Orientations in the Plane as Quantum States

The aim of the present article is to introduce and to discuss some of the most basic fundamental concepts of quantum physics by using orientations or angles in the plane, illustrated through linear polarizations. We start with the Euclidean plane, which is certainly a paradigmatic example of a Hilbert space. The orientations in the plane are identified with the pure states. Associating these quantum orientations with linear polarizations of light in the plane normal to its propagation constitutes a nice illustration of the presented formalism. Another contribution concerns the description of the dynamics of quantum states. The pure states form the unit circle (actually a half of it) and the mixed states form the unit disk (actually a half of it). Rotations in the plane rule time evolution through Majorana-like equations involving only real quantities for closed (Heisenberg-Dirac) and open (Lindblad) systems. Interesting probabilistic aspects are developed. Finally, we present a comprehensible example of quantum measurement with pointer states lying also in the Euclidean plane.