JWKB Method Research Articles

On the semiclassical JWKB and MAF solutions of the uniformly lengthening pendulum via change of dependent variable in the Bessel’s equation

Semiclassical JWKB method and method of modified Airy functions (MAF) are the fundamental conventional tools in quantum mechanical and optical waveguide systems and yet their classical mechanical applicability has somehow escaped much attention. In this work, semiclassical JWKB and MAF solution of a classical mechanical problem regarding the Lengthening Pendulum (LP) is being studied to reinforce the power of the semiclassical methods in this aspect. Here, JWKB and MAF solutions of the associated Bessel’s equation is being obtained by transforming it first into one of the normal forms via change of dependent variable on the contrary to our previous work where change of independent variable was used for the JWKB solution. By the same motivation for the suggested change of dependent variable here, we once again, obtain advantageous results in comparison to the conventional numerical exact solutions given in the literature where adiabatic LP systems with very low lengthening rates are required. We obtain both JWKB and MAF approximated Bessel functions successfully and our semiclassical solutions for the LP system works with a great accuracy even for much higher lengthening rates in comparison to these conventional numerical methods. Here we study under what conditions our semiclassical solutions can give accurate results by using the conventional JWKB and MAF applicability criteria. In effect, the semiclassical JWKB and MAF methods which is normally known to be used in quantum mechanical and optical waveguide systems are being used to solve the pure classical-mechanical LP system successfully and advantageously here and the applicability criteria of these semiclassical methods in terms of the physical LP system parameters are being determined and presented.

Application of JWKB Method on the Effect of Magnetic Field on Alpha Decay

Application of JWKB Method on the Effect of Magnetic Field on Alpha Decay

Macroscopic quantum escape of Bose-Einstein condensates: Analysis of experimentally realizable quasi-one-dimensional traps

The variational JWKB method is used to determine experimentally accessible macroscopic quantum tunneling regimes of quasibound Bose-Einstein condensates in two quasi-one-dimensional trap configurations. The potentials can be created by magnetic and optical traps: a symmetric trap from two offset Gaussian barriers and a tilt trap from a linear gradient and Gaussian barrier. Scaling laws in barrier parameters, ranging from inverse polynomial to square root times exponential, are calculated and used to elucidate different dynamical regimes, such as when classical oscillations dominate tunneling rates in the symmetric trap. The symmetric trap is found to be versatile, with tunneling times at and below 1 s, able to hold ${10}^{3}--{10}^{4}$ atoms, and realizable for atoms ranging from rubidium to lithium, with unadjusted scattering lengths. The tilt trap produces subsecond tunneling times, is able to hold a few hundred atoms of lighter elements such as lithium, and requires the use of Feshbach resonance to reduce scattering lengths. To explore a large parameter space, an extended Gaussian variational ansatz is used, which can approximate large traps with Thomas-Fermi profiles. Nonlinear interactions in the Gross-Pitaevskii equation are shown to produce additional effective mean-field barriers, affecting scaling laws.

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.

On the JWKB solution of the uniformly lengthening pendulum via change of independent variable in the Bessel’s equation

Common recipe for the lengthening pendulum (LP) involves some change of variables to give a relationship with the Bessel’s equation. In this work, conventional semiclassical JWKB solution (named after Jeffreys, Wentzel, Kramers and Brillouin) of the LP is being obtained by first transforming the related Bessel’s equation into the normal form ‘via the suggested change of independent variable’. JWKB approximation of the first-order Bessel functions (ν=1) of both types along with their zeros are being obtained analytically with a very good accuracy as a result of the appropriately chosen associated initial values and they are extended to the neighbouring orders (ν=0 and 2) by the recursion relations. The required initial values are also being studied and a quantization rule regarding the experimental LP parameters is being determined. Although common numerical methods given in the literature require adiabatic LP systems where the lengthening rate is slow, JWKB solution presented here can safely be used for higher lengthening rates and a criterion for its validity is determined by the JWKB applicability criterion given in the literature. As a result, the semiclassical JWKB method which is normally used for the quantum mechanical and optical waveguide systems is applied to the classical LP system successfully.

