Question

derivation of bloch theorem​

Answer 1

we prove Bloch's theorem:

For electrons in a perfect crystal, there is a basis of wavefunctions with the properties:

Each of these wavefunctions is an energy eigenstate

Each of these wavefunctions is a Bloch wave, meaning that this wavefunction {\displaystyle \psi } \psi can be written in the form

{\displaystyle \psi (\mathbf {r} )=\mathrm {e} ^{\mathrm {i} \mathbf {k} \cdot \mathbf {r} }u(\mathbf {r} )} \psi ({\mathbf {r}})={\mathrm {e}}^{{{\mathrm {i}}{\mathbf {k}}\cdot {\mathbf {r}}}}u({\mathbf {r}})

where u has the same periodicity as the atomic structure of the crystal.