Chapter 10 - Position & Momentum.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": 90,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [],
   "source": [
    "init_printing(use_unicode=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [],
   "source": [
    "# SymPy works better if you specify what letters are symbols:\n",
    "x, y, z = symbols('x y z', real=True)\n",
    "\n",
    "# notice we can also put some restrictions on the symbols:\n",
    "a, c = symbols('a c', nonzero=True, real=True) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "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": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{2}$"
      ],
      "text/plain": [
       "1/2"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(x,(x,0,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{3}$"
      ],
      "text/plain": [
       "1/3"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(x**2,(x,0,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle a c^{2}$"
      ],
      "text/plain": [
       "   2\n",
       "a⋅c "
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = (c*cos((pi*x)/(2.0*a)))**2\n",
    "A.integrate((x,-a,a))"
   ]
  },
  {
   "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": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAOCAYAAAASVl2WAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAXUlEQVQYGWP8//8/Aww0NDQYAdmrgdgYyP4AEmcBMgSA9GwgfgfEJkCsBMRwAFIAUhkKEgGyy4AUyBQ4YIKzcDBGFUAChmA4MIIiCxjEoAgCxQkoLkD0PSA+B8SnAeDgGGsGGU5UAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 18,
     "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": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0UlEQVQoFXWR0Q2CQBBEhVgAwQ60BLUCsQO0A+3BP/4MJWgLWgIdGCmBDiB2gG+OW0PIuckyO3Ozu5cj6vt+ZlEURenrFlyRJVojLTIjwht+BZ86ABNA2p66ib14BhMzee0Dqukm7ozggawlTOIFzzTdjBlCNzGJaqoii/1dBvr/m2pi6s+te2y3Lb/V48NQvdBE6woZbFurO9rKJOA0bXhHDBW5DBhtYmXP88C0CRjXaLW2OiPFHaEDczNTa+2RPEmb6+ND3RcMdoUtfAd3f+wLrOY/XeyA6s8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = x*psi**2\n",
    "C.integrate( (x,-a,a), conds='none')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{- \\frac{a^{3}}{6} + \\frac{a^{3}}{\\pi^{2}}}{a} + \\frac{- \\frac{a^{3}}{\\pi^{2}} + \\frac{a^{3}}{6}}{a}$"
      ],
      "text/plain": [
       "     3    3      3    3\n",
       "    a    a      a    a \n",
       "  - ── + ──   - ── + ──\n",
       "    6     2      2   6 \n",
       "         π      π      \n",
       "- ───────── + ─────────\n",
       "      a           a    "
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D = x**2 * psi**2\n",
