### 6.18 Pe­ri­odic Sin­gle-Par­ti­cle States

The sin­gle-par­ti­cle quan­tum states, or en­ergy eigen­func­tions, for non­in­ter­act­ing par­ti­cles in a closed box were given in sec­tion 6.2, (6.2). They were a prod­uct of a sine in each ax­ial di­rec­tion. Those for a pe­ri­odic box can sim­i­larly be taken to be a prod­uct of a sine or co­sine in each di­rec­tion. How­ever, it is usu­ally much bet­ter to take the sin­gle-par­ti­cle en­ergy eigen­func­tions to be ex­po­nen­tials:

 (6.25)

Here is the vol­ume of the pe­ri­odic box, while is the “wave num­ber vec­tor” that char­ac­ter­izes the state.

One ma­jor ad­van­tage of these eigen­func­tions is that they are also eigen­func­tion of lin­ear mo­men­tum. For ex­am­ple. the lin­ear mo­men­tum in the -​di­rec­tion equals . That can be ver­i­fied by ap­ply­ing the -​mo­men­tum op­er­a­tor on the eigen­func­tion above. The same for the other two com­po­nents of lin­ear mo­men­tum, so:

 (6.26)

This re­la­tion­ship be­tween wave num­ber vec­tor and lin­ear mo­men­tum is known as the “de Broglie re­la­tion.”

The rea­son that the mo­men­tum eigen­func­tions are also en­ergy eigen­func­tions is that the en­ergy is all ki­netic en­ergy. It makes the en­ergy pro­por­tional to the square of lin­ear mo­men­tum. (The same is true in­side the closed box, but mo­men­tum eigen­states are not ac­cept­able states for the closed box. You can think of the sur­faces of the closed box as in­fi­nitely high po­ten­tial en­ergy bar­ri­ers. They re­flect the par­ti­cles and the en­ergy eigen­func­tions then must be a 50/50 mix of for­ward and back­ward mo­men­tum.)

Like for the closed box, for the pe­ri­odic box the sin­gle-par­ti­cle en­ergy is still given by

 (6.27)

That may be ver­i­fied by ap­ply­ing the ki­netic en­ergy op­er­a­tor on the eigen­func­tions. It is sim­ply the New­ton­ian re­sult that the ki­netic en­ergy equals since the ve­loc­ity is by the de­f­i­n­i­tion of lin­ear mo­men­tum and in quan­tum terms.

Un­like for the closed box how­ever, the wave num­bers , , and are now con­strained by the re­quire­ment that the box is pe­ri­odic. In par­tic­u­lar, since is sup­posed to be the same phys­i­cal plane as 0 for a pe­ri­odic box, must be the same as . That re­stricts to be an in­te­ger mul­ti­ple of , (2.5). The same for the other two com­po­nents of the wave num­ber vec­tor, so:

 (6.28)

where the quan­tum num­bers , , and are in­te­gers.

In ad­di­tion, un­like for the si­nu­soidal eigen­func­tions of the closed box, zero and neg­a­tive val­ues of the wave num­bers must now be al­lowed. Oth­er­wise the set of eigen­func­tions will not be com­plete. The dif­fer­ence is that for the closed box, is just the neg­a­tive of , while for the pe­ri­odic box, is not just a mul­ti­ple of but a fun­da­men­tally dif­fer­ent func­tion.

Fig­ure 6.17 shows the wave num­ber space for a sys­tem of elec­trons in a pe­ri­odic box. The wave num­ber vec­tors are no longer re­stricted to the first quad­rant like for the closed box in fig­ure 6.11; they now fill the en­tire space. In the ground state, the states oc­cu­pied by elec­trons, shown in red, now form a com­plete sphere. For the closed box they formed just an oc­tant of one. The Fermi sur­face, the sur­face of the sphere, is now a com­plete spher­i­cal sur­face.

It may also be noted that in later parts of this book, of­ten the wave num­ber vec­tor or mo­men­tum vec­tor is used to la­bel the eigen­func­tions:

In gen­eral, what­ever is the most rel­e­vant to the analy­sis is used as la­bel. In any scheme, the sin­gle-par­ti­cle state of low­est en­ergy is ; it has zero en­ergy, zero wave num­ber vec­tor, and zero mo­men­tum.

Key Points
The en­ergy eigen­func­tions for a pe­ri­odic box are usu­ally best taken to be ex­po­nen­tials. Then the wave num­ber val­ues can be both pos­i­tive and neg­a­tive.

The sin­gle-par­ti­cle ki­netic en­ergy is still .

The mo­men­tum is .

The eigen­func­tion la­belling may vary.