Bloch's theorem: An operator-based derivation

We present an operator-based derivation of Bloch's theorem. In this approach, we first construct an explicit operator form of the Hamiltonian by writing the potential part in terms of an infinite sum of the momentum-translation operators. Next, we prove that it commutes with the position-translation operator corresponding to a lattice translation, using the exponential re-ordering identity. We then build on Merzbacher's treatment of simultaneous diagonalization of commuting operators to reproduce Kittel's central equation and also to derive a new formula for the simultaneous eigenket corresponding to a specific energy eigenvalue. Moreover, through the Born rule, we provide a precise account of the position and momentum probability distributions for Bloch states, and we examine the concept of crystal momentum. Finally, we apply our findings to the Kronig–Penney model, where we numerically depict the band structure and illustrate the delocalized nature of the single electron.