Second approximation of the generalized WKBJ method. III

N. I. Zhlrnov, A.A. Sokolov

Н. И. Жирнов, F. B. Nurlygayanov

Roop Narain Kesarwani, Y. P. Varshni

Eigenvalues are determined for the plane pendulum problem by the WKBJ method in one- and four-term approximations, and the results are compared. It is found that at high quantum numbers, the four-term WKBJ approximation can yield eigenvalues of eight-significant-figure accuracy, but for low quantum numbers the results continue to be poor.

Н. И. Жирнов, F. B. Nurlygayanov

John J. Stachel, Jerzy F Plebanski

This is the first of a series of papers developing the classical theory of a spinning particle. The equations of motion will be derived from a Lagrangian, and solutions for the classical trajectory and spin precession in external fields will be given. In this paper an abstract spin vector is introduced to characterize the spin of a classical particle. Lagrangians for the classical trajectories and for the motion of the abstract spin vector are derived from corresponding quantum-mechanical Lagrangians by the WKBJ approximation method for nonrelativistic and relativistic particles. The equations of motion for the trajectory and the abstract spin vector following from the extremalization of these Lagrangians are given. The equations of motion for the precession in an external electromagnetic field of the spin vector (or tensor) in space–time is derived from the equations of motion for the abstract spin vector. In the relativistic case, they are equivalent to the Bargmann–Michel–Telegdi equations [Phys. Rev. Lett. 2, 435 (1959)]. The relationship between the ensemble and single-particle points of view is also elucidated.

Bertha Swirles Jeffreys

ABSTRACT The exact solution for a trapezoidal potential barrier is given in terms of Airy functions and trigonometrical functions. Numerical values are found for the ratio of the amplitudes of the transmitted and reflected waves. These are compared with those given by the WKBJ approximation.