diff --git a/exercises/ex02_solution/anharmonic.ipynb b/exercises/ex02_solution/anharmonic.ipynb index cd7812173f3190735f39c747f001bf71e3f034b0..da89b03e4417821aecfbf575756d74ef77fe800f 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,