Mathematical methods for quantum many-body systems

该项目将与具有多个粒子的系统相关，并将分为两个主要方向。第一个方向是关于太阳能电池，其基本原理是光伏效应，它的定义是材料在阳光下产生电位差和电流。主要的焦点是漂移-扩散方程，它描述了导电电子和空穴的密度受扩散、漂移以及带电载流子的产生和复合的影响。一些太阳能电池(如有机太阳能电池)也可以用所谓的高斯无序模型有效地描述。这既包括量子力学，也包括统计力学。可以利用该模型进行数值计算，也可以进一步研究该模型与微分方程法之间的联系。对于后者，我将使用各种方法来求解微分方程组，使用渐近法、摄动法等。请注意，现在的方程不仅取决于位置和时间，还取决于能量。该项目的这一方向为利用先进的数学方法并将其用于光伏领域提供了机会，在光伏领域，实验工作占主导地位，产生了积极的环境影响。第二个方向是关于多粒子系统的量子力学，主要焦点是描述具有多个粒子的系统的行为的二次量子化。如果出现在该方法中的运算符(创建和湮灭运算符)被变量替换，则该方法会导致经典的极限。这类似于用经典力学中的位置和动量变量来代替量子力学中的位置和动量算符。特别有趣的是，这些算符也可能是混沌的，而多粒子量子混沌是目前数学物理界的热门话题。我也可以从单粒子量子混沌开始，利用现有的工作首先进行量子化，然后将其连接到多粒子量子系统。这些研究方向是基于初步的结果，包括我的主管的工作以及我第三年和第四年的项目，在前者，我使用微扰理论等技术推导和求解漂移-扩散方程，而在后者，我考虑二次量子化，并试图使用正则微扰理论描述具有许多粒子的系统的行为。这些主题之间的平衡将在项目进展过程中加以调整。相应的EPSRC研究领域是数学物理(在数学科学主题中)和太阳能技术(在能源主题中)。

This project will be related to systems with many particles and it will be divided into two main directions. The first direction is about solar cells where the basic principle is the photovoltaic effect, which is defined as the generation of electric potential difference and electric current in a material upon exposure to sunlight. The main focus is the drift-diffusion equations which describe the density of conducting electrons and holes as governed by diffusion, drift, and well as generation and recombination of charged carriers. Some solar cells (such as organic solar cells) can also be effectively described by the so-called Gaussian disorder model. This incorporates quantum mechanics as well as statistical mechanics. Numerical work could be done using this model, or extended research about the connection between this model and the differential equations approach. For the latter, I will use various ways to solve the differential equations using asymptotics, perturbation, etc. Note that now the equations do not only depend on position and time but also on energy. This direction of the project gives the opportunity to take advantage of advanced mathematical methods and use them in the area of photovoltaics, where experimental work is dominant, resulting in a positive environmental impact. The second direction is about quantum mechanics of many-particle systems and the main focus is second quantisation that describes the behaviour of systems with several particles. This approach leads to a classical limit if the operators (the creation and annihilation operators) appearing in it are replaced by variables. This is similar to replacing the position and momentum operators in quantum mechanics with position and momentum variables in classical mechanics. It is particularly interesting because these operators can also be chaotic and many-particle quantum chaos is a currently hot topic in the mathematical physics community. I can also start from single-particle quantum chaos and first quantisation using existing work and then connect it to many-particle quantum systems. These lines of research are based on preliminary results, including work of my supervisor and from my 3rd and 4th year projects, where on the former I derived and solved the drift-diffusion equations using techniques such as perturbation theory and, on the latter, I considered second quantisation and I tried to describe the behavior of systems with many particles using canonical perturbation theory. The balance between these topics will be adapted during the progress of the project. The corresponding EPSRC research areas are Mathematical Physics (in the Mathematical Sciences theme) and Solar Technology (in the Energy theme).