Measurement in Quantum Mechanics and an Application to The Quantum Zeno Effect

Abstract

This thesis explores the quantum Zeno effect, a phenomenon in which frequent measurements can inhibit the evolution of a quantum system. The aim of the thesis is to make this statement mathematically precise. To this end, the first chapter introduces the mathematical language needed throughout the thesis. The following chapter then develops the basic formalism of quantum theory based on [2], explaining how states describe the possible condition of a system and how observables describe the possible outcomes of measurements. This leads to chapter 4, where quantum measurements are discussed as mathematical procedures rather than merely informal acts of observation. In particular, we consider idealized properties that a measurement may satisfy, such as repeatability, and focus on the sharp measurements relevant for the Zeno effect. With these ingredients in place, the thesis culminates in a rigorous formulation of the idea expressed in the opening sentence. The main result concerns the limiting behavior of a quantum system subjected to increasingly frequent measurements of a particularly ideal type, namely L¨uders measurements. Between successive measurements, the system is allowed to evolve according to its usual time evolution. Under suitable assumptions, we prove that in the limit of infinitely frequent measurements the time evolution of the system can freeze.