Griffiths1.5.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem 1.5\n",
    "\n",
    "## Problem statement\n",
    "Consider the wavefunction $$\\Psi(x,t) = Ae^{-\\lambda|x|}e^{i\\omega t},$$ with $A$, $\\lambda$, and $\\omega$ positive real constants.\n",
    "\n",
    "1. Normalize $\\Psi$.\n",
    "2. Find $\\langle x\\rangle$ and $\\langle x^2\\rangle$\n",
    "3. Find $\\sigma_x$. Also sketch $\\Psi^2$ as a function of $x$, mark $\\langle x \\rangle + \\sigma$ and $\\langle x \\rangle - \\sigma$ to illustrate the \"spread\".  What is the probability of finding the particle inside this range?\n",
    "\n",
    "----\n",
    "## Solution\n",
    "\n",
    "This problem is very similar to 1.3, but here we're given $\\Psi$ instead of $\\Psi^2$. Since $\\Psi$ is complex, we need to take the complex conjugate to get $|\\Psi^2|$ (which is what we integrate to do the normalization.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import sympy as sp\n",
    "\n",
    "# make output look nice\n",
    "sp.init_printing()\n",
    "\n",
    "# set up our symbols\n",
    "lam, omega, A = sp.symbols('lambda omega A', positive=True, real=True)\n",
    "x, t = sp.symbols('x t', real=True)\n",
    "\n",
    "# define Psi...\n",
    "Psi = A*sp.exp(-lam*sp.Abs(x))*sp.exp(sp.I*omega*t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGwAAAAmBAMAAADTi2P0AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nGMLCXeu8b0XI9Sqb/gK+RI/p/18hTwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left\\{- \\sqrt{\\lambda}, \\sqrt{\\lambda}\\right\\}$$"
      ],
      "text/plain": [
       "{-√λ, √λ}"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#...and |\\Psi|^2\n",
    "Psisq = Psi*sp.conjugate(Psi)\n",
    "\n",
    "# our normalization condition is that the integral is equal to one\n",
    "normalization_condition = sp.Eq(1, sp.integrate(Psisq, (x, -sp.oo, sp.oo)))\n",
    "\n",
    "# and we can solve it for the normalization constant A.\n",
    "sp.solveset(normalization_condition, A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that we have two options here, one positive and one negative. In fact, any complex phase would work, but we specified that $A$ is real, so we only get the two options. It's customary to take the positive branch, but it doesn't really matter. We work the same magic that we did in problem 1.3 to apply the normalization. This particular method doesn't guarantee either the positive or the negative branch, but like I said, it doesn't really matter."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHEAAAAVBAMAAACDNRyvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\sqrt{\\lambda} e^{- \\lambda \\left|{x}\\right|} e^{i \\omega t}$$"
      ],
      "text/plain": [
       "    -λ⋅│x│  ⅈ⋅ω⋅t\n",
       "√λ⋅ℯ      ⋅ℯ     "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Psi = Psi.subs(A, next(iter(sp.solveset(normalization_condition, A))))\n",
    "\n",
    "Psi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Expectation values and $\\sigma_x$\n",
    "\n",
    "With $\\Psi$ normalized, we're prepared to compute expectation values and use them to calculate $\\sigma_x$. They're computed in exactly the same way as they were in problem 1.3."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expectx = sp.integrate(sp.conjugate(Psi)*x*Psi, (x, -sp.oo, sp.oo))\n",
    "expectxsq = sp.integrate(sp.conjugate(Psi)*x**2*Psi, (x, -sp.oo, sp.oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJZjLNVN0i77ur\nRHZ72Yd1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVElEQVQIHWNgEDIxZWBgSGeQmMDAsoCBOYGB\n+wAD+0cG/gMMvN8Z5BUYeP8xzDdgYP3MMF8BREJEgLLs3xm4NzCwfATpYkpgYGhnkApgYBB+d5QB\nAPogE3QldevOAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expectx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACAAAAAqBAMAAAAgz0ShAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWYiuzKJ\nRN0MreaOAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5ElEQVQoFWNgwAEYlVElTEI+owowsA0VAVan\n7x5obh9o7n80QKJ7ji41QNHBM4GlACbA3Alk8U3g/gmkpkatBpL9F4ACFzj/MjAwOzCcB3EWAEUZ\nGIHxx2LAwKTAAI1KvgQGBqYHDCy/GRg4v4FUHAVi9s9gAYY6IJsjACQK1A+K+f4NDAx3GG6DBe4f\nAFL2Bxi4CwUXgwWegMj9CgxM//9/ADFZQTq5VwCdAAWpIHqfwRcYnzGAYSYDryZDPExgKwNDJsP9\nCwz7BSAinG/OhDzg/QFzPNBhwCB8wJUA9AfIpwwMANKXUgme/lrWAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{1}{2 \\lambda^{2}}$$"