    "E = D.integrate( (x,-a,a), conds='none')\n",
    "E"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{a^{2} \\left(-6 + \\pi^{2}\\right)}{3 \\pi^{2}}$"
      ],
      "text/plain": [
       " 2 ⎛      2⎞\n",
       "a ⋅⎝-6 + π ⎠\n",
       "────────────\n",
       "       2    \n",
       "    3⋅π     "
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.simplify()  # this is a useful method!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.130690966048658 a^{2}$"
      ],
      "text/plain": [
       "                   2\n",
       "0.130690966048658⋅a "
      ]
     },
     "execution_count": 27,
     "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": "markdown",
   "metadata": {},
   "source": [
    "## Example 10.2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [],
   "source": [
    "h = Symbol('hbar', real=True, positive=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": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 x$"
      ],
      "text/plain": [
       "2⋅x"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(x**2, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{0.5 i \\pi \\hbar \\sin{\\left(\\frac{\\pi x}{2 a} \\right)}}{a^{2}}$"
      ],
      "text/plain": [
       "             ⎛π⋅x⎞\n",
       "0.5⋅ⅈ⋅π⋅h̅⋅sin⎜───⎟\n",
       "             ⎝2⋅a⎠\n",
       "──────────────────\n",
       "         2        \n",
       "        a         "
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Solution\n",
    "-1j*h*diff( 1/a*cos((pi*x)/(2*a)) ,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACsAAAAuCAYAAACxkOBzAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADxklEQVRYCe2Z7W3bMBCG7cIDBO0G7gZps0GzQT4maLJBg/6y/xXOBkkmaNMNkg3ysUGyQVtv4L4PTRKUREmkpBQBqgNoksfj8dXLo2yep5vNZtJFlsvljuZ9tXPntv4s/fql/M26OLZzVgJ26uarfaH2g8p7p8usW/29yXQYmp8I4KdAsVJ7Lt1uoMtptvrrwyys3rehEXjC5Vrlo8pv9euYb/dHzA5RFovFSuWpzpfGHlSu68bL+pi/Psx6Uu3WH0jxwSttQ2Po91QID5glXJw8qX/pOq6u8zflifqIHPMm4HAdqh19E9jFzeFT+7lpvSZ/vZi1jjnF+wBQDXtr1WVAHMSYnmleNI8Hr/XXGax1DKNnFiSLckjOaJSEhzGHUbYAB9S92o/OLsWfDwMZf9FEXvKc3jq5lJ15t6r+I6OKrfTT8mTpiDVikzD5rgLzhAUsmphN8WfAytAF/Y0cAIADgVNeN/RvVSay80zQTxHNITQAdqu2CRfmqc2u8F72OvRNMpMxzu5U/8RQNQxfqH5WYctYJBskvqzgAzncVv4TIsqx7QdjDcACJASzL925NYbhyqsl5qhBB3M8MCEQCiTBbrIUvm7l0JzmYPZuZJFgOKkJg4SXF/l0bJvw8gMtjQJY2XJ4iFUnnNrOIlDM9zEfOCIkCDMTBqqxaRUP1k440YzC00rvWGh1FjEwc+UjDDPMjlRMCGiMNZPEg5U1kx41OYwt2smnNbKiideIPmR7p7RmxHyrmgUjOP4W9GkWWC6NpXQJg9gh4gAfCyQl9iUS9e2/FKKjr0wZhsErg1aFM4KtcjKMZmR2GB6rXkZmq5wMo5nqFtnvEjYMjiQv45dCEk0djMYD1oG0pCn/H7P6mcf1J/ZTMImxVKOhmCVL+DZ10a52vcGKUa7u/0R6gWX7hZKrT3gVejHgvcAKFdeSvnmF5IcL72DJkzC02996qGQ3lzn3LNj/pULm50B6l0hRN006MWsBpKQwuYq7BBzZRgDygCTkuOFmSSewWuFUizVuvwXDWwKQYU6LNgmO7DjPBqtFyC+0br9syEzC3g+VUGDbJAFDZUo7C6yAzuWUpETIVN06ZF0KCTnNAzw+CrmvOgdlfe4BY6E9Lcr2hsIrjFwreh6EZAnAYmmjiew6JU+ywNpFKgtJTxYcFk0OVjVAkbtt5T8ZNw8gG/7FIV2VskvGQVYY+CWrDcA5gBMBWKsPKHbCiHTEOn3z34JqdigZKE563RS0GAeN/CthgMD6jfTnKoC/UoHddyqkUnkIDh667Iz6X2gi2AaMF+vVAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\pi^{2} \\hbar^{2}}{4 a}$"
