From probabilities to quantum and classical mechanics

Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that basic mathematical structures of quantum mechanics like the probability amplitudes, Born rule, commutation and uncertainty relations, momentum operator, probability density current, rules for including the scalar and vector potentials and antiparticles can be obtained from the definition of the mean values of powers of the space coordinates and time. Equations of motion of quantum mechanics, the Klein–Gordon equation, Schrödinger equation and Dirac equation, are obtained from the requirement of relativistic invariance of the theory. Limit case of localized probability densities yields the Hamilton–Jacobi equation of classical mechanics. Many particle systems are also discussed.