### 4.3 Proof of the Bloch-Theorem

For each vector $\stackrel{\to }{R}$ of the Bravais lattice we define a translation operator ${T}_{R}$ by

 ${T}_{R}f\left(\stackrel{\to }{r}\right)=f\left(\stackrel{\to }{r}+\stackrel{\to }{R}\right)$ (4.8)

for an arbitrary function $f$.
Since the Hamiltonian is periodic, we find

 ${T}_{R}H\psi \left(\stackrel{\to }{r}\right)=H\left(\stackrel{\to }{r}+\stackrel{\to }{R}\right)\psi \left(\stackrel{\to }{r}+\stackrel{\to }{R}\right)=H\left(\stackrel{\to }{r}\right)\psi \left(\stackrel{\to }{r}+\stackrel{\to }{R}\right)=H{T}_{R}\psi \left(\stackrel{\to }{r}\right)$ (4.9)

Eq. (4.9) holds for all state functions $\psi$; so we can write

 ${T}_{R}H=H{T}_{R}\phantom{\rule{2em}{0ex}}\text{or}\phantom{\rule{2em}{0ex}}\left[{T}_{R},H\right]=0\phantom{\rule{2em}{0ex}}.$ (4.10)

Successively applying the translation leads to

 ${T}_{R}{T}_{{R}^{\prime }}\psi \left(\stackrel{\to }{r}\right)={T}_{{R}^{\prime }}{T}_{R}\psi \left(\stackrel{\to }{r}\right)=\psi \left(\stackrel{\to }{r}+\stackrel{\to }{R}+{\stackrel{\to }{R}}^{\prime }\right)\phantom{\rule{2em}{0ex}},$ (4.11)

i.e.

 ${T}_{R}{T}_{{R}^{\prime }}={T}_{{R}^{\prime }}{T}_{R}={T}_{R+{R}^{\prime }}\phantom{\rule{2em}{0ex}}.$ (4.12)

Following Eq. (4.10) we can choose the Eigenvectorsystem of $H$ to be simultaneously a Eigenvectorsystem of ${T}_{R}$:

 $\begin{array}{cc}\begin{array}{rl}H\psi & =E\psi \\ {T}_{R}\psi & =c\left(\stackrel{\to }{R}\right)\psi \end{array}& \end{array}$ (4.13)

Following Eq. (4.12) the Eigenvalues $c\left(\stackrel{\to }{R}\right)$ of the translation operator obey

 ${T}_{R}{T}_{{R}^{\prime }}\psi \left(\stackrel{\to }{r}\right)=c\left({\stackrel{\to }{R}}^{\prime }\right){T}_{R}\psi \left(\stackrel{\to }{r}\right)=c\left({\stackrel{\to }{R}}^{\prime }\right)c\left(\stackrel{\to }{R}\right)\psi \left(\stackrel{\to }{r}\right)\phantom{\rule{2em}{0ex}}.$ (4.14)

Thus for the Eigenvalues we find

 $c\left(\stackrel{\to }{R}+{\stackrel{\to }{R}}^{\prime }\right)=c\left(\stackrel{\to }{R}\right)c\left({\stackrel{\to }{R}}^{\prime }\right)\phantom{\rule{2em}{0ex}}.$ (4.15)

Let’s have a closer look on the primitive translation vectors of the Bravais lattice ${\stackrel{\to }{a}}_{i}$. For adequate variables ${x}_{i}$ we can write

 $c\left({\stackrel{\to }{a}}_{i}\right)={e}^{i2\pi {x}_{i}}\phantom{\rule{2em}{0ex}}.$ (4.16)

Thus for a general translation vector

 $\stackrel{\to }{R}={n}_{1}{\stackrel{\to }{a}}_{1}+{n}_{2}{\stackrel{\to }{a}}_{2}+{n}_{3}{\stackrel{\to }{a}}_{3}$ (4.17)

we find by successively applying Eq. (4.15)

 $c\left(\stackrel{\to }{R}\right)=c{\left({\stackrel{\to }{a}}_{1}\right)}^{{n}_{1}}*c{\left({\stackrel{\to }{a}}_{2}\right)}^{{n}_{2}}*c{\left({\stackrel{\to }{a}}_{3}\right)}^{{n}_{3}}$ (4.18)

which may be rewritten as

 $c\left(\stackrel{\to }{R}\right)={e}^{i\stackrel{\to }{k}\stackrel{\to }{R}}$ (4.19)

using Eq. (4.16) and the following definitions:

 $\stackrel{\to }{k}={x}_{1}{\stackrel{\to }{b}}_{1}+{x}_{2}{\stackrel{\to }{b}}_{2}+{x}_{3}{\stackrel{\to }{b}}_{3}$ (4.20)

and

 ${\stackrel{\to }{a}}_{i}{\stackrel{\to }{b}}_{j}=2\pi {\delta }_{i,j}\phantom{\rule{1em}{0ex}}.$ (4.21)

The above defined vectors $\stackrel{\to }{b}$ are the basic vectors of the reciprocal lattice.
Summing up, we have shown that for every vector of the real space the Eigenvectors $\psi$ of $H$ can be chosen to fulfill the following relation:

 ${T}_{R}\psi \left(\stackrel{\to }{r}\right)=\psi \left(\stackrel{\to }{r}+\stackrel{\to }{R}\right)=c\left(\stackrel{\to }{R}\right)\psi \left(\stackrel{\to }{r}\right)={e}^{i\stackrel{\to }{k}\stackrel{\to }{R}}\psi \left(\stackrel{\to }{r}\right)\phantom{\rule{2em}{0ex}}.$ (4.22)

This is the definition of the Bloch theorem according to Eq. (4.5). Additional illustrative information you can find in the MaWi II script.

© J. Carstensen (Quantum Mech.)