Quantum Kinematics of Unstable Systems

Abstract

Let us first present a few historical facts and a brief survey of the problems related to unstable quantum systems. This chapter is devoted to a discussion of the kinematical properties (in the quantum-theoretical sense) of unstable systems, especially their time evolution. We begin with the basic notions and formulation of postulates which are expected to be obeyed be any unstable quantum system (Section 2). Since they represent just one of several approaches towards a description of open systems within the quantum theory, we also reviewhere the other existing methods. We then discuss how the decaying system behaves over short intervals (i.e. soon after the instant of preparation), and dependence of this behaviour on the energy distribution of the initial state (Section 3). Since each unstable system can be viewed as a part of some larger isolated system, there arises the natural question: ‘Can one reconstruct a complete description of the decay from the knowledge of the time evolution of the unstable system itself?’ This is the so-called inverse decay problem treated in Section 4: we present there the basic criterion for existence and uniqueness of the solution as well as its simple properties. The most frequently used and practically successful way of describing the decay processes starts from the assumption that the time evolution of an unstable system is governed by an operator semigroup. However, the well-known difficulty of the below unbounded total energy then arises. In Section 5 we discuss th is problem in detail, and at the same time present other properties of the energy spectrum together with some examples. Thus we have to show the sense in which the semigroup description may be used as an approximation to the true reduced evolution. With this purpose, we treat the so-called bounded-energy approximation (Section 6), which has a natural physical meaning. A discussion of the quantitative aspects of the problem is postponed until the next chapter.