Implementation of Second Quantization for Fermionic systems in Python : An application to the Tight-binding and Hubbard models

Sammanfattning

The Quantum-Mechanical description of subatomic particles is the most accurate theory of these particle’s behavior. The first quantization is the standard method for solving the Schrodinger equation, but the dimension of the partial differential equation increases rapidly, especially when dealing with more than one particle. These differential equations become difficult to solve and, although there exist methods in this formalism, a new method is approached. This thesis starts with an introduction where the first quantization is discussed in multi-particle physics and electronic systems. The second quantization formalism is then introduced and discussed with some definitions and notation. Some applications of the new formalism are then discussed, such as some spin operators and the number operator. Some new models and operators are introduced, such as the tight-binding model, the Hubbard model, and the number operator. Some numerical applications are also discussed for these models and operators. The second quantization formalism is then implemented in Python, and some analysis is performed for the tight-binding and Hubbard model. In other words, this thesis focuses on the implementation of the second quantization formalism for fermions in Python and is then used to analyze two models. It is also a first step into the multi-particle quantum mechanics.