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(Solved): 1. Bloch's theorem for band structure of a crystal is given by, \[ \Psi_{k}(\mathbf{r})=u_{k}(\math ...
1. Bloch's theorem for band structure of a crystal is given by, \[ \Psi_{k}(\mathbf{r})=u_{k}(\mathbf{r}) e^{i \mathbf{k} \cdot \mathbf{r}} \] Explain this equation. Applying the momentum operator \( (-i \hbar \nabla) \) on the Bloch wavefunction show that \( \hbar \mathbf{k} \) is not the momentum of the electron. What is the condition for \( \hbar \mathbf{k} \) to become the momentum of the electron? 2. Sketch the Bloch wave function for a 3 s electron in a 1-d chain of sodium atoms. 3. In band theory, the equation \[ \left(\lambda_{k}-\epsilon\right) c(k)+\sum_{G} U(G) c(k-G)=0 \] is called the central equation. What is this equation used for? What is Bloch's theorem and what is a Bloch wavefunction?