文档介绍：Journal of Statistical Physics, Vol. 68, Nos. 1/2, 1992
Quantum Chaos, Classical Randomness, and
Bohmian Mechanics
Detlef Diirr, 1,2 Sheldon Goldstein, 1 and Nino Zanghil' 3
Received February 13, 1992
It is argued that dynamical chaos in quantum mechanics arises solely from the
collapse rule applied in measurements. As such it is quite distinct from classical
(deterministic) chaos, which arises from the dynamical law itself. It is shown,
however, that if the particles of a quantum system are regarded as "real," .,
if their positions are made part of the state description, one obtains a formula-
tion of quantum theory, Bohmian mechanics, in which "quantum chaos" also
arises solely from the dynamical law. Moreover, this occurs in a manner far
simpler than in the classical case.
KEY WORDS: Quantum chaos; quantum randomness; sensitive dependence
on initial conditions; Bohmian mechanics; Bernoulli system; hidden variables.
1. INTRODUCTION
A characteristic feature of chaotic classical dynamical systems is the ran-
domness or unpredictability of their behavior. Randomness and unpredic-
tability are also, of course, characteristic features of quantum phenomena.
However, they are not to be found in the quantum dynamics, the
Schr6dinger evolution, itself. This evolution is very regular, in fact
preserving distances in Hilbert space. Thus, it can have no sensitive
dependence on initial conditions, the hallmark of classical chaos: A (small)
variation 6~o in the initial wave function
4'o --' ~o + 60o
1 Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903.
2 Fakultfit ftir Mathematik, Universit/it M/inchen, 8000 M/inchen 2, Germany.
3 Istituto di Fisica, Universit/t di Genova, INFN, 16146 Genova, Italy.
259
0022-4715/92/0700- 1992 Plenum Publishing Corporation
260 D/irr et at.
leads to the variation 6Or,
~t--, ~,, + ~,,
at later times whose magni