      ],
      "text/plain": [
       " 1  \n",
       "────\n",
       "   2\n",
       "2⋅λ "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expectxsq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABwAAAAvBAMAAAACzbekAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAInarRM2ZVBDdiWbv\nuzJCz3LGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABQElEQVQoFbWSrU8DQRDF332Vu1IuxZCgWgwG\nRAUCFPcfFFkSkm5CUhRpVS0XUCg2wRAMVYgawGJaJAminkCqSFANH+IogWNml21yHka8eb/NzO5k\n7oDfKKUchrAycWysjQzmWhk8zBB2YDUehubMlzhA/t1g/hKrwJ1Bar0F1k31ElCPFboRlUhVdhYD\nyzSAXWZ0xzTOPt0XMCGokPhfwLXChtJmESdsbKGwJKY7bNqwipScJIgphQI2Yzhuk2K+e77GGVs1\n1nqafnDGnlDJiMct/x7qW0zkL57zr8wtc7uPZHuRZr+CC7LBQKMTo9CnxSUaCxLON+37VaOXKMSm\nRtKAC3tDw9UOuSMWFceso77yQE6QCe/Nr7DIx8/xGydgSmAW7g2aGp+ABVQjjFrM1ml3W7qfZkyP\nNiNnytT0AvwAdjdZvBJM1YcAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{\\sqrt{2}}{2 \\lambda}$$"
      ],
      "text/plain": [
       " √2\n",
       "───\n",
       "2⋅λ"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sigma = sp.sqrt(expectxsq - expectx**2)\n",
    "sigma"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plotting\n",
    "\n",
    "The problem asks for a sketch, so as before we have to choose a value for $\\lambda$. It's fine to use $\\lambda=1$. The wrinkle with this plot is we need to add the extra annotations at $\\langle x \\rangle \\pm \\sigma_x$. I'll go a step further and shade the area under the curve between the requested limits.  First, we substitute our chosen value for $\\lambda$ in to both $|\\Psi^2|$ and $\\sigma_x$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADAAAAAUBAMAAAA5EaOUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAzklEQVQYGWNgQAJMfo4MbBA+lIJKqjLsZeCD\nsKEUVKKUYX0AVokyhvgErBIMDPMFgBLKXtYLUI1iYCgG2sEssJcdphFqBwOTA1CCi+EOA5ACA853\nQPCQgeEIROQrXAIizcCmwLYAqJTrAxdMB1Ri7cx5QMvjmQqY0CT6//8HikibKSuCJBiVTQKgOkAU\n1FYglR7AqoBNgrWTQWQBNgnuHuODSOIMLBAOCwP/BmRhJDa/AhIHmckBlDBAFoCxmS8wiCA7FybO\nwGBicwDBQbAA3XUgeGICvEAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$e^{- 2 \\left|{x}\\right|}$$"
      ],
      "text/plain": [
       " -2⋅│x│\n",
       "ℯ      "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Psisq = (Psi*sp.conjugate(Psi)).subs(lam, 1)\n",
    "\n",
    "Psisq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABwAAAAvBAMAAAACzbekAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAInarRM2ZVBDdiWbv\nuzJCz3LGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA8UlEQVQoFbWRsQ7BUBSG/1aJVtJYTewMBgMT\nj2A0SHRiZLJqmGwdbUwGk9kbSLyCxEMYkEidc93T3u6c4T/na869vckH6KrGXEJoJRMPVj+DhVkG\n1xnCBNb4epNvxQgreHdB74g2cBGko2egK9t1YBQqdHq0Eqm1bQg06AF2jdF50XOWdJ/LBLdJUXwD\nJ4VjldMyNjzYgcJqUNrxMIdVppZ7uCE1P4DN6L/mlKgc9h3uGA44R3H85I5FoJpEno/8vZSLJH7+\nO9MPXW76ITT9EJp+CLUfmqTYT1rsxyjlJ+WvH2HtR1D70Sh+NCZ+vpz4+QDG90XydfieogAAAABJ\nRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{\\sqrt{2}}{2}$$"
      ],
      "text/plain": [
       "√2\n",
       "──\n",
       "2 "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sigma_x = sigma.subs(lam, 1)\n",
    "sigma_x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, we use `lambdify` to turn our mathematical expression into a python function, and plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "function_to_plot = sp.lambdify(x, Psisq, \"numpy\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAUBAMAAABCNWFYAAAAG1BMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAB4Gco9AAAACHRSTlMAdt3NMolEZgN4ymIAAAAJcEhZcwAADsQAAA7EAZUrDhsA\nAAAXSURBVAgdYxAyKVZjCGMAIpoQQipCagCy4Q1mVUJFwQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ \\right ]$$"
      ],
      "text/plain": [
       "[]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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auaXU9ekDTzwRZqbMmRN1NZIJBXgJWrUqhPfpp8OVV0ZdjRSDwYPDmjcnngj/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\n/wO8aPSDqTTtewAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10cdc1940>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "%matplotlib inline\n",