A Normal and Standard Form Analysis of the JWKB Asymptotic Matching Rule via the First Order Bessel’s Equation

Traditional first order JWKB method ( = : ( J W K B ) 1 ) is a conventional semiclassical approximation method mainly used in quantum mechanical systems for accurate solutions. ( J W K B ) 1 general solution of the Time Independent Schrodinger’s Equation (TISE) involves application of the conventional asymptotic matching rules to give the accurate wavefunction in the Classically Inaccessible Region (CIR) of the related quantum mechanical system. In this work, Bessel Differential Equation of the first order ( = : ( B D E ) 1 ) is chosen as a mathematical model and its ( J W K B ) 1 solution is obtained by first transforming into the normal form via the change of independent variable. The ( J W K B ) 1 general solution for appropriately chosen initial values in both normal and standard form representations is analyzed via the generalized ( J W K B ) 1 asymptotic matching rules regarding the S ˜ i j matrix elements given in the literature. Instead of applying the common ( J W K B ) 1 asymptotic matching rules relying on the physical nature of the quantum mechanical system, i.e., a physically acceptable (normalizable) wavefunction, a pure semiclassical analysis is studied via the ( B D E ) 1 model mathematically. Finally, an application to a specific case of the exponential potential decorated quantum mechanical bound state problem is presented.

On The Exact and JWKB Solution of 1D Quantum Harmonic Oscillator by Mathematica

Although being the fundamental semiclassical approximation method mainly used in quantum mechanics and optical waveguides, JWKB method along with the application of the associated JWKB asymptotic matching rules is known to give exact solutions for the Quantum Harmonic Oscillator (QHO). Asymptotically matched JWKB solutions are typically accurate or exact in the entire domain except for a narrow domain around the classical turning points where potential energy equals the total energy of the related quantum mechanical system. So, one has to cope with this diverging behavior at the classical turning points since it prohibits us from using continuity relations at the related boundaries to determine the required JWKB coefficients. Here, a computational diagram and related mathematica codes to surmount the problem by applying parity matching for even and odd JWKB solutions rather than boundary continuities are being presented. In effect, JWKB coefficients as well as the conversion factor for the dimensionless form of the Schrodingers equation, which is common to both exact and JWKB solutions, is being successfully obtained.

Riemannζfunction from wave-packet dynamics

We show that the time evolution of a thermal phase state of an anharmonic oscillator with logarithmic energy spectrum is intimately connected to the generalized Riemann $\ensuremath{\zeta}$ function $\ensuremath{\zeta}(s,a)$. Indeed, the autocorrelation function at a time $t$ is determined by $\ensuremath{\zeta}(\ensuremath{\sigma}+i\ensuremath{\tau},a)$, where $\ensuremath{\sigma}$ is governed by the temperature of the thermal phase state and $\ensuremath{\tau}$ is proportional to $t$. We use the JWKB method to solve the inverse spectral problem for a general logarithmic energy spectrum; that is, we determine a family of potentials giving rise to such a spectrum. For large distances, all potentials display a universal behavior; they take the shape of a logarithm. However, their form close to the origin depends on the value of the Hurwitz parameter $a$ in $\ensuremath{\zeta}(s,a)$. In particular, we establish a connection between the value of the potential energy at its minimum, the Hurwitz parameter and the Maslov index of JWKB. We compare and contrast exact and approximate eigenvalues of purely logarithmic potentials. Moreover, we use a numerical method to find a potential which leads to exact logarithmic eigenvalues. We discuss possible realizations of Riemann $\ensuremath{\zeta}$ wave-packet dynamics using cold atoms in appropriately tailored light fields.

