fix typos

book/ApproximateMethods.ipynb

@@ -883,7 +883,7 @@
     "\n",
     "$$\n",
     "\\begin{align}\n",
-    "\\left[\\frac{dE_n}{dF}\\right]_{F=0} &= \\int_{-1}^{1} \\psi_n(x) \\left[ \\frac{d \\hat{H}}{dF} \\right]_{F=0} \\psi_n(x) dx \\\\\n",
+    "\\left[\\frac{dE_n}{dF}\\right]_{F=0} &= \\int_{-1}^{1} \\psi_n^*(x) \\left[ \\frac{d \\hat{H}}{dF} \\right]_{F=0} \\psi_n(x) dx \\\\\n",
     "&= \\int_{-1}^{1} x|\\psi_n(x)|^2 dx  \\\\\n",
     "&= \\int_{-1}^{1} \\text{(even function)} \\text{(odd function) } dx \\\\\n",
     "&= \\int_{-1}^{1}\\text{(odd function) } dx \\\\\n",
@@ -903,7 +903,7 @@
     "To determine the first-order correction to the wavefunction, one needs to evaluate integrals that look like:\n",
     "\n",
     "$$\n",
-    "V_{mn} = \\int_{-1}^{1} \\psi_m(x) (x) \\psi_n(x) dx \n",
+    "V_{mn} = \\int_{-1}^{1} \\psi_m^*(x) (x) \\psi_n(x) dx \n",
     "$$\n",
     "\n",
     "From the properties of odd and even functions, and the fact that $\\psi_n(x)$ is odd if $n$ is even, and *vice versa*, it's clear that $V_mn = 0$ unless $m+n$ is odd. (That is, either $m$ or $n$, but not both, must be odd.) The integrals we need to evaluate all have the form\n",
@@ -1431,7 +1431,7 @@
     "- The Hellmann-Feynman theorem indicates that given the ground-state wavefunction for a molecule, the force on the nuclei can be obtained. Explain how.\n",
     "- What does it mean that perturbation theory is inaccurate when the perturbation is large?\n",
     "- Can you explain why the energy goes down when the electron-in-a-box is placed in an external field?\n",
-    "- For a sufficiently-highly excited state, the effect of an external electric field is negligible. Why is this true intuitively? Can you show it graphically? Can you explain it mathematically?\n",
+    "- For a sufficiently-highly excited state of the particle-in-a-box, the effect of an external electric field is negligible. Why is this true intuitively? Can you show it graphically? Can you explain it mathematically?\n",
     "\n",
     "## 🔁 Recapitulation\n",
     "- What is the secular equation?\n",
@@ -1452,6 +1452,12 @@
     "- [Perturbation theory](https://en.wikipedia.org/wiki/Perturbation_theory)\n",
     "- [Variational method](https://en.wikipedia.org/wiki/Variational_method_(quantum_mechanics))"
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8c4a8946",
+   "metadata": {},
+   "source": []
   }
  ],
  "metadata": {