Stochastic mechanics and neural networks

Abstract

Quantum mechanics is one of the most successful theories of modern physics. Nelson’s stochastic mechanics description offers an alternative approach to non-relativistic quantum mechanics based on Newtonian mechanics. In this thesis, we derive the stochastic mechanical equivalent to the quantum-mechanical Rayleigh-Ritz principle. This principle is then used to build a genetic algorithm that calculates the osmotic velocity together with the ground state energy by first minimizing the derived energy functional and then solving a Riccati equation. This new and efficient algorithm is then used to solve the ground state of two tweezer potentials used in levitodynamics. These are then compared to each other and with the harmonic oscillator by using methods from time series analysis. Additionally, we use the Itô-formula to derive two coupled stochastic differential equations, which yields a phase space description of quantum mechanics without violating the Heisenberg uncertainty principle.