    "\n",
    "domain = np.arange(-2, 2, 0.001)\n",
    "plt.plot(domain, function_to_plot(domain))\n",
    "plt.title(r\"Sketch of $|\\Psi^2|(x)$\")\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"Probability density\")\n",
    "\n",
    "# since it's a sketch, numbers on the axes aren't meaningful\n",
    "ax = plt.gca()\n",
    "ax.set_xticklabels([])\n",
    "ax.set_yticklabels([])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's the plot, but now we need to add the annotations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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0zYZqgcdKri3w5XXf5NN1Islz881hIiXNNlheqqrCcMKCrZ0pZS9dC3wlYdQJ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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x113d68eb8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# the basic plot\n",
    "plt.plot(domain, function_to_plot(domain))\n",
    "plt.title(r\"Sketch of $|\\Psi^2|(x)$\")\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"Probability density\")\n",
    "\n",
    "# since it's a sketch, numbers on the axes aren't meaningful\n",
    "ax = plt.gca()\n",
    "ax.set_xticklabels([])\n",
    "ax.set_yticklabels([])\n",
    "\n",
    "# add vertical lines for <x>+sigma, <x>-sigma\n",
    "minus = (expectx - sigma_x).evalf()\n",
    "plt.plot([minus, minus], [0, 0.4],\"r\")\n",
    "\n",
    "plus = (expectx + sigma_x).evalf()\n",
    "plt.plot([plus, plus], [0, 0.4], \"r\")\n",
    "\n",
    "# add text labels for <x>+sigma, <x>-sigma\n",
    "minus_text = r\"$\\langle x \\rangle - \\sigma_x$\"\n",
    "plt.text(minus, 0.41, minus_text, horizontalalignment='center', color=\"r\")\n",
    "\n",
    "plus_text = r\"$\\langle x \\rangle + \\sigma_x$\"\n",
    "txt = plt.text(plus, 0.41, plus_text, horizontalalignment='center', color=\"r\")\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we want to pretty it up with a fill, that requires some additional work.  There are two matplotlib functions, `fill`, and `fill_between`, which do similar things. It turns out that `fill_between` is the one we want here; if we use `fill`, it doesn't fill all the way down to the x-axis. Also, if you're relatively inexperienced with python, line 27 is worth some attention. I use a *list comprehension* to create a list of true/false values, and then turn it into a *numpy array* for use with `fill_between`. Strictly speaking, that latter step isn't necessary, but it's something I like to do anyway."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Ve64JiUB25FmtqmeIyDpV7SciNYAFqjqkQhcS0QceeODk78nJySQnJ1cmZhMO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\n2AAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x113da8828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# the basic plot\n",
    "plt.plot(domain, function_to_plot(domain))\n",
    "plt.title(r\"Sketch of $|\\Psi^2|(x)$\")\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"Probability density\")\n",
    "\n",
    "# since it's a sketch, numbers on the axes aren't meaningful\n",
    "ax = plt.gca()\n",
    "ax.set_xticklabels([])\n",
    "ax.set_yticklabels([])\n",
    "\n",
    "# add vertical lines for <x>+sigma, <x>-sigma\n",
    "minus = (expectx - sigma_x).evalf()\n",
    "plt.plot([minus, minus], [0, 0.4],\"r\")\n",
    "\n",
    "plus = (expectx + sigma_x).evalf()\n",
    "plt.plot([plus, plus], [0, 0.4], \"r\")\n",
    "\n",
    "# add text labels for <x>+sigma, <x>-sigma\n",
    "minus_text = r\"$\\langle x \\rangle - \\sigma_x$\"\n",
    "plt.text(minus, 0.41, minus_text, horizontalalignment='center', color=\"r\")\n",
    "\n",
    "plus_text = r\"$\\langle x \\rangle + \\sigma_x$\"\n",
    "txt = plt.text(plus, 0.41, plus_text, horizontalalignment='center', color=\"r\")\n",
    "\n",
    "# the fill. Semicolon suppresses output\n",
    "fill_region = np.array([x > minus and x < plus for x in domain])\n",
