diff --git a/notebook/band-theory/free_electron.ipynb b/notebook/band-theory/free_electron.ipynb
index a4bf5aa..560283f 100644
--- a/notebook/band-theory/free_electron.ipynb
+++ b/notebook/band-theory/free_electron.ipynb
@@ -80,7 +80,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 64,
+   "execution_count": 8,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -97,7 +97,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 65,
+   "execution_count": 9,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -121,7 +121,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 66,
+   "execution_count": 10,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -168,7 +168,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 67,
+   "execution_count": 11,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -215,7 +215,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 68,
+   "execution_count": 12,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -239,7 +239,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 69,
+   "execution_count": 13,
    "metadata": {
     "tags": []
    },
@@ -251,7 +251,7 @@
     "\n",
     "output = Output()\n",
     "cell_type = RadioButtons(options=['Simple cubic', 'FCC', 'BCC'], value='Simple cubic', description=\"Cell type:\")\n",
-    "cell_hint =HTML(value=f\"<b>Note that there may be a delay in the figure updating.</b>\"))\n",
+    "cell_hint =HTML(value=f\"<b>Note that there may be a delay in the figure updating.</b>\")\n",
     "nkpt = IntSlider(value=4, min=4, max=11, description=\"Number of k-point:\", style=style)\n",
     "grange = IntSlider(value=0, min=0, max=3, description=\"Gvector range:\", style=style)\n",
     "gcov = FloatSlider(value=0.5, min=0.1, max=1.0, description=\"Guassian covariance:\", style=style)\n",
@@ -342,13 +342,13 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 70,
+   "execution_count": 14,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "application/vnd.jupyter.widget-view+json": {
-       "model_id": "0f200beb6df748f29562228c12d2eeb3",
+       "model_id": "8ebb0a5baa734ca5bdee2e2f3b9afe42",
        "version_major": 2,
        "version_minor": 0
       },
@@ -368,22 +368,16 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<details open>\n",
-    "    <summary style=\"font-size: 22px;\"><b>Legend</b></summary>\n",
+    "## Legend\n",
     "\n",
-    "<p style=\"text-align: justify;font-size:15px\">\n",
-    "    The 1st Brillouin zone of the selected cell is shown on the left. \n",
-    "    The path along which the band structure is calculated is indicated with blue vectors and sampled\n",
-    "    k-points are shown with red dots.\n",
-    "    \n",
-    "    The figure on the right shows the calculated band structure. \n",
-    "    \n",
-    "    We provide three kinds of cell structure: simple cubic, \n",
-    "    face-centered cubic (FCC) and body-centered cubic (BCC). Use the radio \n",
-    "    button to select the cell type. The number of k-points and G-vector ranges \n",
-    "    can be tuned with the sliders.\n",
-    "</p>\n",
-    "    "
+    "The 1st Brillouin zone of the selected cell is shown on the left. The path along which the band structure is calculated is indicated with blue vectors and sampled\n",
+    "k-points are shown with red dots.\n",
+    "\n",
+    "The figure on the right shows the calculated band structure. \n",
+    "\n",
+    "We provide three kinds of cell structure: simple cubic, \n",
+    "face-centered cubic (FCC) and body-centered cubic (BCC). Use the radio \n",
+    "buttons to select the cell type (note that there may be a delay while the figure is updated).\n"
    ]
   },
   {
@@ -410,7 +404,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.10.12"
+   "version": "3.10.9"
   }
  },
  "nbformat": 4,
diff --git a/notebook/band-theory/theory/theory_free_electron.ipynb b/notebook/band-theory/theory/theory_free_electron.ipynb
index 05c9799..6a0a7b6 100644
--- a/notebook/band-theory/theory/theory_free_electron.ipynb
+++ b/notebook/band-theory/theory/theory_free_electron.ipynb
@@ -34,14 +34,14 @@
     "In the empty lattice approximation, the electrons move \"freely\" in a weak, periodic potential. Electron-electron interactions are neglected.\n",
     "The eigenfunctions of the Schrödinger equation for free electrons are:\n",
     "\n",
-    "$$\\large \\psi(\\vec{r}) = e^{i\\vec{k} \\vec{r}}$$   \n",
+    "$$\\large \\psi(\\vec{r}) = e^{i\\vec{k} \\cdot \\vec{r}}$$   \n",
     "\n",
     "When $\\vec{k}$ lies outside the 1st Brillouin zone, the plane wave \n",
     "can be written as:\n",
     "    \n",
-    "$$\\large \\psi(\\vec{r}) = e^{i\\vec{k} \\vec{r}}e^{i\\vec{G} \\vec{r}} = e^{i(\\vec{k}+\\vec{G})\\vec{r}}$$   \n",
+    "$$\\large \\psi(\\vec{r}) = e^{i\\vec{k}\\cdot \\vec{r}}e^{i\\vec{G} \\cdot\\vec{r}} = e^{i(\\vec{k}+\\vec{G})\\cdot \\vec{r}}$$   \n",
     "    \n",
-    "where, $\\vec{k}$ vector is inside the first Brillouin zone and $\\vec{G}$ \n",
+    "where $\\vec{k}$ vector is inside the first Brillouin zone and $\\vec{G}$ \n",
     "is a reciprocal lattice vector. The dispersion is:\n",
     "   \n",
     "$$\\large E = \\frac{\\hbar^2(\\vec{k}+\\vec{G})^2}{2m}$$\n",
@@ -107,7 +107,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.10.12"
+   "version": "3.10.9"
   }
  },
  "nbformat": 4,