      ],
      "text/plain": [
       " 2  2\n",
       "π ⋅h̅ \n",
       "─────\n",
       " 4⋅a "
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Solution\n",
    "B1 = (pi*h/(2*a))**2 * (cos((pi*x)/(2*a)))**2\n",
    "B1.integrate( (x,-a,a), conds='none' )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 10.3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = Symbol('p', real=True) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Solution\n",
    "A = integrate(1/sqrt(2*pi*a*h)*exp(-I*p*x/h)*cos((pi*x)/(2*a)),(x,-a,a), conds='none')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{\\sqrt{2} \\sqrt{\\pi} a \\hbar^{2}}{\\sqrt{a \\hbar} \\left(4 a^{2} p^{2} e^{\\frac{i a p}{\\hbar}} - \\pi^{2} \\hbar^{2} e^{\\frac{i a p}{\\hbar}}\\right)} - \\frac{\\sqrt{2} \\sqrt{\\pi} a \\hbar^{2}}{\\sqrt{a \\hbar} \\left(4 a^{2} p^{2} e^{- \\frac{i a p}{\\hbar}} - \\pi^{2} \\hbar^{2} e^{- \\frac{i a p}{\\hbar}}\\right)}$"
      ],
      "text/plain": [
       "                          2                                           2       \n",
       "                √2⋅√π⋅a⋅h̅                                  √2⋅√π⋅a⋅h̅        \n",
       "- ─────────────────────────────────────── - ──────────────────────────────────\n",
       "          ⎛         ⅈ⋅a⋅p          ⅈ⋅a⋅p⎞           ⎛         -ⅈ⋅a⋅p          \n",
       "          ⎜         ─────          ─────⎟           ⎜         ───────         \n",
       "    _____ ⎜   2  2    h̅     2  2    h̅ ⎟     _____ ⎜   2  2     h̅      2  2 \n",
       "  ╲╱ a⋅h̅⋅⎝4⋅a ⋅p ⋅ℯ      - π ⋅h̅ ⋅ℯ     ⎠   ╲╱ a⋅h̅⋅⎝4⋅a ⋅p ⋅ℯ        - π ⋅h̅\n",
       "\n",
       "         \n",
       "         \n",
       "─────────\n",
       " -ⅈ⋅a⋅p ⎞\n",
       " ───────⎟\n",
       "    h̅  ⎟\n",
       " ⋅ℯ       ⎠"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Solution\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALQAAAA7CAYAAADSK6A/AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAOdUlEQVR4Ae2d7ZUctRKGx3scgFlHAGRg7AhYMjB2BDYZmLO/7H97IAMgAttkcCGCNWRgbgR3cQa+76NVaaWWulvd0zPTPdN1jlZfpSqVqlT6GM3svc+fP2/2AW/evHkkPs8VXtXyU5t7tbgr3uFGQHp6Ku6fFP+xy16I/k8KP3bxuN9VOXHdT6L3Y1+HJuZ5cHKS9z/qxMXYjqj9rCe1+ocxf6f4h7EyDmj3lvFU+K6tzb19eGh1AO/M7GrtSFsHtyn3fP8aSkPtJjEiz/9S8fdD+7AEfMn1QP38U/E