Time-dependent states
In this section we would like to discuss the behavior of states in the one-dimensional lattice in more detail. If the amplitude for an electron to be at $x_n$ is $C_n$, the probability of finding it there is $\abs{C_n}^2$. For the stationary states described by Eq. (13.14), this probability is the same for all $x_n$ and does not change with time. How can we represent a situation which we would describe roughly by saying an electron of a certain energy is localized in a certain region—so that it is more likely to be found at one place than at some other place? We can do that by making a superposition of several solutions like Eq. (13.14) with slightly different values of $k$—and, therefore, slightly different energies. Then at $t=0$, at least, the amplitude $C_n$ will vary with position because of the interference between the various terms, just as one gets beats when there is a mixture of waves of different wavelengths (as we discussed in Chapter 48, Vol. I). So we can make up a “wave packet” with a predominant wave number $k_0$, but with various other wave numbers near $k_0$.1 In our superposition of stationary states, the amplitudes with different $k$’s will represent states of slightly different energies, and, therefore, of slightly different frequencies; the interference pattern of the total $C_n$ will, therefore, also vary with time—there will be a pattern of “beats.” As we have seen in Chapter 48 of Volume I, the peaks of the beats [the place where $\abs{C(x_n)}^2$ is large] will move along in $x$ as time goes on; they move with the speed we have called the “group velocity.” We found that this group velocity was related to the variation of $k$ with frequency by \begin{equation} \label{Eq:III:13:17} v_{\text{group}}=\ddt{\omega}{k}; \end{equation} the same derivation would apply equally well here. An electron state which is a “clump”—namely one for which the $C_n$ vary in space like the wave packet of Fig. 13–5—will move along our one-dimensional “crystal” with the speed $v$ equal to $d\omega/dk$, where $\omega=E/\hbar$. Using (13.16) for $E$, we get that \begin{equation} \label{Eq:III:13:18} v=\frac{2Ab^2}{\hbar}\,k. \end{equation} In other words, the electrons move along with a speed proportional to the typical $k$. Equation (13.16) then says that the energy of such an electron is proportional to the square of its velocity—it acts like a classical particle. So long as we look at things on a scale gross enough that we don’t see the fine structure, our quantum mechanical picture begins to give results like classical physics. In fact, if we solve Eq. (13.18) for $k$ and substitute into (13.16), we can write \begin{equation} \label{Eq:III:13:19} E=\tfrac{1}{2}m_{\text{eff}}\,v^2, \end{equation} where $m_{\text{eff}}$ is a constant. The extra “energy of motion” of the electron in a packet depends on the velocity just as for a classical particle. The constant $m_{\text{eff}}$—called the “effective mass”—is given by \begin{equation} \label{Eq:III:13:20} m_{\text{eff}}=\frac{\hbar^2}{2Ab^2}. \end{equation} Also notice that we can write \begin{equation} \label{Eq:III:13:21} m_{\text{eff}}\,v=\hbar k. \end{equation} If we choose to call $m_{\text{eff}}\,v$ the “momentum,” it is related to the wave number $k$ in the way we have described earlier for a free particle. Don’t forget that $m_{\text{eff}}$ has nothing to do with the real mass of an electron. It may be quite different—although in commonly used metals and semiconductors it often happens to turn out to be the same general order of magnitude, about $0.1$ to $30$ times the free-space mass of the electron. We have now explained a remarkable mystery—how an electron in a crystal (like an extra electron put into germanium) can ride right through the crystal and flow perfectly freely even though it has to hit all the atoms. It does so by having its amplitudes going pip-pip-pip from one atom to the next, working its way through the crystal. That is how a solid can conduct electricity.