6.1 In­tro to Par­ti­cles in a Box

Since most macro­scopic sys­tems are very hard to an­a­lyze in quan­tum-me­chan­ics, sim­ple sys­tems are very im­por­tant. They al­low in­sight to be achieved that would be hard to ob­tain oth­er­wise. One of the sim­plest and most im­por­tant sys­tems is that of mul­ti­ple non­in­ter­act­ing par­ti­cles in a box. For ex­am­ple, it is a start­ing point for quan­tum ther­mo­dy­nam­ics and the quan­tum de­scrip­tion of solids.

It will be as­sumed that the par­ti­cles do not in­ter­act with each other, nor with any­thing else in the box. That is a du­bi­ous as­sump­tion; in­ter­ac­tions be­tween par­ti­cles are es­sen­tial to achieve sta­tis­ti­cal equi­lib­rium in ther­mo­dy­nam­ics. And in solids, in­ter­ac­tion with the atomic struc­ture is needed to ex­plain the dif­fer­ences be­tween elec­tri­cal con­duc­tors, semi­con­duc­tors, and in­su­la­tors. How­ever, in the box model such ef­fects can be treated as a per­tur­ba­tion. That per­tur­ba­tion is ig­nored to lead­ing or­der.

In the ab­sence of in­ter­ac­tions be­tween the par­ti­cles, the pos­si­ble quan­tum states, or en­ergy eigen­func­tions, of the com­plete sys­tem take a rel­a­tively sim­ple form. They turn out to be prod­ucts of sin­gle par­ti­cle en­ergy eigen­func­tions. A generic en­ergy eigen­func­tion for a sys­tem of $I$ par­ti­cles is:

$${\psi^{\rm S}_{{\vec n}_1,{\vec n}_2,\ldots,{\vec n}_i,\ldots,{\v... ...}_2, S_{z2},\ldots,{\skew0\vec r}_i, S_{zi},\ldots,{\skew0\vec r}_I, S_{zI}) =}$$

$$\pp{\vec n}_1/{\skew0\vec r}_1//z1/ \times \pp{\vec n}_2/{\skew0\... ...\vec r}_i//zi/ \times \ldots \times \pp{\vec n}_I/{\skew0\vec r}_I//zI/\qquad%$$ (6.1)

In such a sys­tem eigen­func­tion, par­ti­cle num­ber $i$ out of $I$ is in a sin­gle-par­ti­cle en­ergy eigen­func­tion $\pp{\vec n}_i/{\skew0\vec r}_i//zi/$. Here ${\skew0\vec r}_i$ is the po­si­tion vec­tor of the par­ti­cle, and $S_{zi}$ its spin in a cho­sen $z$-​di­rec­tion. The sub­script ${\vec n}_i$ stands for what­ever quan­tum num­bers char­ac­ter­ize the sin­gle-par­ti­cle eigen­func­tion. A sys­tem wave func­tion of the form above, a sim­ple prod­uct of sin­gle-par­ti­cles ones, is called a “Hartree prod­uct.”

For non­in­ter­act­ing par­ti­cles con­fined in­side a box, the sin­gle-par­ti­cle en­ergy eigen­func­tions, or sin­gle-par­ti­cle states, are es­sen­tially the same ones as those de­rived in chap­ter 3.5 for a par­ti­cle in a pipe with a rec­tan­gu­lar cross sec­tion. How­ever, to ac­count for nonzero par­ti­cle spin, a spin-de­pen­dent fac­tor must be added. In any case, this chap­ter will not re­ally be con­cerned that much with the de­tailed form of the sin­gle-par­ti­cle en­ergy states. The main quan­ti­ties of in­ter­est are their quan­tum num­bers and their en­er­gies. Each pos­si­ble set of quan­tum num­bers will be graph­i­cally rep­re­sented as a point in a so-called “wave num­ber space.” The sin­gle-par­ti­cle en­ergy is found to be re­lated to how far that point is away from the ori­gin in that wave num­ber space.

For the com­plete sys­tem of $I$ par­ti­cles, the most in­ter­est­ing physics has to do with the (anti) sym­metriza­tion re­quire­ments. In par­tic­u­lar, for a sys­tem of iden­ti­cal fermi­ons, the Pauli ex­clu­sion prin­ci­ple says that there can be at most one fermion in a given sin­gle-par­ti­cle state. That im­plies that in the above Hartree prod­uct each set of quan­tum num­bers ${\vec n}$ must be dif­fer­ent from all the oth­ers. In other words, any sys­tem wave func­tion for a sys­tem of $I$ fermi­ons must in­volve at least $I$ dif­fer­ent sin­gle-par­ti­cle states. For a macro­scopic num­ber of fermi­ons, that puts a tremen­dous re­stric­tion on the wave func­tion. The most im­por­tant ex­am­ple of a sys­tem of iden­ti­cal fermi­ons is a sys­tem of elec­trons, but sys­tems of pro­tons and of neu­trons ap­pear in the de­scrip­tion of atomic nu­clei.

The an­ti­sym­metriza­tion re­quire­ment is re­ally more sub­tle than the Pauli prin­ci­ple im­plies. And the sym­metriza­tion re­quire­ments for bosons like pho­tons or he­lium-4 atoms are non­triv­ial too. This was dis­cussed ear­lier in chap­ter 5.7. Sim­ple Hartree prod­uct en­ergy eigen­func­tions of the form (6.1) above are not ac­cept­able by them­selves; they must be com­bined with oth­ers with the same sin­gle-par­ti­cle states, but with the par­ti­cles shuf­fled around be­tween the states. Or rather, be­cause shuf­fled around sounds too much like Las Ve­gas, with the par­ti­cles ex­changed be­tween the states.

Key Points

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

Sys­tems of non­in­ter­act­ing par­ti­cles in a box will be stud­ied.

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

In­ter­ac­tions be­tween the par­ti­cles may have to be in­cluded at some later stage.

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

Sys­tem en­ergy eigen­func­tions are ob­tained from prod­ucts of sin­gle-par­ti­cle en­ergy eigen­func­tions.

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

(anti) sym­metriza­tion re­quire­ments fur­ther re­strict the sys­tem en­ergy eigen­func­tions.