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.     

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.     

Generalized WKB method in the three-dimensional case. V

The three-dimensional generalized WKB method, which is studied in [1–4], is extended to the case in which the standard wave equation for the model system allows the separation of variables in cylindrical or spherical coordinates.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 36–40, December, 1976.     

Generalized WKB method in the three-dimensional case. I

The possibility of using the generalized WKB method for studying the steady states of a single-frequency system in three-dimensional potential fields which do not allow the variables to be separated in the Schrödinger equation is considered. Approximate wave functions are constructed; these transform into the well-known van Horn wave functions in the quasiclassical case.     

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.     

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.     

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.     

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 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.     

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.     

On matching conditions in the WKB method

The modified matching conditions for quasiclassical wave functions on both sides of a turning point for the radial Schrödinger equation have been obtained. They differ significantly from the usual Kramers condition which holds for the one-dimensional case. Namely, the ratio C₂/C₁ in the subbarrier and the classical allowed regions is not a universal constant ( , as usual), but depends on the values of the orbital angular momentum l, energy E and on the behaviour of the potential V(r) at r → 0. The comparison with exact and numerical solutions of the Schrödinger equation shows that the modified matching conditions not only make the quasiclassical approximation in the subbarrier region asymptotically exact within the n → ∞ limit, but also considerably enhances its accuracy even in the case of small quantum numbers, n 1. The power-law, funnel and short-range potentials are considered in detail.     

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.     

Comparison of WKB calculations and experimental results for three-dimensional cochlear models.

The WKB asymptotic method is applied to the calculation of cochlear models with square scala cross section, for which the fluid motion is fully three dimensional. The analysis begins with the exact solution for wave propagation in a duct with constant properties. This solution is somewhat tedious but straightforward, since it requires a Fourier series expansion across the duct. Then with the formulation of Whitham [Linear and Nonlinear Waves (Wiley, New York, 1974)], the approximate solution is readily generated for the duct with properties which vary slowly along the length. Numerical calculations are carried out for the experimental models of Cannel [Ph.D. thesis, Univ. of Warwick (1969)] and Helle [Dr.-Ing. disser., Technische Univ., Müchen (1974)] who furnish quantitative details of both "basilar membrane" response and model parameters. Without any free parameters for adjusting, the present WKB solution shows quite satisfactory agreement with the experimental model results. Computer time is reasonable; the calculation of displacement envelope and phase at a number of stations along the cochlea for a given frequency requires only one second of CPU time. Thus the credibility and practically of the approach is established for the investigation of yet more realistic and more elaborate cochlear models.     

New WKB method in supersymmetry quantum mechanics

In this paper, we combine the perturbation method in supersymmetric quantum mechanics with the WKB method to restudy an angular equation coming from the wave equations for a Schwarzschild black hole with a straight string passing through it. This angular equation serves as a naive model for our investigation of the combination of supersym- metric quantum mechanics and the WKB method, and will provide valuable insight for our further study of the WKB approximation in real problems, like the one in spheroidal equations, etc.     

Theory of the generalized collision integral. IV

In the maximum time interval of kinetic evolution of the single-particle distribution function of a spatially homogeneous system summation of the step diagrams (without intersections of the lines of propagation and interaction) is carried out for an arbitrary nonequilibrium state in the Markov limit, as well as for a local equilibrium state with regard for memory effects. Procedures are indicated for the step renormalization of diagrams of arbitrary order with respect to the density in the Markov limit, with subsequent ladder renormalization. Ring diagrams with preliminary step and ladder renormalizations are summed. A more correct derivation is given for the generalized Boltzmann (Boltzmann-Landau) collision integrals for dense systems with a strong short-range (as well as strong long-range) potential. It is shown that the step, ring, and ladder renormalizations yield a nondivergent two-particle distribution function in a classical equilibrium plasma.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 86–91, August, 1978.     

The WKB method for the Dirac equation with the vector and scalar potentials

The WKB approximation is developed for the Dirac equation with the spherically symmetrical vector and scalar potentials. The relativistic wavefunctions are constructed, new quantization rule containing the spin-orbital interaction is obtained. For spherically symmetrical model of the Stark effect the quasi-classical spectrum of relativistic hydrogen-like atom is calculated. Application of the WKB method to the mass spectrum of the hydrogen-like quark systems was done.