3bf31ODgz9AD+Q4VfVP6P4o3il4po/1HhkwKAnVyp7m+Xi/54/AeKf46KQ3JfHvpSHBFq3wDPb0oDs6eOuFWpxEt9Qrmtii61mWHZe/Xpl7Z+eRkx5O/RgQKOjfxVow10WMmcroT3QWnKvlZIQHW/Knz0sU2AgBMMWghfqdT2J4+VviGv8myWhNYVCU/3K8XJ/mpX/KxLos/gbRRv1X+jNzTu4q86jLlG0UPZ7g1fMmAvjxV3rbpM2D+EE+uAvbYzRE/jnXDw0r9HeNRDvw2YRL8pZCvfGS08YbYEP/jAEkAn/lJ+9P4P2gImiU0UV7Bjfo6H/hxqVYj5J3JbheJORUd4c04i2689HcRQ8bQG2FJwbLKDfxQwXibFtSEpxhk5o4/K4iR8n6rtg7iQtDNoxQzwCwoMhEyHIRp3yKqrYs+QWRyE8A13ws86JX7M7mxVsPpdxxH/2DPFbDsVHSPOOI0MbBOK4HVPndsreyQMl0Mde+DYGNkRxDbyXHk8dxHUFrtkbJ81EcygmTn/NpiACxOYYyBjAMNt7pegsyt+1kcmI+FQgNxF/tEY1yj6UP3v5CsZ3HZOSOx1iyAcjI7gwLdB78j9zNdvFGNbGLfD9XnoF8dP5QbwLm85VIHhmvu3BnEMw0GgjtHmQvHvhYaT8zMenm9pVTCUncZeIa2rg+pRnFMeHVEe5WWKpm7G4PrrZenqJgbHOQxvjuFipM09N7Twtmwh+IwCHA6HYYyULwG3Inj2BO6TU+PM0j2Wm4mqb1s6E2KNTOsedkf8jH1xVRBPBos+MdHagBP0tvepKKS0KsU8TdEcDG8UTNEoqRfUR2RAzljp1yoPziPCMZrcGLxXeVjalbbJBB0bl+cqb72GEx7wRCHm7QqbfzyvwK9Z7/Pw4uBYvIZraUMxNum2Lmob+uIMutRISAhrsypBUR2X6ZxWuzrLjMuuXRJCUaaLX4TWmRQNlMKqkBil8igfwJDAQSFvFZjh5J0cwhszcdX8Fjx/VoeEv9VbrHr4dY2doWax2qITJsILpZ0BK8bLYayssnY9xtmHDzzC1kZp9q+PFH5WQO7sswGVd/ZdbQDaMhGnAPrOhB4Kxp/xCHo766DCgOCxkpmjPJ6OKxMzkoyEx+FqZQgU+Q0hINxsVVBfmJh4L5Y+jIgB4GKfQXDGTNrnVbQVwL/PO2/FQI0ZJww3eGPlkQP4dBs5HK7BgjH7cgwHI2ZMmMxMPmvrUTYxXStrxucqMF7Nuuq8eGND6IOJ1+xHHx3jT18CFD20iGOMLAPJbFUe5lyZPFR4pTyzPcwOlRlw/TfEOxf5GTFi0Ru1Kvj+xX1k8GyS4qn7rp7ibrSmRROFsCp1ehvVVz2eEV72aaXngTFa/11/VI4RmremHj1du8roj/CYuJSwrWCC3yjNZQATnRsLJkFn/4UDICttW0F0quT0BNhjv1SbjJ7KsnHwSEX+mUGLAMQ3ihNj9mVuxqsOL2R7UpbxAL49BloFHn+jOONnBFRnvOBf3N95nE6+wkHZNrMhz4SM85QloHqUt+nDEwrj1rpqQQMQnTYF3SJ0/8VQgf/dRsW/htMlF+MAMJaXChcK9B3vXXOOgPa5QitsKWcr3ajC+CeGfRYhMNh4wa8VB+NSmhO7DZJDVx6B7HI7qVM5M7zK64lOLz/PG3oEDNCUoWwCrAqJ50pqbzPI9TYqb/Y9qnLGB6//KvQaqnDgXyV3wmRYxrYQXauf4biJ2EKeLYuTXTGeGsP+QoHxwVO2jXELuYMUm3zYYoAzS3khniCglfkYo0tmgS83JQd8tQWXZSth4vGTqJaf8Ow68coTwKMkIBy8Y593ZgDASw5jaot3ykDlyILSPyi4VStD8gXCZQXp5N9sS58VeJNA2yoQLuP6t8LjUgPVs+WhHrzm9dhGdSYrfcVog1yq45DPhGTrUqSvcgMmDeM5GMRjsNwtTM59eWKbzqDFBMVx2OAahAOTBcrwPAxQAipDKIyDDppwGJsZXoIfZ4Q/hh99YMBRGu1jqFkVUB57yFgW0pniISw8JibKdRNX6aB86htQszo0mjiDQCk4Ec4jBDO4Jm6cZ4vHqpn0R3n6iTED3yrw4UXT04LDDYfhXSptuqMdQD6Z9K40/cP+vM/o0xZ3OeiPkfuOwm0KG2ASosMA930KwwUhGSRfZ8L7bBIxQCgBo8bAPzQZJNh3mW340UcMmOVxI3540ppVAcNtTrY+xUGfwzGyFSeN6uiPO5ApHgJusqh9cgbpI0BfFL4U3m+K2SqYQnFC9HOjmIlLHdsJq0e/5E1mynEQGP4DxcBDBfbRjo4rKf/BJnB+BKNfxsxLR8mdk3H7f5MlVG/9HloCcSfKYDEIPBPsG4zAfExC9DmNM4m+YDA9/29Jj6FX00a0MVqWaW5IkkFUng8uaj7ZSlh5mhgzAXkw1C7nkbQ/dEZ9/Vd9QN/JePT1ayq5RQe7YxIn55azvg5U1OP1mOEoZKfG7PvitgBKsyrgnWtXBd98eBQNWjgvQMXzr1kdSkxtq/NMlRgF25bq/XSJ4J7L3omfyTCEtbUZLbfGCXtjO0UfEtjaoEWcJQRDTpSdcJkwI34oH0/Gfp1gBq7kTgFPcCH+rEYGVWcGQ27EeGU+ueOajNWF0