D.26 Den­sity of states

This note de­rives the den­sity of states for par­ti­cles in a box.

Con­sider the wave num­ber space, as shown to the left in fig­ure 6.1. Each point rep­re­sents one spa­tial state. The first ques­tion is how many points have a wave num­ber vec­tor whose length is less than some given value . Since the length of the wave num­ber vec­tor is the dis­tance from the ori­gin in wave num­ber state, the points with form an oc­tant of a sphere with ra­dius . In fact, you can think of this prob­lem as find­ing the num­ber of red points in fig­ure 6.11.

Now the oc­tant of the sphere has a vol­ume (in wave num­ber space, not a phys­i­cal vol­ume)

Con­versely, every wave num­ber point is the top-left front cor­ner of a lit­tle block of vol­ume

where , , and are the spac­ings be­tween the points in the , , and di­rec­tions re­spec­tively. To find the ap­prox­i­mate num­ber of points in­side the oc­tant of the sphere, take the ra­tio of the two vol­umes:

Now the spac­ings be­tween the points are given in terms of the sides , , and of the box con­tain­ing the par­ti­cles as, (6.3),

Plug this into the ex­pres­sion for the num­ber of points in the oc­tant to get:
 (D.12)

where is the (phys­i­cal) vol­ume of the box . Each wave num­ber point cor­re­sponds to one spa­tial state, but if the spin of the par­ti­cles is then each spa­tial state still has dif­fer­ent spin val­ues. There­fore mul­ti­ply by to get the num­ber of states.

To get the den­sity of states on a wave num­ber ba­sis, take the de­riv­a­tive with re­spect to . The num­ber of states in a small wave num­ber range is then:

The fac­tor is the den­sity of states on a wave num­ber ba­sis.

To get the den­sity of states on an en­ergy ba­sis, sim­ply elim­i­nate in terms of the sin­gle-par­ti­cle en­ergy us­ing . That gives:

The used ex­pres­sion for the ki­netic en­ergy is only valid for non­rel­a­tivis­tic speeds.

The above ar­gu­ments fail in the pres­ence of con­fine­ment. Re­call that each state is the top-left front cor­ner of a lit­tle block in wave num­ber space of vol­ume . The num­ber of states with wave num­ber less than some given value was found by com­put­ing how many such lit­tle block vol­umes are con­tained within the oc­tant of the sphere of ra­dius .

The prob­lem is that a wave num­ber is only in­side the sphere oc­tant if all of its lit­tle block is in­side. Even if 99% of its block is in­side, the state it­self will still be out­side, not 99% in. That makes no dif­fer­ence if the states are densely spaced in wave num­ber space, like in fig­ure 6.11. In that case al­most all lit­tle blocks are fully in­side the sphere. Only a thin layer of blocks near the sur­face of the sphere are par­tially out­side it.

How­ever, con­fine­ment in a given di­rec­tion makes the cor­re­spond­ing spac­ing in wave num­ber space large. And that changes things.

In par­tic­u­lar, if the -​di­men­sion of the box con­tain­ing the par­ti­cles is small, then is large. That is il­lus­trated in fig­ure 6.12. In this case, there are no states in­side the sphere at all if is less than . Re­gard­less of what (D.12) claims. In the range , il­lus­trated by the red sphere in fig­ure 6.12, the red sphere gob­bles up a num­ber of states from the plate . This num­ber of states can be es­ti­mated as

since the top of this ra­tio is the area of the quar­ter cir­cle of states and the bot­tom is the rec­tan­gu­lar area oc­cu­pied per state.

This ex­pres­sion can be cleaned up by not­ing that

with 1 for the low­est plate. Sub­sti­tut­ing for , , and in terms of the box di­men­sions then gives
 (D.13)

Here is the area of the quan­tum well and 1 is the plate num­ber. For non­rel­a­tivis­tic speeds is pro­por­tional to the en­ergy . There­fore the den­sity of states, which is the de­riv­a­tive of the num­ber of states with re­spect to en­ergy, is con­stant.

In the range a sec­ond quar­ter cir­cle of states gets added. To get the num­ber of ad­di­tional states in that cir­cle, use 2 for the plate num­ber in (D.13). For still larger val­ues of , just keep sum­ming plates as long as the ex­pres­sion be­tween the square brack­ets in (D.13) re­mains pos­i­tive.

If the -​di­men­sion of the box is also small, like in a quan­tum wire, the states in wave num­ber space sep­a­rate into in­di­vid­ual lines, fig­ure 6.13. There are now no states un­til the sphere of ra­dius hits the line that is clos­est to the ori­gin, hav­ing quan­tum num­bers 1. Be­yond that value of , the num­ber of states on the line that is within the sphere is

since the top is the length of the line in­side the sphere and the bot­tom the spac­ing of the states on the line. Clean­ing up, that gives
 (D.14)

with the length of the quan­tum wire. For still larger val­ues of sum over all val­ues of and for which the ar­gu­ment of the square root re­mains pos­i­tive.

For non­rel­a­tivis­tic speeds, is pro­por­tional to the en­ergy. There­fore the above num­ber of states is pro­por­tional to the square root of the amount of en­ergy above the one at which the line of states is first hit. Dif­fer­en­ti­at­ing to get the den­sity of states, the square root be­comes an rec­i­p­ro­cal square root.

If the box is small in all three di­rec­tions, fig­ure 6.14, the num­ber of states sim­ply be­comes the num­ber of points in­side the sphere:

 (D.15)

In other words, to get the to­tal num­ber of states in­side, sim­ply add a 1 for each set of nat­ural num­bers , , and for which the ex­pres­sion in brack­ets is pos­i­tive. The de­riv­a­tive with re­spect to en­ergy, the den­sity of states, be­comes a se­ries of delta func­tions at the en­er­gies at which the states are hit.