Derivation of uncertainty principle

book/knowledge-library/quantum-mechanics.ipynb

@@ -534,7 +534,119 @@
    "metadata": {},
    "source": [
     "$$\n",
-    "\\sigma_x \\sigma_p = \\frac{\\hbar}{2}\n",
+    "\\sigma_x \\sigma_p \\geq \\frac{\\hbar}{2}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "39c7bfea",
+   "metadata": {},
+   "source": [
+    "To derive this, let's consider the position and momentum operators. They are given respectively by:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "dfe8cfa2",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\hat x = x\n",
+    "$$\n",
+    "\n",
+    "$$\n",
+    "\\hat p = -i\\hbar \\frac{\\partial}{\\partial x}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f9b56797",
+   "metadata": {},
+   "source": [
+    "Now, it is important to recognize that these two operators do not commute - if we switch the order we apply these operations, we get different results. That is:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6886c7ff",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\hat x \\hat p \\psi \\neq \\hat p \\hat x \\psi\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "322961fb",
+   "metadata": {},
+   "source": [
+    "We call the difference between applying these two operators in either order the **commutator**, denoted by square brackets:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7fa24891",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "[\\hat x, \\hat p] = \\hat x \\hat p - \\hat p \\hat x\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "168a8710",
+   "metadata": {},
+   "source": [
+    "If we compute this commutator by substituting the operators in, we get:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "541a660f",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "[\\hat x, \\hat p] = i \\hbar\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "24785160",
+   "metadata": {},
+   "source": [
+    "We can now make use of a statistical theorem:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b12c93d4",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\sigma_a \\sigma_b \\geq \\frac{1}{2i} [\\hat a, \\hat b]\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a6950416",
+   "metadata": {},
+   "source": [
+    "Plugging in our result, we get the uncertainty principle:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b4c4c962",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\sigma_x \\sigma_p \\geq \\frac{\\hbar}{2}\n",
     "$$"
    ]
   }