SEMICLASSICAL VIOLATION OF THE EQUIVALENCE PRINCIPLE IN THE GRAVITATIONAL LENSES REALM

An expression for the gravitational deflection angle of a high-energy massless scalar boson is computed using the JWKB method. The resulting expression depends on the energy of the scalar boson or, equivalently, on its wavelength, even to first order in [Formula: see text], where G is Newton's constant, M is the mass of the gravitational source, and b is the impact parameter. On the other hand, combining the expressions for the magnification of two images obtained from a gravitational lens with the classical expression for the cross section, it is found that for purely gravitational scatterings the gravitational cross sections are proportional to the relative intensities. As a consequence of this remarkable proportionality and of the fact that the deflection angle obtained via the JWKB method is energy dependent, we come to the conclusion that chromatic effects in the gravitational lenses at the semiclassical level are possible, which violates the classical equivalence principle. The probability of detecting this tiny effect in the foreseeable future is discussed.

Errors in the application of the JWKB method to calculating the survival factor in dissociative electron attachment using the local complex potential

The accuracy of the JWKB method for determining the survival factor defined for dissociative electron attachment (DEA) processes is examined for a range of electronic resonance lifetimes within the local complex potential approximation. The author concludes that the accuracy is inadequate for molecules with properties commonly found for shape resonance induced DEA. More accurate methods using the uniform Airy function approximation give much better results, but the direct numerical integration of Schrodinger's equation appears simpler still.

Propagation of a Mixture of Modes of a Laser Beam in a Medium with Saturable Nonlinearity

Propagation of a mixture of modes of a laser beam through a saturable nonlinear medium has been studied using JWKB method and the paraxial ray approximation. Two second order nonlinear coupled differential equations for the beam width parameters resembling equations of coupled nonlinear oscillators of unit mass are obtained. A scalar potential of the system has been formulated whose analysis yields some valuable information like existence of two critical values of the potential within which bound state of the system exists. From the stability analysis it has been found that the stable beam propagation depends on the ratio of intensities of the two modes. A threshold of power is defined. When beam power is above the threshold value the propagation is stationary.

The JWKB Method in Central-Field Problems. Planar Radial Wave Equation and Resolution of Kramers' Dilemma

It is well-known that applications of the JWKB method to central-field problems in three dimensions require half-integral quantization of the angular momentum for their success. Thus, the square of the angular momentum must be represented by the term (/ + ½) 2 h 2 rather than /(/ + 1)h 2 . This was first shown by Kramers in 1926 and has subsequently been discussed by several authors including, in particular, Langer (1937). While Kramers based his discussion on the ordinary radial variable r, Langer switched to the variable x defined by r = e x . In this new representation of the central-field problem, the expression (/ + ½) 2 h 2 emerges naturally. The ad hoc character of the Langer transformation has, however, often been emphasized. In the present communication, we choose a different entry to the problem. We keep the variable r and focus on physically equivalent forms of the radial Schrodinger equation in this variable. This leads to a smoother emergence of the (/ + ½) 2 h 2 term. Our analysis is carried out for a general dimension D. For a given D, there are D physically equivalent radial equations, corresponding to the subdimensions d = 1, 2..., D. We show that it is only the d = 2 equation that can be satisfactorily treated by the JWKB approximation. In the past, the focus was always on the d = I equation, and this was the reason behind the problems encountered by Kramers and Langer. As to the d = 2 equation, we finally show that this equation also is the most convenient starting point for determining the exact solutions of a central-field problem for general values of D and angular-momentum quantum number L.

JWKB method as an exact technique

AbstractThe JWKB method involves divergent asymptotic expansions in powers of ħ. Borel summability turns divergence into convergence, leading to exact solutions of the Schrödinger equation. The formulas that connect real solutions across linear turning points, where the terms of the expansion are singular, are complex but nonunidirectional. A series of basic problems constructed from linear potentials—for which the JWKB solutions are simple, in which the boundary conditions and physics differ, but for which the turning points are the same—is used to demonstrate the nondirectionality of the turning points and how “Borel sum approximants” lead to exact energy eigenvalues. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

On eigenfunction approximations for typical non–self–adjoint Schrödinger operators

We construct efficient approximations for the eigenfunctions of non-self-adjoint Schroedinger operators in one dimension. The same ideas also apply to the study of resonances of self-adjoint Schroedinger operators which have dilation analytic potentials. In spite of the fact that such eigenfunctions can have surprisingly complicated structures with multiple local maxima, we show that a suitable adaptation of the JWKB method is able to provide accurate lobal approximations to them.

