Asymptotic expansions in wave mechanics

• Published in: U.S.S.R. computational mathematics and mathematical physics 1964, Vol.4 (5), p.80-110
• Main Authors: Dubrovskii, V.A., Skuridin, G.A.
• Format: Article
• Language: English
• ISSN: 0041-5553
• DOI: 10.1016/0041-5553(64)90142-9

In this paper asymptotic expansions are obtained which describe the wave behaviour of particles. The essence of the method lies in the fact that a new variable (action) is introduced where a small parameter emerges as a constant ratio for the highest derivative. In this case it is convenient to investigate equations of wave mechanics with the help of a single asymptotic method. The new variable is introduced in order to obtain the necessary equation of the characteristics. Section 1 contains necessary information about the asymptotic method. In Section 2 on the basis of physical considerations the wave equations are given in five-dimensional form with action as the fifth coordinate. In Section 3 an asymptotic expansion in h is obtained and a Cauchy problem (dispersion problem) is solved. In Section 4 a connection between the asymptotic expansion and perturbation theory is established and it is shown that the series in perturbation theory so constructed is part of some asymptotic series. A series of perturbation theory, as we can easily see, is to some extent a classical limit, when e and → 0, so that e 2/ c = const. In Section 5 it is shown that, in the case of Schrödinger' s equation, on the basis of the formulae obtained, a bounded wave impulse moves in accordance with equations which, in the limit of an exact impulse, are the Hamiltonian equations of classical dynamics. Also this case is equivalent to the case when → 0. The formulae obtained in Section 5 generalize Ehrenfest' s theorem, and the method enables us to calculate any dynamical variables which characterize the wave behaviour of the particles. Here we show that the statistical explanation of ordinary quantum mechanics can be avoided.