The Definition of Entropy for Quantum Unstable Systems: A View-Point Based on the Properties of Gamow States.

Osvaldo Civitarese,Manuel Gadella

doi:10.3390/e20040231

Abstract

In this paper, we review the concept of entropy in connection with the description of quantum unstable systems. We revise the conventional definition of entropy due to Boltzmann and extend it so as to include the presence of complex-energy states. After introducing a generalized basis of states which includes resonances, and working with amplitudes instead of probabilities, we found an expression for the entropy which exhibits real and imaginary components. We discuss the meaning of the imaginary part of the entropy on the basis of the similarities existing between thermal and time evolutions.

Highlights

The definition of entropy and its interpretation in terms of the evolution to equilibrium of isolated systems was a crucial step in understanding the link between mechanical and thermal features in classical mechanics [1]
In the context of quantum mechanics in the Heisenberg picture, the time evolution of a system is governed by its Hamiltonian
From a conceptual point of view, we are faced to a difficult question, namely: If equilibrium appears in a particular instant of the time evolution of a system and is governed by a Hamiltonian, which is the operator that obeys Equation (1) and has the entropy given by the associated observable so that

Summary

Introduction

The definition of entropy and its interpretation in terms of the evolution to equilibrium of isolated systems was a crucial step in understanding the link between mechanical and thermal features in classical mechanics [1]. The main difference between classical and quantum mechanical counting of states is, the existence of the exclusion principle (for fermions) and other symmetry restrictions (both for fermions and bosons) imposed to quantum states In both cases, fermions and bosons, the definition of the probability assigned to a state remains valid. We are going to show that a comprehensive scheme leading to the definition of entropy for resonances can be rigorously designed by adopting path integration techniques. We shall discuss this method as well as its application to a model for resonances which is analytically solvable.

Entropy and time Evolution

Resonances in Quantum Systems

Complex Entropy

Time-Temperature Plane

Final Remarks