Representations of Vectors and Operators

Abstract

We have managed to get through four chapters of this text without specifying any concrete form of the state vectors, and treating them as some abstractions defined only by the rules of the games that we could play with them. This approach is very convenient and rewarding from a theoretical point of view as it emphasizes the generality of quantum approach to the world and allows to derive a number of important general results with relative ease. However, when it comes to responding to experimentalists’ requests to explain/predict their quantitative experimental results, we do need to have something a bit more concrete and tangible than the idea of an abstract vector. The similar situation actually arises also in the case of our regular three-dimensional geometric vectors. It is often convenient to think of them as purely geometrical objects (arrows, for instance) and derive results independent of any choice of coordinate system. However, at some point, eventually, you will need to get to some “down-to-earth” computations, and to carry them out, you will have to choose a coordinate system and replace the “arrows” with a set of numbers—the vector components.