Classical and Quantum Dynamics 2nd Edition

Author: W. Dittrich et al.
Publisher:
Publish Date: 1998-03-01
Features:
Fragment: That quantum and classical mechanics are, in fact, disjoint physical worlds was clear from the very beginning. Present-day experience is no exception; it is rather embarrassing to find out that an important geometric phase in a cyclic adiabatic quantum process has been overlooked since the dawn of quantum mechanics. This so-called Berry phase signals that in nonrelativistic as well as relativistic quantum theory, geometric methods play an eminent role. The appearance of topology in quantum mechanics is probably the most important new development to occur in recent years. A large portion of this text is therefore devoted to the geometric structure of topologically nontrivial physical systems. Berry phases, Maslov indices, Chern-Simons tensors, and various other topological quantities have clearly demonstrated that quantum mechanics is not, as of yet, a closed book.
1. The Action Principles in Mechanics
We begin this chapter with the definition of the action function as the time integral over the Lagrangian \( L(q(t), \dot{q}(t); t) \) of a dynamical system: Here, \( q_i, i=1,2,...,N \), are points in \( N \)-dimensional configuration space. Thus, \( q_i(t) \) describes the motion of the system, and \( \dot{q}_i(t) = \frac{dq}{dt} \) determines its velocity along the path in configuration space. The endpoints of the trajectory are given by \( q_i(t_1) = q_{i1} \) and \( q_i(t_2) = q_{i2} \). Next, we want to find out what the actual dynamical path of the system is. The answer is contained in the principle of stationary action: in response to infinitesimal variations of the integration path, the action \( S \) is stationary, \( \delta S = 0 \), for variations about the correct path, provided the initial and final configurations are fixed. On the other hand, if we permit infinitesimal changes of \( q_i(t) \) at the initial and final times, including alterations of those times, the only contribution to \( \delta S \) comes from the endpoint variations, or \( \delta S = G(t_2) - G(t_1) \). (1.2) Equation (1.2) is the most general formulation of the action principle in mechanics. The extremal values \( G_1 \) and \( G_2 \) depend only on the endpoint path variables at the respective terminal times. Again, given a system with the action function \( S \), the actual dynamical path in configuration space follows that path about which general variations produce only endpoint contributions. The explicit form of \( G \) depends on the special presentation of the action principle. In the following, we begin with the one that is best known, i.e.