Linear Algebra

Abstract

In this chapter, we attack one of the simplest but ubiquitous problems that a computational physicist typically faces, the solution of an eigenvalues problem. In quantum physics, the eigenvalues problem usually arise to solve a given Hamiltonian describing a single- or many-body quantum systems. Indeed, apart from a few classes of problems for which analytical solutions are known (from the particle in the box to the Bethe Ansatz solutions), the search for the ground and excited states of a quantum system shall be performed numerically. The numerical solution of the eigenvalues problem equals to the diagonalization of a matrix representing the Hamiltonian in a suitable basis, independently from the fact that the original system is discrete, e.g. a collection of qubits or atomic levels, or continuous, e.g. atoms or ions in free space or traps, or Bose-Einstein condensates. The latter scenarios shall be recast into discrete system either via space discretization (see Appendix B) or via second quantization. Finally, a plethora of problems in math, physics, and computer science can be recast in eigenvalues problems, thus hereafter we first introduce the problem and then present, in order of complexity, the state-of-art approaches for global and partial eigensolvers. As we will see, the eigensolver strategies are based on linear algebra operations that, for the sake of completeness, we briefly recall hereafter, also to set the notation: we will use throughout the book standard quantum mechanics notation and assume that the variables are complex. We base the quick overview presented here and in the next chapter (and appendixes) on many different excellent books and reviews, such as, e.g. [120–129].