Operators

Abstract

In this chapter, we recall the material from the previous chapters and summarise how vectors and operators play different roles in quantum mechanics. After an introduction to so-called eigenvalue problems, we discuss observables and evolution operators, as well as projectors and commutators. They provide us with the essential mathematical techniques for later chapters. Mathematically, quantum mechanics is a theory about calculating probabilities using the concepts of states and operators. In this chapter we are going to study the most important aspects of operators for quantum mechanics, and show how it fits in with the theory. First, we will address the most fundamental properties of operators, namely their eigenvalues and eigenstates. In the previous chapter you have encountered the eigenvalue equation of the Hamiltonian $H |E_n\rangle = E_n |E_n\rangle \, ,$ where H is an operator, and $|E_n\rangle$ is an eigenstate of H with eigenvalue $E_n$. The Hamiltonian is the energy operator and the eigenvalues $E_n$ are the energies that the system can possess. The states (or vectors) $|E_n\rangle$ describe the system when it has the energy $E_n$.