This chapter so far has shown that lots can be learned from the simple model of noninteracting particles inside a closed box. The biggest limitation of the model is particle motion. Sustained particle motion is hindered by the fact that the particles cannot penetrate the walls of the box.
One way of dealing with that is to make the box infinitely large. That produces motion in infinite and empty space. It can be done, as shown in chapter 7.9 and following. However, the analysis is nasty, as the eigenfunctions cannot be properly normalized. In many cases, a much simpler approach is to assume that the particles are in a finite, but periodic box. A particle that exits such a box through one side reenters it at the same time through the opposing side.
To understand the idea, consider the one-dimensional case. Studying
one-dimensional motion along an infinite straight line
Similarly a periodic box of dimensions
The biggest difference between the closed box and the periodic box is
linear momentum. For noninteracting particles in a periodic box, the
energy eigenfunctions can be taken to be also eigenfunctions of linear
momentum
Key Points
- A periodic box is a mathematical concept that allows unimpeded motion of the particles in the box. A particle that exits the box through one side reenters it at the opposite side at the same time.
- For a periodic box, the energy eigenfunctions can be taken to be also eigenfunctions of linear momentum.