    "plt.fill_between(domain, function_to_plot(domain), color=\"r\", alpha=0.3, where=fill_region);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Probability\n",
    "\n",
    "The probability of finding the particle between $\\langle x \\rangle - \\sigma_x$ and $\\langle x \\rangle + \\sigma_x$ is just the area of the shaded region, so to calculate it, we integrate $$\\int_{\\langle x \\rangle - \\sigma_x}^{\\langle x \\rangle + \\sigma_x} |\\Psi^2| dx.$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAI0AAABdBAMAAABqTvi5AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nO44wDqPEjvwpYRyihN7Dxew4jwDvbuaTJnnUKgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\lambda \\int_{- \\frac{\\sqrt{2}}{2 \\lambda}}^{\\frac{\\sqrt{2}}{2 \\lambda}} e^{- 2 \\lambda \\left|{x}\\right|}\\, dx$$"
      ],
      "text/plain": [
       "   √2              \n",
       "  ───              \n",
       "  2⋅λ              \n",
       "   ⌠               \n",
       "   ⎮    -2⋅λ⋅│x│   \n",
       "λ⋅ ⎮   ℯ         dx\n",
       "   ⌡               \n",
       "  -√2              \n",
       "  ────             \n",
       "  2⋅λ              "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.integrate(Psi*sp.conjugate(Psi), (x, expectx-sigma, expectx+sigma))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It turns out that, because of the absolute value, sympy doesn't know how to do this integral. Let's offer some help. We can get rid of the absolute value by breaking the integral in two at $x=0$ and being explicit about the sign of $x$. Since there's symmetry, We can do just the positive side and multiply it by two."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFcAAAAtBAMAAAApX+r4AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMlTvq5l2Zoki\nRLvZ+6rzAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABLElEQVQ4EWNgoASwJhKvW7z9I/GKGThHFSOF\n1hAMDY6yT1VIPqALM0yAsDWRECWMFfXEK2ZgmD9EFQsZg4AKA8N/NPABGAyblJTMlJTUwQEyyD3I\nD3I+NHKhMYg7Um5DFYIomOL2fPUFSMII5kPuvkYYD6YYxkenWSfsYSiCCcrAGDhoPgYbBv8DOCTR\nhVcx2DKcv4AuioP/ACgeT0SCZDjKwMAZAFRsicMgZGHe/QIMckAB7gZkUVxspg0MvUC5FbjkYeKM\nO2ccYOD9yFPAwMC7gdcBJoydvnyAZwMDgwb7AQYGp9A4/B7kSWWQABp3/zrQqHx40sBuLgN71syF\nQCmmPzjkUYT5gW4FApYJKKI4OPxABxMN2ICKiTIVZCKnAoPEAaKNnjFrAQG1AD5eWIc56PzWAAAA\nAElFTkSuQmCC\n",
      "text/latex": [
       "$$- \\frac{1}{e^{\\sqrt{2}}} + 1$$"
      ],
      "text/plain": [
       "   -√2    \n",
       "- ℯ    + 1"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# abs(x) = x if x>0\n",
    "pos_side = lam*sp.exp(-2*lam*x)\n",
    "\n",
    "# The negative side is just like the positive, so multiply by 2\n",
    "2*sp.integrate(pos_side, (x, 0, sigma))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that since the limits of integration depend on $\\lambda$, the end result does not. That's nice, because it allows us to say something about probability without worrying about choosing a value for $\\lambda$.\n",
    "\n",
    "We can turn this into a number, too."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$0.756883265565786$$"
      ],
      "text/plain": [
       "0.756883265565786"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(2*sp.integrate(pos_side, (x, 0, sigma))).evalf()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So, there's a 76% probability of finding the particle within $\\sigma$ of the central peak, *independent of our choice of $\\lambda$*."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "## Portfolio references\n",
    "* Problem 1.3\n",
    "\n",
    "## Index terms\n",
    "* Wavefunction\n",
    "* Normalization\n",
    "* Plotting\n",
    "* Probability\n",
    "\n",
    "## Book references\n",
    "\n",
    "## External references\n",
    "* [Matplotlib thumbnail gallery](http://matplotlib.org/gallery.html) for some ideas on what kinds of things you can plot, and how.\n",
    "* To figure out that the `abs` function in `sympy` is actually `Abs`, I looked at the [reference documentation for the Functions module](http://docs.sympy.org/latest/modules/functions/index.html)"
   ]
  }
 ],
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