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The purpose of this research is to better understand the static and dynamical properties of quantum
systems. What makes a quantum system a metal, a material in which particles can move easily? What
makes it an insulator, a material in which particles are trapped to confined regions of space? What
triggers the transition from a metallic phase to an insulating phase? To answer these questions, we
consider quantum systems described by Hamiltonian matrices that have a tridiagonal form. A
tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the first
diagonal below the main diagonal, and the first diagonal above the main diagonal. We study how the
eigenvalues and eigenstates of these matrices depend on the parameters of the Hamiltonian and use
these results to predict the dynamical behavior of the system.
The advantage of using tridiagonal matrices is that they are easy to study numerically and in some
cases can even lead to analytical solutions. Despite their simplicity, they model realistic systems of
nature and provide a pedagogical framework to learn about concepts and tools of quantum mechanics
which include: quantum spins, quantum superposition, delocalization measures, the Schrödinger
equation, and quantum phase transitions.