HV70Wh+8CwrVmmL2texKeRmMhQdydYGTe+lEK769uGdbbD2vSrA1yaOm7iSF8WMWh3UlknB/jO+ZsTjsIwuAtR3nArnC1bJKphQbnRQdKCTGHSVNBMiaWD2uirQdfFkwrI6mFfCwM3IlRwETIaw9xRtllDK3ikdrwCDiB4A+YV4skrVwtZya3zcecbrI+O7SINGCgm071UBts6AxZtbmm3ODOwj48fxbj8pumw73O2N4tmD7++VYrYfNbCV3OLDZOcgGq9sCd/FGnQixZ4yGki8Kp6aZba45FV2BcWwyhhAlxsbDoW1xmFtDxqrz8jBWQDv2wfbys1k77zm3Prark+CY6uX4ljyuL7rHNhjk3sp8qwGvRRNrf2sGoF1y1E1TCvSUkbg3uvXr4e8W12KXGs/T3QE1i3HiSr+WMVetxzHqtkTlev+icpdJbZuMrie46qJV2jEPIYJH4goXwVT0aliduJIJ2PQMip+1ownlUOAJ5p8k2OjmE/zeGE25itUU9GhKyt0jMBJbDlkjHxg0Xzs3jEsoSqeAHhoPskbA1PRGcP7pNocvUHLmPkEa9TDKbWN2/Gx96gPU6aic1KWOVLYo95yyJDYJuCZk0fgNla+HkPllRu47JXDNz+UdiA8Po7m2x7ZA3yV8ckhHpiH/ubBebPAG4cEv4uO8FeYYASO3UO/lBEVH7KoHAOOv/XAYY+tyU08rsLD4J0x+3RcbWkeK4HHc1ImD89bKQtQSSfgzymhvvPUla9mEfiFJgLjNzs4Wg+tAeeGIn4A1Bx8DJD3vLEX5YGQedmN0hg8Hp4vsCpyW5fwKEll7Kv51gRemgfnRgsa1Dnoo2N4M44x5PAKUGlWLMZmdl9IOEqD1oBjTOeK4z1w017cI6Oo8ELp5EpO7ePDXIR6mzT6ipvPIpkE8cTopJMRnl8BKx1e2cYHZ0BZ2y9nHUyCozRojSbLIe+lGXiA/Mbnr5U0xcQGj1GGpVS4wSBp2wOPVR88t9LPFfDcxwJ45w9LEOYoDVrGyNJvyz+GjMfGI3NQc4ZqMUpSGo+Kh2YC8LX+4iFSdRkIF9pMGKNLHno79criS3/5fY7qvgp/FBR4YOCz/Jb62SgJF9RIysCQ7dE8H3Cwtwa4guOwRz1GiIfFSw8FDIvJww/gcKiEzuCfNVCbocAkIuwVJCOTlTHc6YQdK9RReuh4MKQAPFjmxVTOtsO2HnGToWkUy+GyeJsylNic8SUjE59VbB8TdtRQHL1BjxqVYY3w0PH+eVjrA2DLMFlJLhW6PHzyK6RmzIrdKqbYHXwVx+eQA0iTslwNOh2PQTkpE2+F13K/qKS820cPIrJnZN9nuLLlwqCfKLxV4GBL3q1awmueQezDJQwZYB89u4m8GrTTzbg/UjoKnZ1S26RRfzHGa8Xufl4xntp9Mqo0K03zXt5I2SepxAHUJtxNh8IDJ2b3wF+DhJf4U4G4Fvhqe+xRFvMtHPW78/We6vGMGFsTzn3BTbNCeX6ssffdiXD4trZtIfhk84Xynwr0llPEP69fw/LGQF+de6rwaqzu1PaRwntrr/RHSy85XrccA32PPBhXVuybH/p41KP/gWx3gc52gb2zATItHlaDbqhQBsu9NPtiDLV0Fbf4x/qSi+2cyRlGQOUXClNcZQaa+06sBp2POMpmf/pEyuXQBLAnNUXHHyjg1Za458SY2/6bgcmJ3IuD1aBzlXEDwOuy4qFK5fG9K9d2Rbyc7KxKOAheNXq0aEM2WVaDtpG4iy+U5PoKT026+GZB9XYvG25X7kjMO6W+Zx/xq2yJEzMb6LOsZC0wZT/TUOC1sl/Wl/LxzH2P/nc9kmx1lrjd2em4rB46H168Mq/y3PsPxRhNeMiufOej/5zcbkrUj6PYIkw9OqtBRyMqI+GQV/pl/fB1KuHEh8Ko9ZqcwwisW45UC3jn4PlkvLaPXtov66dSnVBuNehU2eyfF//L+qlIp5VbDTrVN1uO+Iu1eOtF/rJ+Ktbp5P4PDeHzQg6+P38AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\frac{2 \\sqrt{2} \\sqrt{\\pi} \\sqrt{\\frac{a}{\\hbar}} \\cos{\\left(\\frac{a p}{\\hbar} \\right)}}{- \\frac{4 a^{2} p^{2}}{\\hbar^{2}} + \\pi^{2}}$"
      ],
      "text/plain": [
       "            ___         \n",
       "           ╱ a     ⎛a⋅p⎞\n",
       "2⋅√2⋅√π⋅  ╱  ─ ⋅cos⎜───⎟\n",
       "        ╲╱   h̅    ⎝ h̅⎠\n",
       "────────────────────────\n",
       "          2  2          \n",
       "       4⋅a ⋅p     2     \n",
       "     - ─────── + π      \n",
       "           2            \n",
       "         h̅             "
