A Path from the Quantization of the Action Variable to Quantum Mechanical Formalism

Abstract

Starting from the quantization of the action variable as a basic principle, I show that this leads one to the probabilistic description of physical quantities as random variables, which satisfy the uncertainty relation. Using such variables I show that the ensemble-averaged action variable in the quantum domain can be presented as a contour integral of a “quantum momentum function,” pq(z), which is assumed to be analytic. The condition that all bound states pq(z) must yield the quantized values of the action variable requires the equation for pq(z) to be the Riccati differential equation, which can be converted to the Schrodinger equation for the wave function. The rule for probabilities in quantum mechanics follows from the physical meaning of pq(z) and its connection with the probability density. The discussion involves the correspondence principle and the assumption that the spacetime symmetries known in classical physics are valid in the quantum domain.