Merge branch 'revert-49de820d' into 'master'

Revert "Exercise list: indentation fix"

See merge request solidstate/lectures!129

docs/11_nearly_free_electron_model.md

@@ -280,57 +280,47 @@ draw_classic_axes(ax, xlabeloffset=4)
 Suppose we have a crystal with lattice vectors $\mathbf{a}_ 1$, $\mathbf{a}_ 2$, and $\mathbf{a}_ 3$.
 
 1. What can be said about the symmetry of the Hamiltonian $\hat{H}$ of this crystal?
-
-To describe the translation through the crystal in terms of the lattice vectors, we define the translation operator $\hat{T}_{\alpha,\beta,\gamma}$ in such a way that
-
-    $$
-    \hat{T}_{\alpha,\beta,\gamma} \psi(\mathbf{r}) = \psi(\mathbf{r} - \alpha \mathbf{a}_1 - \beta \mathbf{a}_2 - \gamma \mathbf{a}_3),
-    $$
-    
-    where $\alpha$, $\beta$, $\gamma$ are integers.
-
-2. Show that $\hat{T}_{\alpha,\beta,\gamma}$ and $\hat{H}$ commute.
+2. To describe the translation through the crystal in terms of the lattice vectors, we define the translation operator $\hat{T}_{\alpha,\beta,\gamma}$ in such a way that $$
+\hat{T}_{\alpha,\beta,\gamma} \psi(\mathbf{r}) = \psi(\mathbf{r} - \alpha \mathbf{a}_1 - \beta \mathbf{a}_2 - \gamma \mathbf{a}_3)
+, $$
+where $\alpha$, $\beta$, $\gamma$ are integers.
+Show that $\hat{T}_{\alpha,\beta,\gamma}$ and $\hat{H}$ commute.
 3. Show that the Bloch wavefunctions defined in the lecture notes are eigenfunctions of $\hat{T}_{\alpha,\beta,\gamma}$. What are the corresponding eigenvalues? What does this say about the eigenfunctions of $\hat{H}$.
-4. By applying $\hat{H}$ to the Bloch wavefunction, show that the Schrödinger equation can be rewritten as 
-
-    $$
-    \left[ \frac{\mathbf{\hat{p}}^2}{2m} + \frac{\hbar}{m} \mathbf{k} \cdot \mathbf{\hat{p}} + \frac{\hbar^2 \mathbf{k}^2}{2m} + V(\mathbf{r}) \right] u_{n,\mathbf{k}}(\mathbf{r}) = E_{n,\mathbf{k}} u_{n,\mathbf{k}}(\mathbf{r}), 
-    $$
-    
-    where $\mathbf{\hat{p}} =-i\hbar\nabla$.
+4. By applying $\hat{H}$ to the Bloch wavefunction, show that the Schrödinger equation can be rewritten as $$
+\left[ \frac{\mathbf{\hat{p}}^2}{2m} + \frac{\hbar}{m} \mathbf{k} \cdot \mathbf{\hat{p}} + \frac{\hbar^2 \mathbf{k}^2}{2m} + V(\mathbf{r}) \right] u_{n,\mathbf{k}}(\mathbf{r}) = E_{n,\mathbf{k}} u_{n,\mathbf{k}}(\mathbf{r})
+, $$
+where $\mathbf{\hat{p}} =-i\hbar\nabla$.
 5. What is $u_{n,\mathbf{k}}(\mathbf{r})$ in case of free electrons? Is your answer consistent with the equation above?
 
 ### Exercise 2: the central equation in 1D
 Let's consider a 1D crystal with a periodicity of $a$.
 Let $k_0$ be any wave number of an electron in the first Brillouin zone.
+
 1. Which $k_n$ are equivalent to $k_0$ in this crystal?
 2. First, we assume that the electrons with these $k_n$ are free. In that case, what are the wavefunctions $\phi_n(x)$ and energies $E_n$ of these electrons?
 3. Make a sketch of the dispersion relation using a repeated Brillouin zone representation. Indicate some $k_n$ and $E_n$ as well as the first Brillouin zone in your sketch.
 
-    We will now introduce a weak periodic potential $V(x) = V(x+na)$ to the system, which causes coupling between eigenstates $\left| \phi_n\right>$ of the free electrons.
-    In order to find the right eigenstates of the system with that potential, we use an 'LCAO-like' trial eigenstate
-
-    $$
-    \left|\psi\right> = \sum_{n=-\infty}^{\infty}C_n \left|\phi_n\right>
-    $$
+We will now introduce a weak periodic potential $V(x) = V(x+na)$ to the system, which causes coupling between eigenstates $\left| \phi_n\right>$ of the free electrons.
+In order to find the right eigenstates of the system with that potential, we use an 'LCAO-like' trial eigenstate
 
-4. Using this trial eigenstate and the Schrödinger equation, show that 
-
-    $$
-    E C_m = \varepsilon_m C_m+\sum_{n=-\infty}^{\infty} C_{n}V_{m-n},
-    $$
+$$
+\left|\psi\right> = \sum_{n=-\infty}^{\infty}C_n \left|\phi_n\right>
+$$
 
-    where $V_n$ are the Fourier components of the potential defined [above](#physical-meaning-of-w).
-    Find an expression for $\varepsilon_m$. _**NB:** This equation is also known as the central equation (in 1D)._
+4. Using this trial eigenstate and the Schrödinger equation, show that $$
+E C_m = \varepsilon_m C_m+\sum_{n=-\infty}^{\infty} C_{n}V_{m-n}
+,$$
+where $V_n$ are the Fourier components of the potential defined [above](#physical-meaning-of-w).
+Find an expression for $\varepsilon_m$. _**NB:** This equation is also known as the central equation (in 1D)._
 