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi_p = sqrt(2*a*pi/h) * 2/(pi**2 - (2*p*a/h)**2) * cos(p*a/h)\n",
    "psi_p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi_p == sqrt(2*a*pi/h)*2/(pi**2 - (2*p*a/h)**2) * cos(p*a/h)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Which agrees with the book.\n",
    "\n",
    "This is about as far as we can go in sympy. Unfortunately, many other momentum integrals choke. There are a few hints to get through the rest here:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Problem 10.3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\sqrt{2} a^{\\frac{3}{2}}}{\\sqrt{\\pi}}$"
      ],
      "text/plain": [
       "    3/2\n",
       "√2⋅a   \n",
       "───────\n",
       "   √π  "
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, y, z = symbols('x y z', real=True)\n",
    "a, c = symbols('a c', nonzero=True, real=True, positive=True)\n",
    "\n",
    "psi = c*1/(a**2 + x**2)  # define the wavefunction with c constant\n",
    "int1 = integrate(psi*psi,(x,-oo,oo), conds='none')  # integrate psi^2\n",
    "solutions = solve(int1 - 1,c)  # solve for c, this returns a list of solutions\n",
    "c2 = simplify(solutions[0])  # simplify the solution for c:\n",
    "c2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\sqrt{2} a^{\\frac{3}{2}}}{\\sqrt{\\pi} \\left(a^{2} + x^{2}\\right)}$"
      ],
      "text/plain": [
       "      3/2   \n",
       "  √2⋅a      \n",
       "────────────\n",
       "   ⎛ 2    2⎞\n",
       "√π⋅⎝a  + x ⎠"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi2 = c2/c*psi\n",
    "psi2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0UlEQVQoFXWR0Q2CQBBEhVgAwQ60BLUCsQO0A+3BP/4MJWgLWgIdGCmBDiB2gG+OW0PIuckyO3Ozu5cj6vt+ZlEURenrFlyRJVojLTIjwht+BZ86ABNA2p66ib14BhMzee0Dqukm7ozggawlTOIFzzTdjBlCNzGJaqoii/1dBvr/m2pi6s+te2y3Lb/V48NQvdBE6woZbFurO9rKJOA0bXhHDBW5DBhtYmXP88C0CRjXaLW2OiPFHaEDczNTa+2RPEmb6+ND3RcMdoUtfAd3f+wLrOY/XeyA6s8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(psi2 * x * psi2,(x,-oo,oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAATCAYAAAByUDbMAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABVElEQVQ4EaWU203DQBBFnYgCQHRgOiCiA+gg0AFJCRZf9q/pAFEBkA5IBzw6IB0guQNzjrVrHItAbEYazezNzN3rmVUmdV0nQ6woikPqb0JPGuI1eHUwhCjUljQuYx/5HfkbfjKN4IC4gOC8U1+Sp2CnY8hU9doh+06d2X88z/MS/5BjjLJWiZ/GYY7PBEeTQZTS77xm5JVkY7aZRCLihSREFVaTEe9MRT6HDI/mUrIxynxTPlxja6hbbikD8FZvdAaf+Aqfg98S/7SWjAYf4hPuQDd2Ep8J4kfkzZDFd1mzTQqVLVFG3hCFBvPNPkTWN2REVyzhI941Vfmpe1kku6J63VUQ1DpDP3Uvm4YmVb33Orwg4fd1D995jMoseOlVXXJuLoDQjaryV1OZW7KpLQZbhHP8dzgD6y7mR9LmaVDoZ97jqjvGH3AvcTFizrM/BuBt+wLykqm6/+nt1QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle a^{2}$"
