Electronic structure of periodic solids Bloch theory

3 Electronic structure of periodic solids Bloch theory 

Inorderto study the implications ofthe periodic structure oflattices on the electronic structure of the corresponding solids we consider a single electron Hamiltonian of the form 

It is convenient to use periodic boundary conditions. For simplicity we consider a cubic lattice, so we take the system to be a rectangular prism with sides L = tVi tzi, 2 = N a, Li = NiUi, that is infinitely reproduced to form infinite space. As in the free particle problem (Section 2.8) this is used just for mathematical convenience, assuming that bulk properties of the system do not depend on the boundary conditions for L, L2, Ly 00. 

In the absence of the periodic potential our problem is reduced again to that of a free particle. Eigenfunctions of H = T that satisfy the periodic boundary conditions are of the form 

This in turn implies that the allowed values of kj are  

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