Mathematical methods for quantum many-body systems

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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).