temp

This indeed has units of charge per unit time, and is a current. If we place this under a electric field, we find that
    $$\hbar k = \hbar k(t=0) + -eEt$$
    \subsection{Effects of Periodic Potential on Electron Gas}
        We have a lattice of electrons, with some periodic potential, with minima at the positions of the electrons on the lattice. We have a theorem, which states that we have translational symmetry, if $x \to x+ na$, where $a$ is the lattice spacing, the physics is the same. Since we cannot measure $\psi$, only $\psi^2$:
        $$\psi(x+a) = e^{iqa}\psi(x)$$
        Using the boundary condition that for a line of $N$ atoms:
        $$\psi(0) = \psi(Na)$$
        Using the previous statement, we have $e^{iNqa} = 1$, which tells us that $Nqa = 2\pi m$, for $m \in \mathbb{Z}$. This tells us that the allowed values of $q$ are
        $$q = \frac{2\pi n}{Na}$$
        This theorem is known as the Bloch theorem.

        We can attempt to model this potential using finite square wells, or delta functions. We place the positive delta functions between each lattice point, blocking off each electron from the ones next to it. Alternatively, we could place negative delta functions directly under every lattice point.

        We have tunnelling happening between every delta barrier. We see that this quickly leads to a ton of reflected and transmitted waves. We can think of the net wavefunction:
        $$\psi = Ae^{iqx} + Be^{-iqx}$$
        Due to the Bloch theorem, we can just solve this in one unit cell, and use the translational symmetry to get the physics everywhere else. Suppose we have 3 bouncing waves inside our cell, generated by 3 deflections. Let us label them 1,2, and 3. The path length of 1 and 2 is $2a$. If 3 must be in phase with 1, then $2a = n\lambda$, for some integer $n$, telling us that $\lambda = \frac{2a}{n}$. This is the condition for constructive interference. In one cell, we have
        $$\psi = Ae^{iqx} + Be^{-iqx} = (A+B)\cos(qx) + i(A-B)\sin(qx)$$