better equations

mvsp/applications/chemistry/examples/circuit_plane_waves.ipynb

@@ -14,7 +14,12 @@
    "source": [
     "We aim to generate a quantum circuit that prepares a plane wavefunction for a given lattice. The plane wavefunction is given by:\n",
     "\n",
-    "$$\\Psi(\\mathbf{r}) = \\sum_{\\mathbf{k}} c_{\\mathbf{k}} e^{i \\pi \\mathbf{k} \\cdot \\mathbf{r}}$$"
+    "$$\\Psi(\\mathbf{r}) = \\sum_{\\mathbf{k}} c_{\\mathbf{k}} e^{i \\pi \\mathbf{k} \\cdot \\mathbf{r}}$$\n",
+    "\n",
+    "where:\n",
+    "- $\\mathbf{r}$ is the position vector.\n",
+    "- $\\mathbf{k}$ is the wavevector.\n",
+    "- $c_{\\mathbf{k}}$ is the coefficient of the wavevector $\\mathbf{k}$.\n"
    ]
   },
   {
@@ -25,6 +30,15 @@
     "\n",
     "$$ H_{\\mathbf{k}, \\mathbf{k}'} = -\\frac{\\hbar^2 \\mathbf{k}^2}{2m} - \\sum_{\\mathbf{r}} \\frac{4 \\pi}{V \\cdot |\\mathbf{k}' - \\mathbf{k}|^2} e^{-i (\\mathbf{k}' - \\mathbf{k}) \\cdot \\mathbf{r}} $$\n",
     "\n",
+    "where:\n",
+    "- $V$ is the volume of the lattice.\n",
+    "- $\\mathbf{r}$ is the position vector.\n",
+    "- $\\mathbf{k}$ is the wavevector.\n",
+    "- $m$ is the mass of the particle.\n",
+    "- The sum is over all lattice sites.\n",
+    "\n",
+    "The eigenvectors of this hamiltonian give the coefficients $c_{\\mathbf{k}}$.\n",
+    "\n",
     "This solution is then used to build the single particl plane wave function in the periodic lattice in real space."
    ]
   },