WKB approximation for a quarkonium equation

A fourth-order differential equation recently proposed for describing quarkonia is studied. The eigenvalue spectrum is self-similar. A WKB approximation reproduces the spectrum and the so-called magic numbers which characterize the self-similarity.     

Bethe-Salpeter type relativistic fermion-antifermion equation in the WKB approximation

We investigate the relativistic fermion-antifermion Bethe-Salpeter type equation whose potential is the sum of Coulomb and linear terms in the WKB approximation. It is shown that in the particular case of an attractive Coulomb potential, the discrete energy spectrum lies in the interval (0,2), and in the case of a repulsive linear potential, in the interval (2, ).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 21–27, September, 1989.     

Semiclassical approximation of the Dirac equation with supersymmetry

The general scheme of the successive construction of semiclassical approximation for the classical Dirac equation in a background Yang-Mills field, where the usual Dirac operator is replaced by that with supersymmetry, is suggested. The first two terms of the semiclassical expansion in Planck’s constant are derived in an explicit form. It is shown that supersymmetry of the initial Dirac operator leads to appearance of new additional terms in the classical equation of motion for spin of a particle and ipso facto requires appropriate modification for the Lagrangian of the spinning particle. The result obtained is used for the construction of one-to-one mapping between two Lagrangians of a classical color-charged spinning particle, one of which possesses local supersymmetry, and another doesn’t. It is demonstrated that for recovery of the one-to-oneness the additional terms obtained above in the semiclassical approximation of the Dirac operator with supersymmetry should be added to the Lagrangian without supersymmetry.     

Anomalies of the free loop wave equation in the WKB approximation

We derive a well-defined, reparametrization invariant expression for the next to leading term in the small ? expansion of the euclidean loop Green functional ψ($\text{C}$). To this order in ?, we then verify that ψ($\text{C}$) satisfies a renormalized loop wave equation, which involves a number of local, but non-harmonic anomalous terms. Also, we find that the quantum fluctuations of the string give rise, in 3 + 1 dimensions, to a correction of the static quark potential by an attractive Coulomb potential of universal strength $\text{α}{}_{\mathrm{<mtext>string</mtext>}}\text{=}\text{1}\text{12}\text{π}$.     

Curve-crossing and the WKB approximation

The transition probabilities and phase changes associated with passage through a potential curve crossing are derived in the form of connection formulae between WKB solutions on either side of the crossing point. The formulae are expressed in terms of three integrals v and ε_± which may be evaluated for arbitrary potential curves and interaction function without knowledge of the associated wavefunctions. The theory, which is fully developed in the chemical energy region (< 100 ev), is applied to a model for the covalent-ionic crossing responsible for inelastic scattering of alkali atoms from neutral targets and its extension to the higher energy region for this model indicated.     

Classical paths and the WKB approximation

Recently, new connection formulas for the WKB method have been proposed, without justification, for quantum tunneling problems. We show that these formulas can be associated with diagrammatic rules within the complex time framework of the path integral formalism and then we express the relevant Green functions in terms of a sum of contributions coming from (easily interpreted) classical paths. The method is applied to barrier penetration and the double well. Received: 6 June 2000 / Published online: 27 October 2000     

WKB approximation in multidimensional problems

Unidimensional WKB-formulas for the solution of multidimensional, non separable, problems are derived and the limits of their applicability are discussed.     

WKB approximation in complex time

The WKB approximation to the one particle Schrödinger equation in time is used to obtain the wavefunction at a given point as a sum of semiclassical terms, each corresponding to a different classical trajectory (real or complex) but ending up at the same point. A method to find out reflection coefficient for processes involving one and two turning points is developed and it is shown that the semiclassical complex analysis reproduces exactly the reflection coefficient that is obtained through the exact solution of the problem. The connection between pair production and reflection amplitude is also shown. The pair production amplitude in a time dependent gravitational background is calculated and it is shown that the vacuum considered in complex trajectory WKB analysis refers to adiabatic vacuum.     

WKB approximation in complex time II

The non-perturbative method, developed recently, of WKB approximation in complex time is applied to some known curved space time. Three cases namely (1) static in and out region, (2) non-static in and out region, (3) static in and non static out region are considered here. We find non-trivial particle production corresponding to the quantum vacuum definition of Castagnino and Mazzitelli.     

Energy levels and level ordering in the WKB approximation

Energy levels and level orderings for a particle in a non-relativistic potential are examined in the WKB approximation. In particular, power-law potentials (V(r) = ar^γ, ?2 < γ < ∞) are discussed in some detail. The energy levels are shown to be determined in terms of a single function G(η, γ) of a variable η. Expansions of this function, valid for small (large) angular momentum quantum numbers (l) and large (small) radial quantum numbers (n), approximate the energy levels well. The ordering of the levels follows from the monotonic behavior of (?/?η)G(η, γ). The values γ = 2 (harmonic oscillator potential) and γ = ?1 (Coulomb potential) for which the WKB approximation gives the exact (i.e. Schrödinger) results lead to degenerate levels. It is about these values of γ that the monotonic behavior of (?/?η)G(η, γ) changes sign (as a function of γ). We also find an ordering theorem for arbitrary central potentials which is valid for large l and small n and is possibly correct for smaller l. The ordering depends on various sums of derivatives of the potential. Similar theorems, which follow from the Schrödinger equation, have been obtained recently for low-lying levels and are compared to our results.     

Semiclassical approximation for relativistic potentials

We consider the application of semiclassical approximation to relativistic potentials for massless particles where the kinetic energy is a nontrivial, nonlocal operator. Quantization rules are derived for an arbitrary confining potential and compared to some exact results forS-waves. These results admit of a partial generalization to smalll values.     

Semiclassical approximation in Batalin-Vilkovisky formalism

The geometry of supermanifolds provided with aQ-structure (i.e. with an odd vector fieldQ satisfying {Q, Q}=0), aP-structure (odd symplectic structure) and anS-structure (volume element) or with various combinations of these structures is studied. The results are applied to the analysis of the Batalin-Vilkovisky approach to the quantization of gauge theories. In particular the semiclassical approximation in this approach is expressed in terms of Reidemeister torsion.     

New approaches to the generalized WKB approximation. V

The problem of bound states in one-dimensional and spherically symmetric potential well is treated within the new formalism of the generalized WKB method, discussed in [1–3]. Exact quantization conditions for the binding energy are derived, and the errors in evaluating energy eigenvalues and wave functions in the zeroth approximation of the method are estimated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 81–86, April, 1975.     

The supersymmetric WKB approximation for Hartmann potential

Using the method of supersymmetric WKB approximation, the energy spectrum of some noncentral separable potentials can be exactly obtained in r and θ dimensions. We take the Hartmann potential as an important example for its validation, and the result is consistent with that obtained by using the supersymmetric quantum mechanics and shape invariance.     

New approaches to the generalized WKB approximation. III

The problem of particle transmission through a one-dimensional potential barrier of arbitrary shape is considered within the new generalized WKB method, developed in [1, 2]. A number of properties of the matrix F(S, S_t), needed to solve specific quantum-mechanical problems, is indicated.     

Second approximation in the generalized WKB method. II

Methods are discussed for calculating the improper integrals in the basic equations found in the first part of this study for the second approximation in the generalized WKB method.The author thanks A. S. Vasilevskii for a valuable discussion of this study.