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.     

Bianchi I Quantum Cosmology in the Bergmann–Wagoner Theory

The Wheeler–DeWitt equation is considered in the context of generalized scalar-tensor theories of gravitation for the Bianchi type I cosmology. Exact solutions are found for two selfinteracting potentials and arbitrary coupling function. The WKB wavefunctions are obtained and a family of solutions satisfying the Hawking–Page regularity conditions of wormholes are found.     

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.     

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.     

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.     

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.     

Towards a theory of the field emission of electrons

Application of a generalized WKB method to the investigation of the field emission of electrons from conductors and semiconductors is considered. Formulas are obtained for estimating the coefficients of transmission of electrons through a potential barrier created by an external electric field and the image force.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 11, pp. 7–13, November, 1972.     

The Boltzmann equation from quantum field theory

We show from first principles the emergence of classical Boltzmann equations from relativistic nonequilibrium quantum field theory as described by the Kadanoff–Baym equations. Our method applies to a generic quantum field, coupled to a collection of background fields and sources, in a homogeneous and isotropic spacetime. The analysis is based on analytical solutions to the full Kadanoff–Baym equations, using the WKB approximation. This is in contrast to previous derivations of kinetic equations that rely on similar physical assumptions, but obtain approximate equations of motion from a gradient expansion in momentum space. We show that the system follows a generalized Boltzmann equation whenever the WKB approximation holds. The generalized Boltzmann equation, which includes off-shell transport, is valid far from equilibrium and in a time dependent background, such as the expanding universe.     

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.     

Particle production in curved spacetime

Particle production in curved spacetime has been discussed through the method of complex time WKB approximation. We consider Dirac equation in non-flat spacetime to understand particle production as particle-antiparticle rotation. The method is also generalized to understand particle production through parametric resonance. To understand the method of CWKB we consider particle production in Kasner spacetime as an example.     

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 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.     

利用广义不确定关系计算Gibbons-Maeda黑洞的统计力学熵

利用广义不确定关系修正的态密度方程并采用WKB近似方法, 计算了Gibbons-Maeda黑洞时空中标量场的统计力学熵. 所得的统计力学熵与该黑洞的视界面积成正比, 与Brick-Wall方法所得的结果相同, 但不需要取任何截断就能避免Brick-Wall方法的发散问题. 关键词： 广义不确定关系 Gibbons-Maeda黑洞 态密度 统计力学熵     

Statistical Entropy of Reissner-Nordstrom Black Hole Computed by Generalized Uncertainty Principle

The equation of state density is corrected by the generalized uncertainty principle. Statistical entropy of scalar field outside Reissner-Nordstrom black hole is computed by WKB approximation method. The result shows that this black hole entropy is proportional to its horizon area, which is the same as that given by brick-wall method. The difference from the brick-wall method is that the present result is convergent without any cutoff.     

The generalized Jeffreys-Born integral

The Jeffreys-Born integral in the WKB approximation is generalized to include the Coulomb potential. The Weyl-transform is used to obtain phase functions for potentials frequently used in fitting modified Coulomb scattering data.     

Reflection at a complex potential barrier in the semiclassical theory of scattering

The reflection of spherical waves at a complex potential barrier is discussed in the semiclassical approximation. We study the complex WKB method and the Uniform Approximation in the special case of weakly absorptive barriers, typical of surface transparent optical potentials used in heavy-ion reactions. It is found that the complex WKB results lead to a very accuratecross- section despite their inaccuracy in the most importantphase shifts. Thereby, the amazing stamina of the WKB has been confirmed once more. One of us (Y.A.) spent the summer of 1976 at the Max-Planck Institute at Heidelberg where the present work has been done. He would like to thank J. Hüfner and H. Weidenmüller for the hospitality extended to him.     

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.     

Quasilinear Approximation and WKB

Quasilinear solutions of the radial Schrödinger equation for different potentials are compared with corresponding WKB solutions. For this study, the Schrödinger equation is first cast into a nonlinear Riccati form. While the WKB method generates an expansion in powers of , the quasi-linearization method (QLM) approaches the solution of the Riccati equation by approximating its nonlinear terms by a sequence of linear iterates. Although iterative, the QLM is not perturbative and does not rely on the existence of any kind of smallness parameters. If the initial QLM guess is properly chosen, the usual QLM solution, unlike the WKB, displays no unphysical turning-point singularities. The first QLM iteration is given by an analytic expression. This allows one to estimate analytically the role of different parameters, and the influence of their variation on the boundedness or unboundedness of a critically stable quantum system, with much more precision than provided by the WKB approximation, which often fails miserably for systems on the border of stability. It is therefore demonstrated that the QLM method is preferable over the usual WKB method.     

Proton Radioactivity Within a Hybrid Method

The proton radioactivity half-lives are investigated theoretically within a hybrid method.The potential barriers preventing the emission of protons are determined in the quasimolecular shape path within a generalized liquid drop model (GLDM). The penetrability is calculated with the Wentzel-Kramers-Brillouin (WKB) approximation. The spectroscopic factor has been taken into account in halflife calculation, which is obtained by employing the relativistic mean field (RMF) theory combined with the Bardeen-Cooper-Schrieffer (BCS) method. The half-lives within the present hybrid method reproduced the experimental data very well. Some predictions for proton radioactivity are made for future experiments.