{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Problem 2.15 code, Quantum Mechanics PHY 341, UMass Dartmouth, J Wang\n",
    "# tidier if assuming a.u., m=hbar=omega=1\n",
    "from sympy import *\n",
    "init_printing()  # pretty printing\n",
    "m, hbar, omega = symbols('m hbar omega', positive=True)\n",
    "x = symbols('x')\n",
    "c = m*omega/hbar\n",
    "E = hbar*omega/2\n",
    "a = sqrt(2*E/m)/omega"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{m^{0.25} \\omega^{0.25}}{\\pi^{0.25} \\hbar^{0.25}} e^{- \\frac{m \\omega x^{2}}{2 \\hbar}}$$"
      ],
      "text/plain": [
       "                                  2 \n",
       "                            -m⋅ω⋅x  \n",
       "                            ────────\n",
       " -0.25   -0.25  0.25  0.25    2⋅h̅  \n",
       "π     ⋅h̅     ⋅m    ⋅ω    ⋅ℯ        "
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi0 = exp(-c*x*x/2)*(c/pi)**(1/4.)\n",
    "psi0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANgAAAArCAMAAAAt+LZoAAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{1}{\\pi^{0.5}} \\left(- \\sqrt{\\pi} \\operatorname{erf}{\\left (1 \\right )} + \\sqrt{\\pi}\\right)$$"
      ],
      "text/plain": [
       " -0.5                  \n",
       "π    ⋅(-√π⋅erf(1) + √π)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 2*integrate(psi0**2,(x,a,oo))\n",
    "p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKoAAAAQCAMAAACx+RSjAAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$0.157299207050285$$"
      ],
      "text/plain": [
       "0.157299207050285"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# numerical value\n",
    "N(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# arbitrary n\n",
    "from sympy.physics.qho_1d import E_n, psi_n\n",
    "n = symbols('n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGoAAAAyCAMAAACDIr4/AAAAOVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACXHtMAAAAEnRSTlMAPl5+\nHt+fDm7PTr8uj++vh8fgRUyJAAADL0lEQVRYCZ1Y68KzIAj2lLbZ7P28/4v9ECdhaw31x6ZJPJwE\nTCnheD2FhD3Z9urXgpX3AqIrkuEXzX7F5uZZINF2c0P2uaWX8Pnw+xNn/ZLbdlh0mwr+QxqhVipo\nZwlK6TQg507WEMhVSRiU8nLr6+zEEI2QQ7ksNkocV0pxKOVjk+DH/4xSPZRYrSiViYvcaaWELFwe\nOxgVsIcyMm+veSBWSbEeKuSVdm4mjxn79b5SKj5uENpWyLZNR/57rQBZYJqn/FBwUU5QOmNh0OZO\n7pdEHg5S5jGm/IgsnEIuxcTZZTmTsnVMbDE/TdXjy102yFNR8SFTLOnX+OyttUxbThfynRxIuUoC\n2aMfbs+X/h2A9s7VTWyL0eUX5TTGIgXk1ihUB1WooBbRZp0MQC1+NfoBwRhaXVlraBZOK4v1VSf3\nXPV+0kIEpUu+cDmCsBa875u8WyJuVXEU3xmVI+irjwKLz4VQwLK6CkpKOPqooxNjUMbVzGumocBV\nMCDgTVHKoRFt0w8yClVQ16SiY2hjGcuCf2QIlY+Bapcf9Dmeqg28g0oZjO3jNQYFFkaQRNo7XYb3\n+EfSEXs+KVDVKK+kwl/ZsiVXMVN2UCjVM2+qYyv2VXVVsd+/EsqpQO0HKw5VpSpnozvvYqhqFLCf\n3Xe9JpvMcycL9cFepYovtdIBREMc5i7L61GC/YHOiat1W8qgk89dKuJH+IWust5gQSCWIq2KdVyV\nsCaCMt+OVAErDhWqXbfDvAh3ARUgNk+qH5FMMp4mgnTby1beDxYiJyYuc023J+an5UwRwbQTMp0/\nYIlF5MT6vJwpjRl7sQingsa7NNL6ajJT8He8I0Xmnlrwr/izZ5NtDHBYWKvwbmMY34vpZHNWYpdV\nZ1FzpuZaTvU+sG/hRS0nFMcuN1wofvmou24LG+m564Fh1ms18VKk7qHwxtK9Y0pCdpRWpCwmrnIb\npn5KTeKrnBq+oG4RGktr6fYhvqBC2JIlOit9XaRa8VtfLFcKSuXPDvcratkY+JgAtZl6mVue15tD\nn0jU4IefDnLsww8cjNb4dlxEi8HPWWDvWXdNvHg0vCJdGhH79PgfzEEVGq+C2rsAAAAASUVORK5C\nYII=\n",
