6. Macro­scopic Sys­tems

Ab­stract

Macro­scopic sys­tems in­volve ex­tremely large num­bers of par­ti­cles. Such sys­tems are very hard to an­a­lyze ex­actly in quan­tum me­chan­ics. An ex­cep­tion is a sys­tem of non­in­ter­act­ing par­ti­cles stuck in a rec­tan­gu­lar box. This chap­ter there­fore starts with an ex­am­i­na­tion of that model. For a model of this type, the sys­tem en­ergy eigen­func­tions are found to be prod­ucts of sin­gle-par­ti­cle states.

One thing that be­comes quickly ob­vi­ous is that macro­scopic sys­tem nor­mally in­volve a gi­gan­tic num­ber of sin­gle-par­ti­cle states. It is un­re­al­is­tic to tab­u­late them each in­di­vid­u­ally. In­stead, av­er­age sta­tis­tics about the states are de­rived. The pri­mary of these is the so-called den­sity of states. It is the num­ber of sin­gle-par­ti­cle states per unit en­ergy range.

But know­ing the num­ber of states is not enough by it­self. In­for­ma­tion is also needed on how many par­ti­cles are in these states. For­tu­nately, it turns out to be pos­si­ble to de­rive the av­er­age num­ber of par­ti­cles per state. This num­ber de­pends on whether it is a sys­tem of bosons, like pho­tons, or a sys­tem of fermi­ons, like elec­trons. For bosons, the num­ber of par­ti­cles is given by the so-called Bose-Ein­stein dis­tri­b­u­tion, while for elec­trons it is given by the so-called Fermi-Dirac dis­tri­b­u­tion. Ei­ther dis­tri­b­u­tion can be sim­pli­fied to the so-called Maxwell-Boltz­mann dis­tri­b­u­tion un­der con­di­tions in which the av­er­age num­ber of par­ti­cles per state is much less than one.

Each dis­tri­b­u­tion de­pends on both the tem­per­a­ture and on a so-called chem­i­cal po­ten­tial. Phys­i­cally, tem­per­a­ture dif­fer­ences pro­mote the dif­fu­sion of ther­mal en­ergy, heat, from hot to cold. Sim­i­larly, dif­fer­ences in chem­i­cal po­ten­tial pro­mote the dif­fu­sion of par­ti­cles from high chem­i­cal po­ten­tial to low.

At first, sys­tems of iden­ti­cal bosons are stud­ied. Bosons be­have quite strangely at very low tem­per­a­tures. Even for a nonzero tem­per­a­ture, a fi­nite frac­tion of them may stay in the sin­gle-par­ti­cle state of low­est en­ergy. That be­hav­ior is called Bose-Ein­stein con­den­sa­tion. Bosons also show a lack of low-en­ergy global en­ergy eigen­func­tions.

A first dis­cus­sion of elec­tro­mag­netic ra­di­a­tion, in­clud­ing light, will be given. The dis­cussed ra­di­a­tion is the one that oc­curs un­der con­di­tions of ther­mal equi­lib­rium, and is called black­body ra­di­a­tion.

Next, sys­tems of elec­trons are cov­ered. It is found that elec­trons in typ­i­cal macro­scopic sys­tems have vast amounts of ki­netic en­ergy even at ab­solute zero tem­per­a­ture. It is this ki­netic en­ergy that is re­spon­si­ble for the vol­ume of solids and liq­uids and their re­sis­tance to com­pres­sion. The elec­trons are nor­mally con­fined to a solid de­spite all their ki­netic en­ergy. But at some point, they may es­cape in a process called thermionic emis­sion.

The elec­tri­cal con­duc­tion of met­als can be ex­plained us­ing the sim­ple model of non­in­ter­act­ing elec­trons. How­ever, elec­tri­cal in­su­la­tors and semi­con­duc­tors can­not. It turns out that these can be ex­plained by in­clud­ing a sim­ple model of the forces on the elec­trons.

Then semi­con­duc­tors are dis­cussed, in­clud­ing ap­pli­ca­tions such as diodes, tran­sis­tors, so­lar cells, light-emit­ting diodes, solid state re­frig­er­a­tion, ther­mo­cou­ples, and ther­mo­elec­tric gen­er­a­tors. A some­what more gen­eral dis­cus­sion of op­ti­cal is­sues is also given.

Sub­sec­tions