H2020Индивидуална стипендия2020–2022

COCONUT · Computational Complexity in Quantum Mechanics

„Хоризонт 2020“ — Действия „Мария Склодовска-Кюри“

Период: 2020-10-01 → 2022-09-30
Финансиране от ЕС: 212 934 €
Участници: 1
Схема: MSCA-IF

Линиите свързват координатора с партньорите.

Накратко на български

Изчислителната сложност в квантовата механика се анализира чрез йерархия, която проследява стъпките за решаване на задачи, като например определянето на енергийните нива в атома на водорода. Това помага за по-доброто разбиране на границите на изчислимостта и точността при работа с математически модели.

Този кратък обзор е генериран от изкуствен интелект

Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.

Резултати накратко

Computational Complexity in Quantum Mechanics

This project studies computational problems in quantum mechanics in the framework of the Solvability Complexity (SCI) Hierarchy. The SCI Hierarchy is a novel theory concerned with the fundamental boundaries of computability. It started out as a way to rigorously track the "number of successive limits" that are needed to solve a computational problem using a computer algorithm. Since the beginning of the project, the field has evolved into a sophisticated and comprehensive classification hierarchy that combines information about solvability, convergence and error estimation. While the theoretical concepts are deliberately kept generic enough to encompass almost any computational problem in mathematics, the main focus of applications so far has been in the mathematical fields of spectral theory and partial differential equations. These equations are of particular relevance in Quantum Mechanics, where they describe the time evolution of particle states. The famous Schrödinger equation forms the basis of modern quantum mechanics. It describes nonrelativistic particles in a force field described by a so-called potential function V. As a prime example, the reader may consider the movement of an electron in the electric field of a proton. Together, the two particles form a hydrogen atom, which can have different states of excitation (depending on the amount of energy the system absorbs from its environment). These excited states are discrete (or quantised) - they only occur at very specific energies. The Schrödinger equation is able to correctly predict these energies: they are given by the so-called eigenvalues (or the spectrum) of the equation. Applications of these quantised energy states are vast: for example, they form the basis for the entire field of spectroscopy - a branch of Physics, which allows us to learn what the stars in our galaxy are made of, or to determine the chemical composition of our atmosphere via its absorption of sunlight. Each chemical element has its own energy spectrum, which identifies it like a fingerprint.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз

Цел на проекта

The goal of this Fellowship is to derive quantitative estimates on the computational complexity of spectral problems in quantum mechanics. The theoretical framework for this task is provided by the so-called Solvability Complexity Index, which roughly speaking, is the number of successive limits needed to solve the computational problem. I will approach this task by combining techniques from numerical analysis with modern methods from spectral approximation theory.The project is divided into three concise work projects:WP1: NONRELATIVISTIC QUANTUM SYSTEMS.In this project, the spectral problem for Schrödinger operators with various types of potentials is studied. New sharp estimates on the computational complexity are derived. This will contribute to a comprehensive understanding of the nonrelativistic theory.WP2: RESONANCES.In this second project, complexity issues are considered for the computation of scattering resonances in quantum mechanics. I will introduce new mathematical tools, which have not been used in complexity theory before to construct algorithms which compute the set of resonances of Schrödinger operators in one limit.WP3: EXTENSION TO RELATIVISTIC THEORY.The purpose of the final project is to extend the above results to the relativistic setting, in which the Schrödinger operator is replaced by a Dirac operator. This task is far from trivial, as methods from the Schrödinger case are generally not useful for Dirac operators.I also have robust career development and public outreach agendas, to complement the scientific aspects of this proposal. Combined, all these elements will establish me as a prominent research leader upon my return to Germany, with extensive links throughout Europe and the US.

Оригинален текст от CORDIS (на английски).

Участници

CARDIFF UNIVERSITY · CARDIFFКоординаторОбединеното кралство