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:

\begin{displaymath}
\fbox{$\displaystyle
\pp{n_xn_yn_z}/{\skew0\vec r}///
...
...l V}^{-\frac12} e^{{\rm i}{\vec k}\cdot{\skew0\vec r}}
$} %
\end{displaymath} (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:

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

\begin{displaymath}
\fbox{$\displaystyle
{\vphantom' E}^{\rm p}= \frac{\hbar...
...} k^2
\qquad
k \equiv \sqrt{k_x^2 + k_y^2 + k_z^2}
$} %
\end{displaymath} (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:

\begin{displaymath}
\fbox{$\displaystyle
k_x = n_x \frac{2\pi}{\ell_x} \qqua...
...c{2\pi}{\ell_y} \qquad
k_z = n_z \frac{2\pi}{\ell_z}
$} %
\end{displaymath} (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.
\begin{figure}
\centering
\setlength{\unitlength}{1pt}
\begin{picture}(...
...
\put(-12,297){$k_y$}
\put(-58.5,120){$k_z$}
\end{picture}
\end{figure}

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:

\begin{displaymath}
\pp{n_xn_yn_z}/{\skew0\vec r}/// = \pp{k_xk_yk_z}/{\skew0\vec r}/// = \pp{p_xp_yp_z}/{\skew0\vec r}///
\end{displaymath}

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
$\begin{picture}(15,5.5)(0,-3)
\put(2,0){\makebox(0,0){\scriptsize\bf0}}
\put(12...
...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}}
\end{picture}$
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.

$\begin{picture}(15,5.5)(0,-3)
\put(2,0){\makebox(0,0){\scriptsize\bf0}}
\put(12...
...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}}
\end{picture}$
The single-particle kinetic energy is still $\hbar^2k^2$$\raisebox{.5pt}{$/$}$$2m$.

$\begin{picture}(15,5.5)(0,-3)
\put(2,0){\makebox(0,0){\scriptsize\bf0}}
\put(12...
...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}}
\end{picture}$
The momentum is $\hbar{\vec k}$.

$\begin{picture}(15,5.5)(0,-3)
\put(2,0){\makebox(0,0){\scriptsize\bf0}}
\put(12...
...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}}
\end{picture}$
The eigenfunction labelling may vary.