diff --git a/docs/5_atoms_and_lcao_solutions.md b/docs/5_atoms_and_lcao_solutions.md index 58e39c7ddf35094f1218c9d68fa96cb749bfba68..6cd93be3b3365a7be9c84793dd14d97788b72820 100644 --- a/docs/5_atoms_and_lcao_solutions.md +++ b/docs/5_atoms_and_lcao_solutions.md @@ -72,19 +72,19 @@ $$ $$ \hat{H} = \begin{pmatrix} - E_0 & -t\\ - -t & E_0 + E_0 & -t \\ + -t & E_0 \end{pmatrix} +e\mathcal{E}\begin{pmatrix} - \langle 1|\hat{x}|1\rangle & \langle 1|\hat{x}|2\rangle\\ + \langle 1|\hat{x}|1\rangle & \langle 1|\hat{x}|2\rangle \\ \langle 2|\hat{x}|1\rangle & \langle 2|\hat{x}|2\rangle \end{pmatrix} = \begin{pmatrix} - E_0 - \gamma & -t\\ - -t & E_0 + \gamma + E_0-\gamma & -t \\ + -t & E_0+\gamma \end{pmatrix}, $$ where we defined $\gamma = e d \mathcal{E}/2$ and used $$ - ⟨1|\hat{x}|1⟩ = -\frac{d}{2}. + \langle 1|\hat{x}|1\rangle = -\frac{d}{2}. $$ #### Question 3. @@ -114,5 +114,5 @@ $$ #### Question 4. We find the polarization using $$ -P=2e\langle\psi|\hat{x}|\psi\rangle = 2e\phi_1^2\langle 1|\hat{x}| 1 \rangle + \phi_1^2\langle 2|\hat{x}| 2 \rangle = ed(\phi_2^2-\phi_1^2) =ed\frac{\eta}{1-\eta} +P=2e\langle\psi|\hat{x}|\psi\rangle = 2e\left(\phi_1^2\langle 1|\hat{x}| 1 \rangle + \phi_1^2\langle 2|\hat{x}| 2 \rangle \right) = ed(\phi_2^2-\phi_1^2) =ed\frac{\eta}{1-\eta} $$