-    ??? hint
-        - Apply $\left<\phi_m\right|$ to the Schrödinger equation.
-        - To evaluate $\left<\phi_m\right| \hat{H} \left| \phi_n\right>$, it may be helpful to separate the kinetic energy and potential energy of the Hamiltonian.
+??? hint
+    - Apply $\left<\phi_m\right|$ to the Schrödinger equation.
+    - To evaluate $\left<\phi_m\right| \hat{H} \left| \phi_n\right>$, it may be helpful to separate the kinetic energy and potential energy of the Hamiltonian.
 
 5. Why is the dispersion relation only affected near $k=0$ and at the edge of the Brillouin zone (see also figures [above](#repeated-vs-reduced-vs-extended-brillouin-zone))?
 
-    ??? hint
-        To answer this question, only consider consider two free electron wavefunctions in the Hamiltonian and ignore all the others. Between what two of free electron wavefunctions does the coupling give significant contribution to the energy levels of the free electron wavefunctions?
+??? hint
+    To answer this question, only consider consider two free electron wavefunctions in the Hamiltonian and ignore all the others. Between what two of free electron wavefunctions does the coupling give significant contribution to the energy levels of the free electron wavefunctions?
 
 ### Exercise 3: the tight binding model vs. the nearly free electron model
 Consider a 1D crystal with a periodic potential given by delta peaks:
@@ -346,41 +336,34 @@ where $\lambda>0$. In this exercise, we will find the band structure of this cry
 
 1. We first find the band structure using the nearly free electron model. To this end, we consider the effect of the potential on the free electron wavefunctions given by $\psi_1(x) \propto e^{ikx}$ and $\psi_2(x) \propto e^{i[k-2\pi/a]x}$ on the interval $k=[0,\pi/a]$. Derive a dispersion relation of the lower band using the Schödinger equation and the trial eigenstate
 
-    $$
-    \Psi(x) = \alpha \psi_1(x) + \beta \psi_2(x).
-    $$
+$$
+\Psi(x) = \alpha \psi_1(x) + \beta \psi_2(x).
+$$
 
-    ??? hint
+??? hint
 
-        Using the Schrödinger equation and the trial eigenstate, first derive a 2×2 eigenvalue problem given by
+    Using the Schrödinger equation and the trial eigenstate, first derive a 2×2 eigenvalue problem given by
 
-        $$
-        E \begin{pmatrix}\alpha \\ \beta\end{pmatrix} = \begin{pmatrix}\varepsilon_0(k)+V_0 & V_1^* \\ V_1 & \varepsilon_0(k - 2\pi/a) + V_0\end{pmatrix} \begin{pmatrix}\alpha \\ \beta\end{pmatrix}.
-        $$
+    $$
+    E \begin{pmatrix}\alpha \\ \beta\end{pmatrix} = \begin{pmatrix}\varepsilon_0(k)+V_0 & V_1^* \\ V_1 & \varepsilon_0(k - 2\pi/a) + V_0\end{pmatrix} \begin{pmatrix}\alpha \\ \beta\end{pmatrix}.
+    $$
 
-        What are $\varepsilon_0(k)$, $V_0$ and $V_1$?
+    What are $\varepsilon_0(k)$, $V_0$ and $V_1$?
 
 2. Make a sketch of the lower band.
 3. We now use a tight binding model approach to derive the dispersion relation.
-We know from that the corresponding dispersion is 
-
-    $$
-    E = \varepsilon_0 - 2 t \cos (ka). 
-    $$
-
-    Find an expression for $\varepsilon_0=\left<n\right| \hat{H} \left|n\right>$ and $-t=\left<n-1\right| \hat{H} \left| n \right>$, using the bound state wavefunction around a single $\delta$-peak, centered at site $n$: 
-    
-    $$
-    |n\rangle = \kappa e^{- \kappa | x-na | },
-    $$
-    
-    where $\kappa = -\frac{m \lambda}{\hbar^2}$.
+We know from that the corresponding dispersion is $$
+E = \varepsilon_0 - 2 t \cos (ka). $$
+Find an expression for $\varepsilon_0=\left<n\right| \hat{H} \left|n\right>$ and $-t=\left<n-1\right| \hat{H} \left| n \right>$, using the bound state wavefunction around a single $\delta$-peak, centered at site $n$: $$
+|n\rangle = \kappa e^{- \kappa | x-na | }
+, $$
+where $\kappa = -\frac{m \lambda}{\hbar^2}$.
 
-    ??? hint
+??? hint
 
-        To ease the calculating $\epsilon_0$ and $t$, calculate them for $| n = 0 \rangle $ and $ | n = 1 \rangle $.
+    To ease the calculating $\epsilon_0$ and $t$, calculate them for $| n = 0 \rangle $ and $ | n = 1 \rangle $.
 
-        You may also make use of the results obtained in [exercise 2 of lecture 5](/5_atoms_and_lcao/#exercise-2-application-of-the-lcao-model) or found on the [wikipedia](https://en.wikipedia.org/wiki/Delta_potential).
+    You may also make use of the results obtained in [exercise 2 of lecture 5](/5_atoms_and_lcao/#exercise-2-application-of-the-lcao-model) or found on the [wikipedia](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?