Complex Vector Spaces and Quantum Mechanics

Abstract

Up to now, we have occasionally used the terms ‘vector space’ or ‘state space’. In this chapter, we will address this concept in more detail. For reasons of simplicity, we will rely heavily on the example of polarization, where the basic formulations are of course independent of the specific realization and are valid for all two-dimensional state spaces (such as polarization states, electron spin states, a double-well potential, the ammonia molecule, etc.). Moreover, the concepts introduced here retain their meaning in higher-dimensional state spaces, as well. Therefore, we can introduce and discuss many topics by using the example of the simple two-dimensional state space. From the technical point of view, this chapter is about the discussion of some of the elementary facts of complex vector spaces.