Braun D. Dissipative quantum chaos and decoherence (STMP 172, Springer, 2001)( 3540411976)(125s).pdf

1. Introduction
The notion of “chaos” emerged in classical physics about a century ago with
the pioneering work of Poincar´e. After two and a half centuries of application
of Newton’s laws to more and plicated astronomical problems, he
was privileged to discover that even in very simple systems -
plicated and unstable forms of motion are possible [1]. It seems that this first
appeared a curiosity to his contemporaries. Moreover, quantum mechanics
and relativistic mechanics were soon to be discovered and distracted most of
the attention from classical problems. In any case, classical chaos interested
mostly only mathematicians, from G. Birkhoff in the 1920s to Kolmogorov
and his coworkers in the 1950s. Only Einstein, as early as 1917, . even
before Schr¨odinger’s equation was invented, clearly saw that chaos in classi-
cal mechanics also posed a problem in quantum mechanics [2]. The rest of
the world started to realize the importance of chaos only puters
allowed us to simulate simple physical systems. It then became obvious that
integrable systems, with their predictable dynamics, that had been the back-
bone of physics for by then three centuries were an exception. Almost always
there are at least some regions in phase space where the dynamics es
irregular and very sensitive to the slightest changes in the initial conditions.
The in principle perfect predictability of classical systems over arbitrary time
intervals given a precise knowledge of all initial positions and momenta of all
particles involved is entirely useless for such “chaotic” systems, as initial
conditions are never precisely known.
The understanding of quantum mechanics naturally developed first of all
with the solution of the same integrable systems known from classical me-
chanics, such as the hydrogen atom (as a variant of Kepler’s problem) or the
harmonic oscillator. With the growing conviction that integrable systems are
a rare exception, it became natural to ask how