Editing Wannier function (section)

=== Properties ===

On the basis of this definition, the following properties can be proven to hold:<ref name=Bohm>{{cite book |title=The Geometric Phase in Quantum Systems |author=A Bohm, A Mostafazadeh, H Koizumi, Q Niu and J Zqanziger |isbn=978-3-540-00031-0 |publisher=Springer |year=2003 |pages=§12.5, p. 292 ff|doi=10.1007/978-3-662-10333-3 |url=https://cds.cern.ch/record/737299 }}</ref>

* For any lattice vector ''' R' ''',
:<math>\phi_{\mathbf{R}}(\mathbf{r}) = \phi_{\mathbf{R}+\mathbf{R}'}(\mathbf{r}+\mathbf{R}')</math>
In other words, a Wannier function only depends on the quantity ('''r''' − '''R'''). As a result, these functions are often written in the alternative notation
:<math>\phi(\mathbf{r}-\mathbf{R}) := \phi_{\mathbf{R}}(\mathbf{r})</math>

* The Bloch functions can be written in terms of Wannier functions as follows:
:<math>\psi_{\mathbf{k}}(\mathbf{r}) = \frac{1}{\sqrt{N}} \sum_{\mathbf{R}} e^{i\mathbf{k}\cdot\mathbf{R}} \phi_{\mathbf{R}}(\mathbf{r})</math>,
where the sum is over each lattice vector '''R''' in the crystal.

* The set of wavefunctions <math>\phi_{\mathbf{R}}</math> is an [[orthonormal basis]] for the band in question.
:<math>\begin{align}
\int_\text{crystal}  \phi_{\mathbf{R}}(\mathbf{r})^* \phi_{\mathbf{R'}}(\mathbf{r}) d^3\mathbf{r} & = \frac{1}{N} \sum_{\mathbf{k,k'}}\int_\text{crystal} e^{i\mathbf{k}\cdot\mathbf{R}} \psi_{\mathbf{k}}(\mathbf{r})^*  e^{-i\mathbf{k'}\cdot\mathbf{R'}} \psi_{\mathbf{k'}}(\mathbf{r}) d^3\mathbf{r} \\
& =  \frac{1}{N} \sum_{\mathbf{k,k'}} e^{i\mathbf{k}\cdot\mathbf{R}} e^{-i\mathbf{k'}\cdot\mathbf{R'}} \delta_{\mathbf{k,k'}} \\
& = \frac{1}{N} \sum_{\mathbf{k}} e^{i\mathbf{k}\cdot\mathbf{(R-R')}} \\
& =\delta_{\mathbf{R,R'}}
\end{align} </math>

Wannier functions have been extended to nearly periodic potentials as well.<ref name=Kohn0>[http://www.physast.uga.edu/~mgeller/4.pdf MP Geller and W Kohn] ''Theory of generalized Wannier functions for nearly periodic potentials'' Physical Review B 48, 1993</ref>