      "text/latex": [
       "$$\\hbar \\omega \\left(n + \\frac{1}{2}\\right)$$"
      ],
      "text/plain": [
       "ℏ⋅ω⋅(n + 1/2)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E_n(n,omega)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAVAAAABECAMAAADgBp3SAAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{\\sqrt[4]{m} \\sqrt[4]{\\omega} \\sqrt{\\frac{2^{- n}}{n!}}}{\\sqrt[4]{\\hbar} \\sqrt[4]{\\pi}} e^{- \\frac{m \\omega x^{2}}{2 \\hbar}} H_{n}\\left(\\frac{\\sqrt{m} x}{\\sqrt{\\hbar}} \\sqrt{\\omega}\\right)$$"
      ],
      "text/plain": [
       "                              2                     \n",
       "                 _____  -m⋅ω⋅x                      \n",
       "                ╱  -n   ────────                    \n",
       "4 ___ 4 ___    ╱  2       2⋅ℏ           ⎛   √m⋅√ω⋅x⎞\n",
       "╲╱ m ⋅╲╱ ω ⋅  ╱   ─── ⋅ℯ        ⋅hermite⎜n, ───────⎟\n",
       "            ╲╱     n!                   ⎝      √ℏ  ⎠\n",
       "────────────────────────────────────────────────────\n",
       "                    4 ___ 4 ___                     \n",
       "                    ╲╱ ℏ ⋅╲╱ π                      "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi_n(n,x,m,omega)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKQAAAA4CAMAAABEz6NLAAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{\\sqrt{2} m^{\\frac{3}{4}} \\omega^{\\frac{3}{4}} x}{\\hbar^{\\frac{3}{4}} \\sqrt[4]{\\pi}} e^{- \\frac{m \\omega x^{2}}{2 \\hbar}}$$"
      ],
      "text/plain": [
       "                      2 \n",
       "                -m⋅ω⋅x  \n",
       "                ────────\n",
       "    3/4  3/4      2⋅ℏ   \n",
       "√2⋅m   ⋅ω   ⋅x⋅ℯ        \n",
       "────────────────────────\n",
       "        3/4 4 ___       \n",
       "       ℏ   ⋅╲╱ π        "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi_n(1,x,m,omega)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANMAAAAwCAMAAAC8Cb/HAAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{1}{\\sqrt{\\pi}} \\left(- \\sqrt{\\pi} \\operatorname{erf}{\\left (1 \\right )} + \\sqrt{\\pi}\\right)$$"
      ],
      "text/plain": [
       "-√π⋅erf(1) + √π\n",
       "───────────────\n",
       "       √π      "
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n=0\n",
    "a = sqrt(2*E_n(n,omega)/m)/omega\n",
    "p = 2*integrate(psi_n(n,x,m,omega)**2,(x,a,oo))\n",
    "p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKoAAAAQCAMAAACx+RSjAAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$0.157299207050285$$"
      ],
      "text/plain": [
       "0.157299207050285"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N(p)"
   ]
  },
  {
   "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",
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 },
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