Griffiths1.3.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem 1.3\n",
    "\n",
    "## Problem statement\n",
    "\n",
    "Consider the **gaussian** distribution\n",
    "$$ \\rho(x) = A e^{-\\lambda(x-a)^2}, $$\n",
    "where $A$, $a$, and $\\lambda$ are positive real constants.\n",
    "\n",
    "1. use equation 1.16 to determine $A$.\n",
    "2. Find $\\langle x\\rangle$, $\\langle x^2\\rangle$, and $\\sigma$.\n",
    "3. Sketch the graph of $\\rho(x)$.\n",
    "\n",
    "----\n",
    "\n",
    "## Solution\n",
    "\n",
    "Given values for $a$ and $\\lambda$, we could pretty easily solve this numerically (and I'll add that at the end for the sake of completeness), but numerical solutions don't provide the same kind of insight that analytical ones do, so we're going the analytical route this time. Also, this is a problem that wouldn't be that hard to do by hand (you have to look up the integrals, but that's not a big deal) so it's a reasonable case for getting familiar with `sympy`'s quirks.\n",
    "\n",
    "The first thing we need to do is let `sympy` know what we're using as mathematical symbols, so that it can distinguish between them and normal python variables. (`lambda` has a special use in creating anonymous functions, and I don't want to clobber that, so I'm using `lam` instead.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import symbols, exp\n",
    "\n",
    "A, a, lam = symbols('A a lambda', real=True, positive=True)\n",
    "x = symbols('x')\n",
    "\n",
    "rho = A*exp(-lam*(x-a)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "A*exp(-lambda*(-a + x)**2)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rho"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A couple of notes here. First, since the problem specifies that $\\lambda$, $A$, and $a$ are real and positive, I went ahead and told `sympy` about that when I declared them as symbols. Second, I didn't declare $\\rho$ as a symbol, but I don't need to since I have assigned to it an expression consisting of symbols.\n",
    "\n",
    "Lastly, while this works, I'd like to make it more readable (i.e., look more like what I recognize as math), so I'm going to use the pretty-printing feature."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import sympy as sp\n",
    "sp.init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHAAAAAXBAMAAAAhP9aaAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$A e^{- \\lambda \\left(- a + x\\right)^{2}}$$"
      ],
      "text/plain": [
       "              2\n",
       "   -λ⋅(-a + x) \n",
       "A⋅ℯ            "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rho"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Much better. Now, we want to use equation 1.16, $$1 = \\int_{-\\infty}^{+\\infty}\\rho(x) dx,$$ to get $A$. We can take the integral using `sympy` and take the reciprocal by hand:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC4AAAAyBAMAAADRg/TkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n7vCRR9+E390nXtvS5Rsm7e4/GVlytLsy9rEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{\\sqrt{\\pi} A}{\\sqrt{\\lambda}}$$"
      ],
      "text/plain": [
       "√π⋅A\n",
       "────\n",
       " √λ "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import integrate, oo\n",
    "\n",
    "integrate(rho, (x, -oo, oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "so $$A=\\sqrt{\\lambda\\over\\pi},$$\n",
    "but this seems awkward. We'd really like `sympy` to do all of the work for us.  We can do this using the `sympy` function `solveset`. However, it's important to know that we **can't** use the `=` symbol to denote algebraic equality. We must instead use the `sympy` function `Eq`. (This is to make `sympy` play well with the rest of python.) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAD8AAAA/BAMAAAClcQ3sAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
      "text/latex": [
       "$$\\left\\{\\frac{\\sqrt{\\lambda}}{\\sqrt{\\pi}}\\right\\}$$"
      ],
      "text/plain": [
       "⎧√λ⎫\n",
       "⎨──⎬\n",
       "⎩√π⎭"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import Eq, solveset\n",
    "\n",
    "solveset(Eq(1, integrate(rho, (x, -oo, oo))), A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Okay, now we're equipped to get the expectation values for part (b). We do need, though, to modify $\\rho(x)$ so that it includes the value we've found for $A$.\n",
    "\n",
    "Once we do that, we use the definitions on page 10 for our expectation values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# set rho equal to what it was, but substitute in the first element of the set of solutions to equation 1.19 for A\n",
    "rho = rho.subs(A, next(iter(solveset(Eq(1, integrate(rho, (x, -oo, oo))), A))))\n",
    "\n",
    "expectx = integrate(x*rho, (x, -oo, oo))\n",
    "expectxsq = integrate(x**2*rho, (x, -oo, oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAJBAMAAAAWSsseAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpmJdu8QRM1mu90y\nVKvMIHo8AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAARklEQVQIHWNgYFRWCWNgYCgV4PjAwMAWycC0\ngIGB6RsDtwMDA+sChvwDDAz5Bgz3GYDUBYYVvAIMHA6c+5iBKp/YyLkzAABmaAxT1tVPwQAAAABJ\nRU5ErkJggg==\n",
      "text/latex": [
       "$$a$$"
      ],
      "text/plain": [
       "a"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expectx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEcAAAAqBAMAAAAT9FuoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n3QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$a^{2} + \\frac{1}{2 \\lambda}$$"
      ],
      "text/plain": [
       " 2    1 \n",
       "a  + ───\n",
       "     2⋅λ"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expectxsq"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Remember that the definition in equation 1.19 is $\\sigma^2$, and we want $\\sigma$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import sqrt\n",
    "sigma = sqrt(expectxsq - expectx**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACgAAAA0BAMAAAAKxGe+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nfzZNeDDes2tYTF78COM3dttxnERjGF4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{\\sqrt{2}}{2 \\sqrt{\\lambda}}$$"
      ],
      "text/plain": [
       " √2 \n",
       "────\n",
       "2⋅√λ"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sigma"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plotting\n",
    "\n",
