shows why a description of the interference in terms of adiabatic quasiclassical phases fails if the phase difference accumulated during the adiabatic motion of the two atoms between the centre of the coupling region and the turning points is small. The vexatious problem of the choice of branch cuts and the determination of Stokes constants via the comparison equa- tion method, were largely resolved by Crothers (1971). Equally well, it was shown by Coveney et al . (1985) that the classically forbidden region is less amenable to generalized phase-integral analysis. The classical approximation of the nuclear motion leads to the impact parameter treatment (Gaunt 1927; Landau 1932; Zener 1932). The impact parameter approx- imation (IPA) offers a simple physical picture in which fast electronic transitions occur as the slow nuclei follow a classical trajectory. By its nature, the IPA neglects

New images of the Earth's upper mantle from measurements of surface wave phase velocity anomalies

The sensitivity of a surface wave to the lateral structure of the Earth's mantle depends on the local structure and thickness of the crust; particularly at short periods, the strong lateral heterogeneity of the crust has a significant nonlinear effect on the surface wave “sensitivity functions.” Thus far, this effect has not been quantified, and tomographic studies of the upper mantle based on surface wave observations have not taken it into account. On the basis of a Jeffreys‐Wentzel‐Kramers‐Brillouin (JWKB) description of surface wave propagation we find and discuss the lateral variations of the sensitivity of Love and Rayleigh waves to mantle seismic anomalies, associated with a realistic model of the crust [Mooney et al., 1998]. We then use the resulting laterally varying sensitivity kernels to determine a new tomographic image of horizontally and vertically polarized shear velocity in the upper mantle from Love and Rayleigh wave phase velocity measurements (periods from 35 to 300 s). In terms of large‐scale patterns of high and low velocity our model is similar to previously published ones, confirming, among other features, the anomalous anisotropy underlying the central Pacific [Ekström and Dziewonski, 1998]; we find, however, that the lateral dependence of the surface wave sensitivity kernels has a significant effect on the absolute amplitude of the anomalies, which should not be neglected as we attempt to obtain higher‐resolution images of the upper mantle.