Dynamics and averages in Hilbert space

Abstract

In the previous chapter the concept of typicality has been introduced for a rather abstract space of the states of a system. Also the properties of the dynamics which are imperative for the applicability of the concept have been denoted quite formally. Here we introduce the Hilbert space of pure quantum states as our concrete state space and formulate a representation of Hilbert space. Within this representation the dynamics as described by the Schrödinger equation indeed meets the above requirements. Furthermore the averages and variances that occur in the context of typicality are mathematically concretized for this quantum case and thus accordingly called Hilbert space average and Hilbert space variance, respectively. © 2009 Springer-Verlag Berlin Heidelberg.