PX262 - K5 - block wavefunctions, electronic energy bands and gaps

considering a 1D lattice

attaching a potential to the lattice:

V (x) = V (x + R_{n}) = \sum_{h} V_{h} \exp (i G_{h} x)

since $\exp (i G_{h} x) = 1$
for an electron moving in such a periodic potential, the wavefunction must have a form:

ϕ_{n, k} = e^{i k x} u_{n, k} (x)

where, $u_{n, k} (x) = u_{n, k} (x + R_{n})$ , ie. it is periodic

the probability will also be periodic:

ϕ_{n, k}^{*} (x) ϕ (x) = e^{- i k x} u_{n, k}^{*} (x) e^{i k x} u_{n, k} (x) = u_{n, k}^{*} (x) u_{n, k} (x)

this is bloch's theorem
similarly, for 3D, the crystal lattice with ${\vec{R}}_{n_{1}, n_{2}, n_{3}}$ and ${\vec{G}}_{h, k, l}$
the potential:

V (\vec{r}) = V (\vec{r} + {\vec{R}}_{n_{1}, n_{2}, n_{3}}) = \sum_{h, k, l} V_{h, k, l} \exp (i {\vec{G}}_{h, k, l} \cdot \vec{r})

since $\vec{\exp} (i {\vec{G}}_{h, k, l} \cdot \vec{r}) = 1$
bloch's theorem:

ϕ_{h, k} (\vec{r}) = e^{i \vec{k} \cdot \vec{r}} u_{n, k} (\vec{r})

$u_{h, k}$ and $ϕ_{h, k} ϕ_{h, k}^{*}$ are periodic
$n$ is a band index and $\vec{k}$ is the wavevector