Chapter 10 - Position & Momentum_blank.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Chapter 10 - Position and Momentum\n",
    "We can start using sympy to handle symbolic math (integrals and other calculus):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from sympy import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "init_printing(use_unicode=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x, y, z = symbols('x y z', real=True)\n",
    "a, c = symbols('a c', nonzero=True, real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "integrate?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are two ways to use the `integrate` function. In one line, like `integrate(x,(x,0,1))` or by naming an expression and then integrating it over a range:\n",
    "\n",
    "```\n",
    "A = (c*cos((pi*x)/(2.0*a)))**2\n",
    "A.integrate((x,-a,a),conds='none')\n",
    "```\n",
    "\n",
    "We'll use both, at different times. For longer expressions, the second form can be easier to read and write.\n",
    "\n",
    "First, just try the following, then we'll re-create some examples in the book."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAqBAMAAACXcryGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWYiuzKJ\nRN0MreaOAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAcElEQVQYGWNggAEhAyCL2dUfRDEw5FODCtEv\naQCbRhbxHwQ+EKV1atRqoDpmB4bzFxgYWAwYmBQYGJgeMLD8ZmBg/wymgNJ8n8Fm3T8App6ASdYA\nMJUKJhkDGGYCGVsZGDIZGDjfnAl5ANQOdMMDBgC/syNtm3aMLwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{1}{2}$$"
      ],
      "text/plain": [
       "1/2"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(x,(x,0,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAqBAMAAACXcryGAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAVO8Qq5l2zWYiibvdRH6G\nTkUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAB0SURBVBgZY2CAASEDIIvZ1Q9EMTDkUYMK0StpAJtG\nFvEOBB4QpXXWqo1AdZwBDHINDAxsDxn4DjAw8Ooy2BWAdYMEgV47BSI51wSDKAaGRRfAFJMyUGwC\nA/dDBga+x2CKaQMDz0sGBo4GBrkEoJKroXsZGAAuYx+6av2YygAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{1}{3}$$"
      ],
      "text/plain": [
       "1/3"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(x**2,(x,0,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The cell below will return an odd set of conditions on the result. This is because the solver doesn't want to assume anything about `a` and there is a special case where the answer would be different. If you look closely though, that special case isn't physically realistic so to igore these special conditions, we add `conds='none'`. The next cell down does what you'd expect. From here on out, just add this to the `integrate` function and we'll get what we expect."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAc0AAAA/BAMAAABwRo6IAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$- c^{2} \\begin{cases} - a & \\text{for}\\: \\frac{0.5 \\pi}{a} = 0 \\\\- 0.5 a & \\text{otherwise} \\end{cases} + c^{2} \\begin{cases} a & \\text{for}\\: \\frac{0.5 \\pi}{a} = 0 \\\\0.5 a & \\text{otherwise} \\end{cases}$$"
      ],
      "text/plain": [
       "     ⎛⎧            0.5⋅π    ⎞      ⎛⎧           0.5⋅π    ⎞\n",
       "   2 ⎜⎪  -a    for ───── = 0⎟    2 ⎜⎪  a    for ───── = 0⎟\n",
       "- c ⋅⎜⎨              a      ⎟ + c ⋅⎜⎨             a      ⎟\n",
       "     ⎜⎪                     ⎟      ⎜⎪                    ⎟\n",
       "     ⎝⎩-0.5⋅a    otherwise  ⎠      ⎝⎩0.5⋅a    otherwise  ⎠"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = (c*cos((pi*x)/(2.0*a)))**2\n",
    "A.integrate((x,-a,a))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADUAAAAVBAMAAAAUZLt1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWbdiTIi\nu0T8UsK3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABE0lEQVQoFWNgwA64YoOxSwBF1zC445Sbw7D/\nAC7JeQznH+CSY2CQN8AtNw0hxRzVtQHBY2DgCgDyGJXBQuwCzHOQ5VqBHJOQz2ChJgaG5UhyfAv4\ngMawQeSuoNq+XVAW6Bao3BcGhvcCCI36//8DORA55q9AuQSGo0uTgMqXpQHNAwGIHM8vBob9C/h0\nGOoZGE4KsBegyAH17V9g1MAQy8CgxcACNR3ZzHgDhmkMrBDHIcxkALpFXkCXgfk3A9MFiIFw+xju\nMjD0G/9iYPzAYL8AXQ7o93Dmjwy8BTL8DQwMSPbZFzBwCjBrM5QxdDosYHVgMN0A1srq9N2DgdOB\ngXlZygYGk6w9bg8YVnUkwE3FxQAAfclHFjGyPEEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1.0 a c^{2}$$"
