From 2fe41b12643a5608d7c8a71728e95b1684c56abc Mon Sep 17 00:00:00 2001
From: Georg Wolfgang Winkler <winklerg@itp.phys.ethz.ch>
Date: Tue, 8 Mar 2016 11:07:25 +0100
Subject: [PATCH] fixed latex rendering for nbviewer in 2.2 solution
---
exercises/ex02_solution/anharmonic.ipynb | 10 +++++-----
1 file changed, 5 insertions(+), 5 deletions(-)
diff --git a/exercises/ex02_solution/anharmonic.ipynb b/exercises/ex02_solution/anharmonic.ipynb
index cd78121..da89b03 100644
--- a/exercises/ex02_solution/anharmonic.ipynb
+++ b/exercises/ex02_solution/anharmonic.ipynb
@@ -26,22 +26,22 @@
"\\end{equation}\n",
"\n",
"Let us now expand the $(a+a^{\\dagger})^4$ using the commutator relation $[a,a^{\\dagger}] = a a^{\\dagger} - a^{\\dagger} a = 1$ and the identity $a^{\\dagger} a = \\hat n$:\n",
- "\\begin{align}\n",
+ "$$\\begin{align}\n",
" ( a + a^{\\dagger})^4 &= (a a + a a^{\\dagger} + a^{\\dagger} a + a^{\\dagger} a^{\\dagger})^2 \\\\\n",
" &= ( a a + 2 a^{\\dagger} a + 1 + a^{\\dagger} a^{\\dagger})^2 \\\\\n",
" &= a^4 + 2 a a \\hat n + 2 a a + 2 \\hat n a a + 6 \\hat n^2 + 6 \\hat n + 3 + 2 \\hat n a^{\\dagger} a^{\\dagger} \\\\\n",
" &\\quad + 2 a^{\\dagger} a^{\\dagger} \\hat n + 2 a^{\\dagger} a^{\\dagger} + (a^{\\dagger})^4\n",
- "\\end{align}\n",
+ "\\end{align}$$\n",
"\n",
"Using $a | n \\rangle = \\sqrt{n} | n -1 \\rangle$, $ a^{\\dagger} | n \\rangle = \\sqrt{n+1} | n+1 \\rangle$ and $ \\langle m | n \\rangle = \\delta_{m,n}$\n",
"we obtain following non-zero matrix elements:\n",
- "\\begin{align}\n",
+ "$$\\begin{align}\n",
" \\langle n | 6 \\hat n^2 + 6 \\hat n + 3 | n \\rangle &= 6 n^2 + 6 n + 3 \\\\\n",
" \\langle n + 2 | 2 \\hat n a^{\\dagger} a^{\\dagger} + 2 a^{\\dagger} a^{\\dagger} + 2 a^{\\dagger} a^{\\dagger} \\hat n | n \\rangle & = \\langle n | 2 a a \\hat n + 2 a a + 2 \\hat a a | n + 2\\rangle \\\\\n",
" & =(4 n + 6) \\sqrt{ (n+1) (n+2) } \\\\\n",
" \\langle n + 4 | (a^{\\dagger})^4 | n \\rangle & = \\langle n | a^4 | n+4 \\rangle \\\\\n",
" & = \\sqrt{(n+1) \\cdot (n+2) \\cdot (n+3) \\cdot (n+4)}\n",
- "\\end{align}\n",
+ "\\end{align}$$\n",
"\n",
"A solution code setting up and diagonalizing a matrix for a given cutoff $N$ is given below.\n",
"The dependence of the energy spectrum on the anharmonicity $K$ is shown in figure at the bottom.\n",
@@ -188,7 +188,7 @@
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
- "version": "3.4.3"
+ "version": "3.4.4"
}
},
"nbformat": 4,
--
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