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:

$$\fbox{$\displaystyle \pp{n_xn_yn_z}/{\skew0\vec r}/// = {... ...cal V}^{-\frac12} e^{{\rm i}{\vec k}\cdot{\skew0\vec r}} $} %$$ (6.25)

Here ${\cal V}$ is the vol­ume of the pe­ri­odic box, while ${\vec k}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $(k_x,k_y,k_z)$ 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 $x$-​di­rec­tion equals $p_x$ $\vphantom0\raisebox{1.5pt}{$=$}$ ${\hbar}k_x$. That can be ver­i­fied by ap­ply­ing the $x$-​mo­men­tum op­er­a­tor $\hbar\partial$$\raisebox{.5pt}{$/$}$​${\rm i}\partial{x}$ on the eigen­func­tion above. The same for the other two com­po­nents of lin­ear mo­men­tum, so:

$$\fbox{$\displaystyle p_x= \hbar k_x \quad p_y= \hbar k_y \quad p_z= \hbar k_z \qquad {\skew0\vec p}= \hbar {\vec k} $} %$$ (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

$$\fbox{$\displaystyle {\vphantom' E}^{\rm p}= \frac{\hbar^2}{2m} k^2 \qquad k \equiv \sqrt{k_x^2 + k_y^2 + k_z^2} $} %$$ (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 $\frac12mv^2$ since the ve­loc­ity is $v$ $\vphantom0\raisebox{1.5pt}{$=$}$ $p$$\raisebox{.5pt}{$/$}$​$m$ by the de­f­i­n­i­tion of lin­ear mo­men­tum and $p$ $\vphantom0\raisebox{1.5pt}{$=$}$ ${\hbar}k$ in quan­tum terms.

Un­like for the closed box how­ever, the wave num­bers $k_x$, $k_y$, and $k_z$ are now con­strained by the re­quire­ment that the box is pe­ri­odic. In par­tic­u­lar, since $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\ell_x$ is sup­posed to be the same phys­i­cal plane as $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 for a pe­ri­odic box, $e^{{{\rm i}}k_x\ell_x}$ must be the same as $e^{{{\rm i}}k_x0}$. That re­stricts $k_x\ell_x$ to be an in­te­ger mul­ti­ple of $2\pi$, (2.5). The same for the other two com­po­nents of the wave num­ber vec­tor, so:

$$\fbox{$\displaystyle k_x = n_x \frac{2\pi}{\ell_x} \qquad ... ...frac{2\pi}{\ell_y} \qquad k_z = n_z \frac{2\pi}{\ell_z} $} %$$ (6.28)

where the quan­tum num­bers $n_x$, $n_y$, and $n_z$ 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, $\sin(-k_xx)$ is just the neg­a­tive of $\sin(k_xx)$, while for the pe­ri­odic box, $e^{-{{\rm i}}k_xx}$ is not just a mul­ti­ple of $e^{{{\rm i}}k_xx}$ but a fun­da­men­tally dif­fer­ent func­tion.

Fig­ure 6.17: Ground state of a sys­tem of non­in­ter­act­ing elec­trons, or other fermi­ons, in a pe­ri­odic box.

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:

$$\pp{n_xn_yn_z}/{\skew0\vec r}/// = \pp{k_xk_yk_z}/{\skew0\vec r}/// = \pp{p_xp_yp_z}/{\skew0\vec r}///$$

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 $\pp000/{\skew0\vec r}///$; it has zero en­ergy, zero wave num­ber vec­tor, and zero mo­men­tum.

Key Points

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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.

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The sin­gle-par­ti­cle ki­netic en­ergy is still $\hbar^2k^2$$\raisebox{.5pt}{$/$}$​$2m$.

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The mo­men­tum is $\hbar{\vec k}$.

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The eigen­func­tion la­belling may vary.