Quantum Mechanics of Stochastic Systems

TL;DR

This paper introduces a quantum framework for stochastic systems, showing how classical probability distributions emerge from quantum harmonic oscillators through specific perturbations, enabling quantum probability engineering and true random number generation.

Contribution

It develops a novel operator algebra and perturbation method that connect quantum mechanics with classical stochastic processes, providing a new foundation for probability realization and randomness generation.

Findings

Classical distributions arise from quantum harmonic oscillator perturbations.

A complete operator algebra for moment and information analysis is constructed.

Supports true random number generation without external whitening processes.

Abstract

We develop a fundamental framework for the quantum mechanics of stochastic systems (QMSS), showing that classical discrete stochastic processes emerge naturally as perturbations of the quantum harmonic oscillator (QHO). By constructing exact perturbation potentials that transform QHO eigenstates into stochastic representations, we demonstrate that canonical probability distributions, including Binomial, Negative Binomial, and Poisson, arise from specific modifications of the harmonic potential. Each stochastic system is governed by a Count Operator (N), with probabilities determined by squared amplitudes in a Born-rule-like manner. The framework introduces a complete operator algebra for moment generation and information-theoretic analysis, together with modular projection operators (R_M) that enable finite-dimensional approximations supported by rigorous uniform convergence theorems. This mathematical structure underpins True Uniform Random Number Generation (TURNG) [Kuang, Sci. Rep., 2025], eliminating the need for external whitening processes. Beyond randomness generation, the QMSS framework enables quantum probability engineering: the physical realization of classical distributions through designed quantum perturbations. These results demonstrate that stochastic systems are inherently quantum-mechanical in structure, bridging quantum dynamics, statistical physics, and experimental probability realization.