Uncertainties, Localization, Stability and Decay of Quantum Systems

Abstract

Uncertainties, localization and stability are significant features concerning quantum systems. These are aspects which distinguish the latter systems from classical ones. This chapter is involved with such key points including related ones as the boundedness of the spectra of Hamiltonians of physical systems from below. This property of boundedness from below is important as otherwise a system would collapse to such a level as to release, in the process, an infinite amount of energy which is physically meaningless and the system, in question, would be unstable. The first section deals with analyses related to investigating the probability of the “fall” of a particle into another in a bound state as well as in determining a lower bound for the average spatial extension of such a system and in deriving an upper bound for the average kinetic energy into consideration in addition to other details. This includes investigating the nature of the resistance of a system with a large number of Fermi particles to the increase in its density. The boundedness of the spectra of Hamiltonians from below is the subject of §3.2–§3.4 for several classes of interactions and §3.4 is involved with multi-particle systems. The final section §3.5 is concerned with the decay of quantum systems and special emphasis is put on the celebrated Paley-Wiener Theorem for describing a physically consistent theory of quantum decay in which the underlying Hamiltonians are bounded from below.