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.     

New approaches to the study of properties of the generalized WKB approximation. I

It is shown that the new approach which has been successfully developed by Froman and Froman [1] during recent years for the study of the properties of quasiclassical solutions of the one-dimensional Schrödinger equation expressed in the form of uniformly converging series can also be extended to the generalized WKB method of Petrashen'-Fock.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 58–65, September, 1972.     

New approaches to the study of properties of the generalized WKB approximation. IV

The problem of evaluating partial scattering phases of particles by spherically symmetric potentials is treated by the new formulation of the generalized WKB method. The error committed in calculating scattering phases in the zeroth approximation is estimated.     

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.     

The generalized WKB method in the three-dimensional case. III

We consider simple examples of applying the generalized WKB method [1, 2] to the study of bound states of multidimensional quantum-mechanical systems.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 128–134, March, 1976.     

Second approximation of the generalized WKBJ method. III

The problem of the passage of particles across a one-dimensional, single-humped potential barrier of arbitrary shape is solved in the second approximation of the generalized WKBJ method. The problem is reduced to terms of the order of ⁴ for particles having an energy equal to the barrier peak. The maximum error involved in the calculation of the transmission and subbarrier-reflection coefficients in the second approximation of the generalized WKBJ method is shown not to exceed 1%.Translated from Izvestiya VUZ. Fizika, Vol. 11, No. 11, pp. 108–118, November, 1968.     

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.     

New approaches to properties of the generalized WKB-approximation. II

Continuing the study started in [1], the possibility of applying the Frömans' formalism to investigate properties of the generalized WKB-approximation is explored. Estimates are obtained and investigated for the absolute value, limiting properties, and symmetry relations of the F matrix, representing approximate solutions of the Schrödinger equation in the form of absolutely and uniformly convergent series.     

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.     

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     

Semiclassical kinetic equation and the WKB approximation

We study the connection between the classical response function of a nucleus described by the Vlasov equation and the corresponding quantum response function. In the limit of large quantum numbers, the Fourier coefficients which appear in the Vlasov theory correspond to the quantum matrix elements evaluated in the WKB approximation. The classical frequencies of motion give the quantum excitation energies according to the correspondence principle.For a central potential, we identify the normal modes of the classical systems with the corresponding shell-model transitions. Thus we can improve the Vlasov theory by excluding excitation modes which correspond to forbidden quantum transitions. Also, we study the effect of a spin-orbit force in the context of the Vlasov theory.Finally, always by exploiting the close analogy between the classical and the quantum response functions, we introduce exchange (Fock) term in the semiclassical RPA equation given by the Vlasov theory.     

A new method for calculating potential curves in the zero-order approximation of a generalized WKB method

A new method, different from that of [1 ], is proposed for plotting potential curves from vibrational-rotational levels in the zero-order approximation in Planck's constant of a generalized WKB method.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, Vol. 16, No. 7, pp. 127–129, July, 1973.     

The generalized WKB method in the three-dimensional case. II

The study of the three-dimensional generalized WKB method, initiated in [1], is continued. A new form of wave functions obtained by this method, which is more convenient for practical purposes, is indicated. The problem of selecting the integration path under quantization conditions and the procedure of calculating the transformation function x _i ⁽⁰⁾ (r) are discussed in detail.Translated from Izvestiya Vysslikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 123–128, March, 1976.     

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.     

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.     

The generalized WKB method in the three-dimensional case. IV

A method of successive approximations is developed for obtaining solutions of the basic equations of the WKB method in the three-dimensional case. Explicit expressions are found for the first approximations to the energy levels and wave functions of the bound states of a particle in a three-dimensional potential well which were obtained in zeroth approximation in [1, 2].Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 53–60, November, 1976.     

On the nature of the generalized WKB method

The nature and the conditions of applicability of the generalized WKB method (the Petrashen-Miller-Good method) are investigated. It is shown that the generalized WKB method is a new approximate method for quantum mechanics, differing essentially from the WKB method.In conclusion I wish to express my sincere appreciation to Academicians V. A. Fok and M. I. Petrashen for their interest in this work and their valuable advice in the process of its completion.