      ],
      "text/plain": [
       "       2\n",
       "1.0⋅a⋅c "
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = (c*cos((pi*x)/(2.0*a)))**2\n",
    "A.integrate((x,-a,a), conds='none')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So this tells us the normalization constant should be $c=\\frac{1}{\\sqrt{a}}$. Check that it is normalized if we do that:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABkAAAAPBAMAAADjSHnWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWbdiTIi\nu0T8UsK3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAdklEQVQIHWNggALmqK4NQCajMpjPLsA8h4HB\nJOQzmNfEwLAcyGCD8K4wMMgbwHlfGBjeC8B4zF+BvAQYj+cXA8P+BXAeUA7BQ1XJADRFHm4Kw10G\nhn6EDUDbw6G22xcwcAowazMwsDp992DgdGBgXpayASiHDACNXiNTPqhPAgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$1.0$$"
      ],
      "text/plain": [
       "1.00000000000000"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi = 1/sqrt(a)*cos((pi*x)/(2.0*a))  # notice we can name the expression something useful.\n",
    "B = psi**2\n",
    "B.integrate( (x,-a,a), conds='none')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because `psi` is a real function, we can calculate expectation values by integrating over $x$ or $x^2$ with `psi**2`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "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": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = x*psi**2\n",
    "C.integrate( (x,-a,a), conds='none')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "D = x**2 * psi**2\n",
    "E = D.integrate( (x,-a,a), conds='none')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAL0AAAAVBAMAAADhtvKAAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$0.130690966048658 a^{2}$$"
      ],
      "text/plain": [
       "                   2\n",
       "0.130690966048658⋅a "
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.n()  # the .n() method approximates the numerical part. You can look at the full expression below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAsMAAAA0BAMAAACdqij5AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nXSIjJFLbAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- \\frac{1}{a} \\left(- 0.166666666666667 a^{3} + \\frac{1.0 a^{3}}{\\pi^{2}}\\right) + \\frac{1}{a} \\left(- \\frac{1.0 a^{3}}{\\pi^{2}} + 0.166666666666667 a^{3}\\right)$$"
      ],
      "text/plain": [
       "                                   3             3                        \n",
       "                          3   1.0⋅a      - -1.0⋅a                        3\n",
       "  - - -0.166666666666667⋅a  + ──────   - ────────── + 0.166666666666667⋅a \n",
       "                                 2            2                           \n",
       "                                π            π                            \n",
       "- ────────────────────────────────── + ───────────────────────────────────\n",
       "                  a                                     a                 "
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 10.2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "h = Symbol('hbar', real=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use the `diff` function to take a derivative of a symbolic expression. For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAAOBAMAAADd6iHDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAf0lEQVQIHWNgVDJ2YICAMAb2H1BmKgPDTChz\nFgNDvgOEvT8AzgQKrA9gPZPYUwNkcXxnCGd4dWA1kMmVwFDKUB9wEchUYmAIYNgMZDDwKIDIPyDi\nEgOjAAPLFyCLRYGBR+AqzweGowxS5UVZHH+ZPzA+YNj///8v1pyjh94xAABbhSE1xl33oAAAAABJ\nRU5ErkJggg==\n",
      "text/latex": [
       "$$2 x$$"
      ],
      "text/plain": [
       "2⋅x"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(x**2, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 10.3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "p = Symbol('p', real=True) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}