    "Plotting and sketching are two different things (with different aims), but we can use a couple of techniques to sketch by plotting.  The biggest difference is that you can sketch without choosing values for unknown constants, but that doesn't work for plotting. We need to choose values for $\\lambda$ and $a$, and then see if we can generalize from the plots.\n",
    "\n",
    "We'll start with $\\lambda = 1$ and $a=0$.\n",
    "\n",
    "To actually carry this out, I substitute values for $\\lambda$ and $a$ into my expression for $\\rho$ using `rho.subs`, and then use `lambdify` to turn it into a python function that can be used to generate a plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from sympy import lambdify\n",
    "\n",
    "function_to_plot = lambdify(x, rho.subs({lam: 1, a:0}), \"numpy\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I need to set up plotting in this notebook, and then choose a domain and plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x10d7c5908>]"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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YOBH22CN2Jb2niVGpNAW6pNqKFVBfH7uKZCjQpdIU6JJqK1fmJ9Dr68PrEakUBbqkWp5G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\niOSEAl1EJCcU6CIiOaFAFxHJif8Pn2JQmbIOHC4AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10d243588>"
      ]
     },
     "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(-5, 5, 0.001)\n",
    "plt.plot(domain, function_to_plot(domain))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Okay, what happens if we vary $\\lambda$? Let's choose some extreme values (10, 0.1) and plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1147d04e0>]"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10d33f780>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "function_to_plot = lambdify(x, rho.subs({lam: 10, a:0}), \"numpy\")\n",
    "plt.plot(domain, function_to_plot(domain))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1140e6828>]"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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4AHefCbQws1YAZtYW6AHcnbWoC9yjj8K0aeqJKZINZvDnP8OkSfDMM7GjyU+Z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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1147ebc88>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "function_to_plot = lambdify(x, rho.subs({lam: 0.1, a:0}), \"numpy\")\n",
    "plt.plot(domain, function_to_plot(domain))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So, $\\lambda$ affects the width in an inverse way.  Let's take a look at $a$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x114b34588>]"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x114a11d30>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "function_to_plot = lambdify(x, rho.subs({lam: 10, a:-2}), \"numpy\")\n",
    "plt.plot(domain, function_to_plot(domain))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x10cde1748>]"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x114a35550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "function_to_plot = lambdify(x, rho.subs({lam: 10, a:2}), \"numpy\")\n",
    "plt.plot(domain, function_to_plot(domain))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's pretty clear that $a$ localizes the peak."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As an aside, if I want my plots to look more like sketches, I can apply the `xkcd` style (based on the webcomic of the same name)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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H6kgB4Ufk+RbkfGJoCnLSE1KzU5Yg93q9AAC73S7H6kiBCAZZEKrVuT1QVmf4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\nHAU5IYTkOQpyQgjJcxTkhBCS5/4/fpF+AhIeD8cAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10d75e1d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "function_to_plot = lambdify(x, rho.subs({lam: 1, a:0}), \"numpy\")\n",
    "with plt.xkcd():\n",
    "    plt.plot(domain, function_to_plot(domain))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Aside: Numerical solutions\n",
    "\n",
    "I mentioned above that, given values for the constants, numerical solutions aren't that hard to compute. I'll illustrate that by computing the expectation value of x for $a=1$, $\\lambda=1$. This is a two step process. First, you have to write a function that calculates the integrand (in this case, $x\\rho(x)$. Then, you feed that function to the numeric integrator.\n",
    "\n",
    "First, the function (written more for clarity than efficiency):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def integrand(x):\n",
    "    \"\"\"Integrand to find the expectation value of x for a gaussian distribution with lambda=1, a=1\"\"\"\n",
    "    lam = 1.0\n",
    "    a = 1.0\n",
    "    A = np.sqrt(lam/np.pi)\n",
    "    \n",
    "    return x*A*np.exp(-lam*(x-a)**2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, integrate it. We use the `scipy.integrate.quad` function (which stands for *quadrature*, another term for integration). This computes a definite integral, but will accept negative or positive infinity as limits."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAc0AAAAUBAMAAAAdLXt9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n3C5SZVMzlaxmJt9w7qham5E83d1t7cB/lJ/hTrf90NgAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left ( 0.9999999999999998, \\quad 4.0968521985254706e-11\\right )$$"
      ],
      "text/plain": [
       "(0.9999999999999998, 4.0968521985254706e-11)"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.integrate import quad\n",
    "\n",
    "quad(integrand, -np.inf, np.inf)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's about as close to the analytic value as we could hope to get, but it's nowhere near as satisfying as taking the symbolic approach like we did above. Also, you'd have to do quite a few examples of $a$ and $\\lambda$ before you could say with any confidence that $\\langle x \\rangle = a$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "\n",
    "## Portfolio references\n",
    "\n",
    "## Index terms\n",
    "* Gaussian\n",
    "* Normalization\n",
    "* Expectation value\n",
    "* sympy\n",
    "* plotting\n",
    "* lambdify\n",
    "* numerical integration\n",
    "\n",
    "## Book references\n",
    "\n",
    "* Equations 1.16--1.19, p. 10\n",
    "\n",
    "## External references\n",
    "* [Sympy tutorial](http://docs.sympy.org/latest/tutorial/index.html)\n",
    "* [Extracting one element from a solveset](http://stackoverflow.com/questions/38418205/get-a-value-from-solution-set-returned-as-finiteset-by-sympy)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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