On the Concept of Entropy for Quantum Decaying Systems

Göran Lindblad

Osvaldo Civitarese, Manuel Gadella

The concept of entropy was initially defined for systems with thermodynamical equilibrium. We try to extend this notion for quantum non-relativistic decaying states. We use a technique based on path integration on coherent states in order to obtain an approximation to the entropy of a decaying state.

Ronnie Kosloff, Stuart A. Rice

An extension of the concept of Kolmogorov entropy to quantum mechanical systems is given. Using the Kolmogorov entropy as a common basis for discussion, the onset of chaotic motion in classical mechanical and quantum mechanical systems is compared. It is found that if the spectrum of the system is discrete, the Kolmogorov entropy is zero, and therefore that a bounded quantum mechanical system cannot have chaotic motion like that observed for a corresponding classical mechanical system. This analysis leads to the prediction that if the distributions of states are similar for two bounded systems which in the classical limit have, respectively, quasiperiodic and chaotic motion, wave packets for the two systems decay at similar rates as found by Brumer and Shapiro. The relationship between the defined Kolmogorov entropy, previous interpretations of ’’KAMlike’’ onset of chaos in quantum mechanical systems, and the role played by preparation and observation of a system in influencing the intramolecular dynamics, are discussed.

Makoto Ohya, D�nes Petz

Ugo Farinelli, A. Gamba

Dannenberg, O

We study a simplified Heisenberg spin model in order to clarify the idea of decoherence in closed quantum systems. For this purpose, we define a new concept: the coherence function $\\Xi(t)$, which describes the dynamics of coherence in the whole system, and which is linked with the total coarse-grained (von Neumann) entropy of all particles. We discuss in some detail a general coherence theory and its elementary results. For example, in the particular setup, decoherence diagonalises reduced density matrices in all possible basis sets. As expected, decoherence is understood as a statistical process that is caused by the dynamics of the system, similar to entropy. Moreover, the concept of decoherence time is applicable in closed systems. However, in most cases, the decay of off-diagonal elements differs from the usual $\\exp(-t/\\tau_d)$ behaviour. We have solved the form of decoherence time $\\tau_d$ in an infinite Heisenberg model with respect to density $\\rho$, spatial dimension $D$ and $\\epsilon$ in a $1/r^{\\epsilon}$ -type of potential.