      ],
      "text/plain": [
       " 2\n",
       "a "
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(psi2 * x**2 * psi2,(x,-oo,oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So $\\Delta x^2 = a^2 - 0^2$ therefore $\\Delta x = a$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 10.17:\n",
    "Now find the momentum representation of the state from 10.3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{\\sqrt{a} \\left(2 \\left(- i \\operatorname{Shi}{\\left(\\frac{a p}{\\hbar} \\right)} - \\frac{\\pi}{2}\\right) \\cosh{\\left(\\frac{a p}{\\hbar} \\right)} + 2 i \\left(\\operatorname{Chi}\\left(\\frac{a p}{\\hbar}\\right) - \\frac{i \\pi}{2}\\right) \\sinh{\\left(\\frac{a p}{\\hbar} \\right)}\\right)}{2 \\pi \\sqrt{\\hbar}} - \\frac{\\sqrt{a} \\left(2 \\left(i \\operatorname{Shi}{\\left(\\frac{a p}{\\hbar} \\right)} - \\frac{\\pi}{2}\\right) \\cosh{\\left(\\frac{a p}{\\hbar} \\right)} - 2 i \\left(\\operatorname{Chi}\\left(\\frac{a p}{\\hbar}\\right) + \\frac{i \\pi}{2}\\right) \\sinh{\\left(\\frac{a p}{\\hbar} \\right)}\\right)}{2 \\pi \\sqrt{\\hbar}}$"
      ],
      "text/plain": [
       "     ⎛  ⎛       ⎛a⋅p⎞   π⎞     ⎛a⋅p⎞       ⎛   ⎛a⋅p⎞   ⅈ⋅π⎞     ⎛a⋅p⎞⎞      ⎛ \n",
       "  √a⋅⎜2⋅⎜- ⅈ⋅Shi⎜───⎟ - ─⎟⋅cosh⎜───⎟ + 2⋅ⅈ⋅⎜Chi⎜───⎟ - ───⎟⋅sinh⎜───⎟⎟   √a⋅⎜2\n",
       "     ⎝  ⎝       ⎝ h̅⎠   2⎠     ⎝ h̅⎠       ⎝   ⎝ h̅⎠    2 ⎠     ⎝ h̅⎠⎠      ⎝ \n",
       "- ──────────────────────────────────────────────────────────────────── - ─────\n",
       "                                2⋅π⋅√h̅                                       \n",
       "\n",
       " ⎛     ⎛a⋅p⎞   π⎞     ⎛a⋅p⎞       ⎛   ⎛a⋅p⎞   ⅈ⋅π⎞     ⎛a⋅p⎞⎞\n",
       "⋅⎜ⅈ⋅Shi⎜───⎟ - ─⎟⋅cosh⎜───⎟ - 2⋅ⅈ⋅⎜Chi⎜───⎟ + ───⎟⋅sinh⎜───⎟⎟\n",
       " ⎝     ⎝ h̅⎠   2⎠     ⎝ h̅⎠       ⎝   ⎝ h̅⎠    2 ⎠     ⎝ h̅⎠⎠\n",
       "─────────────────────────────────────────────────────────────\n",
       "                        2⋅π⋅√h̅                              "
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = symbols('p', nonzero=True, real=True, positive=True)\n",
    "B = integrate(sqrt(1/(2*pi*h))*exp(-I*p*x/h)*psi2,(x,-oo,oo))\n",
    "B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\sqrt{a} e^{- \\frac{a p}{\\hbar}}}{\\sqrt{\\hbar}}$"
      ],
      "text/plain": [
       "    -a⋅p \n",
       "    ─────\n",
       "      h̅ \n",
       "√a⋅ℯ     \n",
       "─────────\n",
       "   √h̅   "
