From 6f7e8422c46828bcbe6b17d4d8f2bcb0fe6b61ba Mon Sep 17 00:00:00 2001
From: "T. van der Sar" <t.vandersar@tudelft.nl>
Date: Thu, 24 Feb 2022 22:26:22 +0000
Subject: [PATCH] Update 5_atoms_and_lcao_solutions.md - more fixes
---
docs/5_atoms_and_lcao_solutions.md | 12 ++++++------
1 file changed, 6 insertions(+), 6 deletions(-)
diff --git a/docs/5_atoms_and_lcao_solutions.md b/docs/5_atoms_and_lcao_solutions.md
index e1020733..58e39c7d 100644
--- a/docs/5_atoms_and_lcao_solutions.md
+++ b/docs/5_atoms_and_lcao_solutions.md
@@ -65,7 +65,7 @@ Where $\beta = -\frac{mV_0^2}{\hbar^2}$ and $α = \frac{mV_0}{\hbar^2}|x_2-x_1|$
#### Question 1.
$$
- H_{\mathcal{E}} = ex\mathcal{E},
+ \hat{H}_{\mathcal{E}} = e\mathcal{E}\hat{x},
$$
#### Question 2.
@@ -74,17 +74,17 @@ $$
\hat{H} = \begin{pmatrix}
E_0 & -t\\
-t & E_0
- \end{pmatrix} +\begin{pmatrix}
- ⟨1|ex\mathcal{E}|1⟩ & ⟨1|ex\mathcal{E}|2⟩\\
- ⟨2|ex\mathcal{E}|1⟩ & ⟨2|ex\mathcal{E}|2⟩
+ \end{pmatrix} +e\mathcal{E}\begin{pmatrix}
+ \langle 1|\hat{x}|1\rangle & \langle 1|\hat{x}|2\rangle\\
+ \langle 2|\hat{x}|1\rangle & \langle 2|\hat{x}|2\rangle
\end{pmatrix} = \begin{pmatrix}
E_0 - \gamma & -t\\
-t & E_0 + \gamma
\end{pmatrix},
$$
-where $\gamma = e d \mathcal{E}/2$ and we have used
+where we defined $\gamma = e d \mathcal{E}/2$ and used
$$
- ⟨1|ex\mathcal{E}|1⟩ = -e d \mathcal{E}/2⟨1|1⟩ = -e d \mathcal{E}/2
+ ⟨1|\hat{x}|1⟩ = -\frac{d}{2}.
$$
#### Question 3.
--
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