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THE ENERGY CONSERVATION LAW IN QUANTUM DISTURBANCE THEORY

V. L. Skopich

Published Dec 1, 2012• 0 citations
OpenAlex

Abstract

It is demonstrated, that usual application of certain formulas in limits of the first approximation of the quantum perturbation theory is unsubstantial. Besides, it has been refused current opinion about the energy conservation law’s violation for little interval of time between initial and final states considered system.

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In this chapter, we present some fundamental issues about approximation methods that are often used when a quantum-mechanical system is perturbed and about the relationship between classical and quantum mechanics. In Sec. 10.1 we introduce the stationary perturbation theory, while Sec. 10.2 is devoted to time-dependent perturbations. In Sec. 10.3 we briefly examine the adiabatic theorem. In Sec. 10.4 we introduce the variation method, an approximation method that is not based on perturbation theory. In Sec. 10.5 we discuss the classical limit of the quantum-mechanical equations, whereas in Sec. 10.6 we deal with the semiclassical approximation, in particular the WKB method. On the basis of the previous approximation methods in Sec. 10.7 we present scattering theory. Finally, in Sec. 10.8 we treat a method that has a wide range of applications: the path-integral method. Stationary perturbation theory Perturbation theory is a rather general approximation method that may be applied when a small additional force (the perturbation) acts on a system (the unperturbed system), whose quantum dynamics is fully known. If the disturbance is small, it modifies both the energy levels and the stationary states. This allows us to make an expansion in power series of a perturbation parameter, which is assumed to be small. Perturbation theory may be applied both to the case where the additional force is time-independent (in which case a stationary treatment suffices – the subject of the present section) as well as to the case where it explicitly depends on time.
It is pointed out that the property of a constant energy characteristic for the circular motions of macroscopic bodies in classical mechanics does not hold when the quantum conditions for the motion are applied. This is so because any macroscopic body—lo-cated in a high-energy quantum state—is in practice forced to change this state to a state having a lower energy. The rate of the energy decrease is usually extremely small which makes its effect uneasy to detect in course of the observations, or experiments. The energy of the harmonic oscillator is thoroughly examined as an example. Here our point is that not only the energy, but also the oscillator amplitude which depends on energy, are changing with time. In result, no constant positions of the turning points of the oscillator can be specified; consequently the well-known variational procedure concerning the calculation of the action function and its properties cannot be applied.
If the initial and the final states of a system have classical limits, and if the classical limits cannot be chosen independently from each other without violating classical conservations laws, the JWKB approximation of the probability amplitude for the transition between such initial and final states requires special care. We compute it explicity for momentum to momentum transitions and for angular momentum to angular momentum transitions. It is shown how the classical conservation laws make their appearance in the limit $\ensuremath{\hbar}=0$. The statements are illustrated by applications to potential scattering theory.