Mathematical methods for quantum many-body systems

这个项目将与多粒子系统有关，它将分为两个主要方向。第一个方向是关于太阳能电池的，其基本原理是光伏效应，即材料在阳光照射下产生电位差和电流。主要的焦点是漂移-扩散方程，它描述了由扩散、漂移以及带电载流子的产生和重组所控制的导电电子和空穴的密度。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。这结合了量子力学和统计力学。利用该模型可以进行数值计算，也可以对该模型与微分方程方法之间的联系进行扩展研究。对于后者，我将使用各种方法来解微分方程，使用渐近性，摄动等。注意，现在方程不仅依赖于位置和时间，还依赖于能量。该项目的这个方向提供了利用先进的数学方法并将其用于光伏领域的机会，在该领域，实验工作占主导地位，从而产生积极的环境影响。第二个方向是关于多粒子系统的量子力学，其主要焦点是描述具有多个粒子的系统行为的二次量子化。如果其中出现的算符（创造和湮灭算符）被变量取代，这种方法将导致经典极限。这类似于用经典力学中的位置和动量变量代替量子力学中的位置和动量算子。这是特别有趣的，因为这些算子也可以是混沌的，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，首先利用现有的工作进行量子化，然后将其与多粒子量子系统联系起来。这些研究是基于初步结果，包括我的导师的工作，从我的第三和第四年的项目，在前者我推导和解决漂移扩散方程使用技术，如摄动理论，在后者，我考虑二次量化，我试图描述系统的行为与许多粒子使用规范摄动理论。这些主题之间的平衡将在项目的进展过程中进行调整。相应的EPSRC研究领域是数学物理（数学科学主题）和太阳能技术（能源主题）。

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).