Stochastic differential equations in Hubert space find fruitful applications both for collapse models and for the theory of open quantum systems. Collapse models are one of the few mathematically well-defined and physically coherent solutions of the celebrated measurement problem of quantum mechanics.
By modifying the standard SchrÃdinger equation adding appropriate stochastic terms, it is possible to embody within one single equation both the (quantum) dynamics of microscopic systems and the (classical) dynamics of macroscopic objects. The resulting equation is a non unitary, non linear, but norm preserving, Ito stochastic differential equation in Hubert space. The general mathematical features of collapse models are well known, and simple physical systems have been studied in detail.
Yet, the literature lacks of a complete and mathematically rigorous treatment of important physical systems, e.g. the harmonic oscillator, the hydrogen atom... Such a kind of analysis is particularly difficult because of the non linear character of the equations of motion, but important for the understanding of the reduction mechanism.
The aim of this research project is to study in full details the following systems:
- The free particle and the system of two "free" identical particles.
- The hydrogen atom interacting with the radiation field.
- A model for a rigid body consisting of particles interacting through harmonic potentials.
Although some properties (like asymptotic behaviours) can be obtained by analytical tools, a complete analysis of such systems can be achieved only by resorting to numerical simulations. Several analytical and numerical techniques will be employed to tackle these problems.
The training objective of the proposal is to acquire a solid knowledge of stochastic processes, stochastic differential equations and of numerical algorithms for solving stochastic differential equations.
Call for proposal
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