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B.simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This agrees with the book after we notice that we had to force $p$ to be positive in order to get the integral to converge. The book has $|p|$ in the argument of the exponent to reflect this constraint."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 10.13"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{\\sqrt{2} \\tanh{\\left(\\frac{x}{a} \\right)} \\operatorname{sech}{\\left(\\frac{x}{a} \\right)}}{2 a^{\\frac{3}{2}}}$"
      ],
      "text/plain": [
       "        ⎛x⎞     ⎛x⎞ \n",
       "-√2⋅tanh⎜─⎟⋅sech⎜─⎟ \n",
       "        ⎝a⎠     ⎝a⎠ \n",
       "────────────────────\n",
       "          3/2       \n",
       "       2⋅a          "
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi = 1/sqrt(2*a)*sech(x/a)\n",
    "dpsi = diff(psi,x)\n",
    "dpsi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\sqrt{2} \\left(\\frac{1}{\\cosh{\\left(\\frac{x}{a} \\right)}} - \\frac{2}{\\cosh^{3}{\\left(\\frac{x}{a} \\right)}}\\right)}{2 a^{\\frac{5}{2}}}$"
      ],
      "text/plain": [
       "   ⎛   1         2    ⎞\n",
       "√2⋅⎜─────── - ────────⎟\n",
       "   ⎜    ⎛x⎞       3⎛x⎞⎟\n",
       "   ⎜cosh⎜─⎟   cosh ⎜─⎟⎟\n",
       "   ⎝    ⎝a⎠        ⎝a⎠⎠\n",
       "───────────────────────\n",
       "            5/2        \n",
       "         2⋅a           "
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ddpsi = diff(dpsi,x)\n",
    "ddpsi = ddpsi.simplify()\n",
    "ddpsi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\int\\limits_{-\\infty}^{\\infty} - \\frac{2 \\operatorname{sech}{\\left(\\frac{x}{a} \\right)}}{\\cosh^{3}{\\left(\\frac{x}{a} \\right)}}\\, dx + \\int\\limits_{-\\infty}^{\\infty} \\frac{\\operatorname{sech}{\\left(\\frac{x}{a} \\right)}}{\\cosh{\\left(\\frac{x}{a} \\right)}}\\, dx}{2 a^{3}}$"
      ],
      "text/plain": [
       "∞                   ∞            \n",
       "⌠                   ⌠            \n",
       "⎮         ⎛x⎞       ⎮      ⎛x⎞   \n",
       "⎮  -2⋅sech⎜─⎟       ⎮  sech⎜─⎟   \n",
       "⎮         ⎝a⎠       ⎮      ⎝a⎠   \n",
       "⎮  ─────────── dx + ⎮  ─────── dx\n",
       "⎮        3⎛x⎞       ⎮      ⎛x⎞   \n",
       "⎮    cosh ⎜─⎟       ⎮  cosh⎜─⎟   \n",
       "⎮         ⎝a⎠       ⎮      ⎝a⎠   \n",
       "⌡                   ⌡            \n",
       "-∞                  -∞           \n",
       "─────────────────────────────────\n",
       "                  3              \n",
       "               2⋅a               "
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expect_p = integrate(psi*ddpsi,(x,-oo,oo))\n",
    "expect_p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "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": "3.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}