@@ -229,6 +229,6 @@ Consider a 1D crystal with a periodic potential given by delta peaks: $$V(x) = -
Using the Schrödinger equation and the trial eigenstate, first derive a 2×2 eigenvalue problem given by $$E \begin{pmatrix}\alpha \\ \beta\end{pmatrix} = \begin{pmatrix}\varepsilon_0(k)+V_0 & V_1^* \\ V_1 & \varepsilon_0(k - 2\pi/a) + V_0\end{pmatrix} \begin{pmatrix}\alpha \\ \beta\end{pmatrix}.$$ What are $\varepsilon_0(k)$, $V_0$ and $V_1$?
2. Make a sketch of the lower band.
3. We now use the tight binding model, where we know that the dispersion relation can be described by $$E = \varepsilon_0 - 2 t \cos (ka).$$ Find an expression for $\varepsilon_0=\left<n\right|\hat{H}\left|n\right>$ and $-t=\left<n-1\right|\hat{H}\left|n\right>$, where $\left<x|n\right>$ represent the wavefunction of a single deltapeak well at site $n$. You may make use of the results obtained in [exercise 2 of lecture 5](/5_atoms_and_lcao/#exercise-2-application-of-the-lcao-model) or [look up the wavefunction](https://en.wikipedia.org/wiki/Delta_potential).
3. We now use the tight binding model, where we know that the dispersion relation can be described by $$E = \varepsilon_0 - 2 t \cos (ka).$$ Find an expression for $\varepsilon_0=\left<n\right|\hat{H}\left|n\right>$ and $-t=\left<n-1\right|\hat{H}\left|n\right>$, where $|n\right>$ is the wavefunction of a single $\delta$-peak well at site $n$. You may make use of the results obtained in [exercise 2 of lecture 5](/5_atoms_and_lcao/#exercise-2-application-of-the-lcao-model) or [look up the wavefunction](https://en.wikipedia.org/wiki/Delta_potential).
4. Compare the bands obtained in exercise 1 and 2: what are the minima and bandwidths (difference between maximum and minimum) of those bands?
5. For what $a$ and $\lambda$ is the nearly free electron model more accurate? And for what $a$ and $\lambda$ is the tight binding model more accurate?
