Second Quantization

TL;DR: Second quantization formalism for interacting systems of many identical particles. Introduction of second quantization formalism for non-interacting and interacting systems.

Abstract: Our goal here is to introduce a formalism known as second quantization, which is extremely useful when dealing with interacting systems of many identical particles. The formalism is also commonly used even for non-interacting systems. We discussed systems of identical particles last semester; here we will review the basic facts about these systems that are needed to introduce second quantization. Let’s start by supposing we have a system of N spinless bosons or fermions with mass m. The particles have positions ri and momenta pi (i = 1, . . . , N). We will see later how to add in spin without any difficulty. Let’s consider the following Hamiltonian: