Localized Electron Waves

What is solid state physics [1–8]? It is not easy to give a simple and complete answer, but let’s try. Take a solid material like a metal, which conducts electricity. The metal has atoms arranged in a periodic fashion. We are taught in high school physics, that there are mobile electrons in the metal. How do they move when, say I apply an electric field? Closer inspection will tell answering this is a daunting task. After all, there is a forest of atoms so-called lattice, which will attract (pull) the electron. We can write the electric field the electron sees due to all of the lattice and the electric field that we apply and try to solve for the Newton equations describing the motion of an electron.