A quantum mechanical toy model for black holes

Abstract

The main aim of this dissertation is to investigate properties of the entropy of black holes. Our primary goal is to investigate the microstates of black holes. Our secondary goal is to study decoherence within the context of black holes. The methodology employed is to study simple, exactly solvable quantum models. These quantum models should serve as toy models for black holes. We consider exactly solvable quantum systems which have a non-degenerate energy spectrum. The energy levels of these quantum systems should not be equally spaced. By choosing an appropriate class of observables, we calculate the expectation values of these observables for different states within a suitably chosen ensemble. This is where the notion of quantum typicality arises. By comparing the expectation values of the chosen observable for several states within the ensemble, we discover that it is not always possible to distinguish among the several states. These findings are then generalised to the microstates of black holes, i.e. no measurement can distinguish black hole microstates. We then study the coherent and squeezed states of a simple quantum system. We deduce that even for such states, distinguishability is not possible. Finally, we study decoherence within the context of black holes. We find a simple quantum model that exhibits decoherence. We conclude that spacetime fluctuations can cause decoherence in quantum systems. Furthermore, by treating Hawking radiation as an effect of decoherence, we conclude that no information is lost in a black hole.