A solid material subjected to mechanical and thermal loading will change its shape and develop internal stress and temperature variations. What is the best way to describe this behavior? In principle, the response of a material (neglecting relativistic effects) is dictated by that of its atoms, which are governed by quantum mechanics. Therefore, if we could solve Schrödinger's equation for all of the atoms in the material (there are about 1022=10 000 000 000 000 000 000 000 atoms in a gram of copper) and evolve the dynamics of the electrons and nuclei over “macroscopic times” (i.e. seconds, hours and days), we would be able to predict the material behavior. Of course, when we say “material,” we are already referring to a very complex system. In order to predict the response of the material we would first have to construct the material structure in the computer, which would require us to use Schrödinger's equation to simulate the process by which the material was manufactured. Conceptually, it may be useful to think of materials in this way, but we can quickly see the futility of the approach: the state of the art of quantum calculations involves just hundreds of atoms over a time of nanoseconds.

Fortunately, in many cases it is not necessary to keep track of all the atoms in a material to describe its behavior. Rather, the overall response of such a collection of atoms is often much more readily amenable to an elegant, mathematical description.