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 $\vphantom{0}\raisebox{1.5pt}{$-$}$$\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.