6.17 Intro to the Periodic Box

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 $\vphantom0\raisebox{1.5pt}{$-$}$$\infty$ $\raisebox{.3pt}{$<$}$ $x$ $\raisebox{.3pt}{$<$}$ $\infty$ is typically nasty. One-di­mensional motion along a circle is likely to be easier. Unlike the straight line, the circumference of the circle, call it $\ell_x$, is finite. So you can define a coordinate $x$ along the circle with a finite range 0 $\raisebox{.3pt}{$<$}$ $x$ $\raisebox{.3pt}{$<$}$ $\ell_x$. Yet despite the finite circumference, a particle can keep moving along the circle without getting stuck. When the particle reaches the position $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\ell_x$ along the circle, it is back at its starting point $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. It leaves the defined $x$-range through $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\ell_x$, but it reenters it at the same time through $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. The position $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\ell_x$ is physi­cally exactly the same point as $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0.

Similarly a periodic box of dimensions $\ell_x$, $\ell_y$, and $\ell_z$ assumes that $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\ell_x$ is physi­cally the same as $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0, $y$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\ell_y$ the same as $y$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0, and $z$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\ell_z$ the same as $z$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. That is of course hard to visualize. It is just a mathematical trick, but one that works well. Typically at the end of the analysis you take the limit that the box dimensions become infinite. That makes this artificial box disappear and you get the valid infinite-space solution.

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 ${\skew 4\widehat{\skew{-.5}\vec p}}$. They then have definite linear momentum in addition to definite energy. In fact, the linear momentum is just a scaled wave number vector; ${\skew0\vec p}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\hbar{\vec k}$. That is discussed in more detail in the next section.

Key Points

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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.

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For a periodic box, the energy eigen­functions can be taken to be also eigen­functions of linear momentum.