Semi-Classical States for Non-Self-Adjoint Schrödinger Operators

We prove that the spectrum of certain non-self-adjoint Schrödinger operators is unstable in the semi-classical limit h→ 0. Similar results hold for a fixed operator in the high energy limit. The method involves the construction of approximate semi-classical modes of the operator by the JWKB method for energies far from the spectrum.

MAF solution for bounded potential problems

We present here the solutions of a bounded linear harmonic potential and abounded quartic potential using the modified Airy function (MAF) method. Resultsobtained by the MAF method have been compared with the analytical (numerical)results and with those obtained by the JWKB method. The comparison showsthat the MAF method gives very accurate results and is, in general, the moreaccurate of the two methods. The MAF method also gives an accurate descriptionof the eigenfunction. A perturbation correction when applied to MAF helps usto get very accurate eigenvalues. The method should be useful in determiningthe eigenvalues and the eigenfunctions of any smoothly varying arbitrarypotential confined by infinite walls.PACS No. 03.65

Theory of field ionization

We describe calculations of tunnelling rate constants for the field ion microscope (FIM) using various model potentials, and potentials obtained from self-consistent electronic structure calculations incorporating the full screening of an applied electric field. For our tunnelling calculations we have used the JWKB method of Haydock and Kingham. The expression for the tunnelling rate constant for a flat surface given by Haydock and Kingham is inaccurate and we have derived a new and reliable formula. We have also introduced a new tunnelling potential for the FIM which includes explicitly the effects of image potentials. None of the potentials for smooth surfaces give ionization zone widths which are narrow enough to account for experimental values. However, the width of the ionization zone is found to be a strongly decreasing function of the corrugation of the surface potential, and for large corrugations we can obtain results in plausible agreement with experimental values. For corrugated model surfaces we have found cross-surface variations in the tunnelling rate constant which may be sufficient to account for the atomic resolution of the FIM under normal operating conditions at liquid nitrogen temperatures.

Model-potential calculations of tunnelling rate constants for the field-ion microscope

The authors have performed tunnelling calculations for the field-ion microscope using a the JWKB method of Haydock and Kingham (1981) and the model potentials introduced by Homeier and Kingham (1981). They have found that Homeier and Kingham overestimated the cross-surface variations in the tunnelling rate constant by a factor of up to five. The reasons for these inaccuracies are analysed and they give the results of more accurate calculations.

Calculations of ionization rate-constants for the field-ion microscope

Abstract Ionization rate-constants are calculated for the field-ion microscope using the JWKB method and tunnelling potential introduced by Haydock and Kingham [Surf. Sci. 103 (1981) 239]. The expression for the tunnelling rate-constant given by Haydock and Kingham is inaccurate and we present a new and reliable formula. The prefactor for the tunnelling rate-constant is also calculated by comparing with the ionization rate-constant in a uniform electric field. The resulting rate-constants for helium atoms are significantly larger than those obtained by Kingham and coworkers [J. Phys. (Paris) 45 (1984) 77]. Our results make it possible to calculate ionization rate-constants in the field-ion microscope with greater accuracy than was previously possible.

Calculations of ionization rate-constants for the field-ion microscope

Ionization rate-constants are calculated for the field-ion microscope using the JWKB method and tunnelling potential introduced by Haydock and Kingham [Surf. Sci. 103 (1981) 239]. The expression for the tunnelling rate-constant given by Haydock and Kingham is inaccurate and we present a new and reliable formula. The prefactor for the tunnelling rate-constant is also calculated by comparing with the ionization rate-constant in a uniform electric field. The resulting rate-constants for helium atoms are significantly larger than those obtained by Kingham and coworkers [J. Phys. (Paris) 45 (1984) 77]. Our results make it possible to calculate ionization rate-constants in the field-ion microscope with greater accuracy than was previously possible.