6.19 DOS for a Pe­ri­odic Box

The den­sity of states is the num­ber of sin­gle-par­ti­cle states per unit en­ergy range. It turns out that the for­mu­lae for the den­sity of states given in sec­tion 6.3 may be used for the pe­ri­odic box as well as for the closed box. A box can hold about the same num­ber of par­ti­cles per unit vol­ume whether the bound­ary con­di­tions are pe­ri­odic or not.

It is not that hard to ver­ify. For a pe­ri­odic box, the wave num­bers can be both pos­i­tive and neg­a­tive, not just pos­i­tive like for a closed box. On the other hand, a com­par­i­son of (6.3) and (6.28) shows that the wave num­ber spac­ing for a pe­ri­odic box is twice as large as for a cor­re­spond­ing closed box. That can­cels the ef­fect of the ad­di­tional neg­a­tive wave num­bers and the to­tal num­ber of wave num­ber vec­tors in a given en­ergy range re­mains the same. There­fore the den­sity of states is the same.

For the pe­ri­odic box it is of­ten con­ve­nient to have the den­sity of states on a lin­ear mo­men­tum ba­sis. It can be found by sub­sti­tut­ing $k$ $\vphantom0\raisebox{1.5pt}{$=$}$ $p$$\raisebox{.5pt}{$/$}$​$\hbar$ into (6.5). That gives the num­ber of sin­gle-par­ti­cle states ${\rm d}{N}$ in a mo­men­tum range of size ${\rm d}{p}$ as:

$$\fbox{$\displaystyle {\rm d}N = {\cal V}{\cal D}_p { \rm d}p \qquad {\cal D}_p = \frac{2s+1}{2\pi^2\hbar^3} p^2 $} %$$ (6.29)

Here ${\cal D}_p$ is the den­sity of states per unit mo­men­tum range and unit vol­ume. Also, $s$ is again the par­ti­cle spin. Re­call that $2s+1$ be­comes $2s$ for pho­tons.

The stag­ing be­hav­ior due to con­fine­ment gets some­what mod­i­fied com­pared to sec­tion 6.12, since zero wave num­bers are now in­cluded. The analy­sis is how­ever es­sen­tially un­changed.

Key Points

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The den­sity of states is es­sen­tially the same for a pe­ri­odic box as for a closed one.