How Does Bloch's Theorem Simplify Complex Calculations?
Bloch's theorem states that the wavefunction of an electron in a periodic potential can be written as the product of a plane wave and a function with the same periodicity as the potential. Mathematically, this is expressed as: \[ \psi_k(r) = e^{ik \cdot r} u_k(r) \] where \( \psi_k(r) \) is the Bloch function, \( e^{ik \cdot r} \) is the plane wave, and \( u_k(r) \) is a periodic function. This decomposition greatly simplifies the analysis of electron behavior in periodic structures, as it reduces the problem to solving for the periodic component \( u_k(r) \).
