Time-Dependent Perturbation Theory

Abstract

In many physical problems it is necessary to consider the collision of a particle with another particle or system of particles. The treatment based on Classical Mechanics is given in Sects. 3.53.6with reference to the motion’s asymptotic conditions, without considering the form of the interaction, while Sect. 3.8shows a detailed treatment of the Coulomb interaction. Here the approach based on Quantum Mechanics is shown, dealing with the following problem: a particle in a conservative motion enters at t = 0 an interaction with another particle or system of particles; such an interaction has a finite duration tP, at the end of which the particle is in a conservative motion again. The perturbation produced by the interaction, which is described by a suitable Hamiltonian operator, may change the total energy of the particle; the analysis carried out here, called time-dependent perturbation theory, allows one to calculate such an energy change. The other particle or system, with which the particle under consideration interacts, is left unspecified. However, it is implied that the larger system, made of the particle under consideration and the entity with which it interacts, form an isolated system, so that the total energy is conserved: if the particle’s energy increases due to the interaction, then such an energy is absorbed from the other entity, or vice versa. As in Classical Mechanics, other dynamical properties of an isolated system are conserved; an example of momentum conservation is given in Sect. 14.8.3. The discussion is carried out first for the case where the eigenvalues prior and after the perturbation are discrete and non degenerate. Starting from the general solution of the perturbed problem, a first-order approximation is applied, which holds for small perturbations, and the probability per unit time of the transition from a given initial state to another state is found. The analysis is repeated for the degenerate case (still for discrete eigenvalues) and, finally, for the situation where both the initial and final state belong to a continuous set. The last section shows the calculation of the perturbation matrix for a screened Coulomb perturbation. The complements deal with the important problems of a perturbation constant in time and a harmonic perturbation; a short discussion about the Fermi golden rule and the transitions from discrete to continuous levels follows.