Triangle Geometry of Spin States and Nonlinear Superposition of Probabilities Describing These States†

Abstract

On the spin-1/2 example, we demonstrate that quantum states can be described by standard probability distributions, which contain the same information that the wave function and the density matrix do. Within the framework of this approach, called for spin-1/2 the quantum suprematism representation, the probability distributions are illustrated by simplex or triangle geometry or by the Triada of Malevich’s Squares (black, red, and white) associated with the triangles, and new quantum relations for areas of the squares are obtained. The superposition principle for spin states and quantum interference phenomenon are expressed as an explicit new nonlinear addition rule for the probability distributions describing the quantum states and illustrated as the addition of two Triadas of Malevich’s squares. We discuss some analogy of the triangle geometry of spin-1/2 states related to the O(3) symmetry group and the pyramide geometry related to the hydrogen-atom dynamical symmetry O(2, 4).