Lecture4-ex.ipynb

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     "end_time": "2019-01-21T19:14:02.202856Z",
     "start_time": "2019-01-21T19:13:58.005438Z"
    }
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T19:14:16.497065Z",
     "start_time": "2019-01-21T19:14:16.490070Z"
    }
   },
   "outputs": [],
   "source": [
    "# Example 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T19:18:41.525066Z",
     "start_time": "2019-01-21T19:18:41.515072Z"
    }
   },
   "outputs": [],
   "source": [
    "hbar, m, L, g = symbols(\"hbar, m, L, g\", positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T19:15:10.019148Z",
     "start_time": "2019-01-21T19:15:10.011153Z"
    }
   },
   "outputs": [],
   "source": [
    "n, ell = symbols('n, ell', integer=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T19:15:27.616326Z",
     "start_time": "2019-01-21T19:15:27.608330Z"
    }
   },
   "outputs": [],
   "source": [
    "x = symbols('x', real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T19:16:34.590134Z",
     "start_time": "2019-01-21T19:16:34.580141Z"
    }
   },
   "outputs": [],
   "source": [
    "def phi0(x,n):\n",
    "    return sqrt(2/L)*sin(n*pi*x/L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T19:17:40.346691Z",
     "start_time": "2019-01-21T19:17:37.931177Z"
    }
   },
   "outputs": [
    {
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      "text/latex": [
       "$$\\frac{\\sqrt{2} \\sin{\\left (\\frac{\\pi n x}{L} \\right )}}{\\sqrt{L}}$$"
      ],
      "text/plain": [
       "      ⎛π⋅n⋅x⎞\n",
       "√2⋅sin⎜─────⎟\n",
       "      ⎝  L  ⎠\n",
       "─────────────\n",
       "      √L     "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phi0(x,n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T19:17:18.079388Z",
     "start_time": "2019-01-21T19:17:18.072393Z"
    }
   },
   "outputs": [],
   "source": [
    "def E0(n):\n",
    "    return hbar**2*n**2*pi**2/(2*m*L**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T19:17:25.608759Z",
     "start_time": "2019-01-21T19:17:23.994749Z"
    }
   },
   "outputs": [
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      "text/latex": [
       "$$\\frac{\\pi^{2} \\hbar^{2} n^{2}}{2 L^{2} m}$$"
      ],
      "text/plain": [
       " 2   2  2\n",
       "π ⋅h̅ ⋅n \n",
       "─────────\n",
       "     2   \n",
       "  2⋅L ⋅m "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E0(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T19:19:09.087115Z",
     "start_time": "2019-01-21T19:19:09.076121Z"
    }
   },
   "outputs": [],
   "source": [
    "#first order energy corrections\n",
    "def E1(n):\n",
    "    return integrate(phi0(x,n)*(m*g*x)*phi0(x,n),(x,0,L))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T19:19:17.066207Z",
     "start_time": "2019-01-21T19:19:14.721647Z"
    }
   },
   "outputs": [
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      "text/latex": [
       "$$\\begin{cases} \\frac{L g m}{2} & \\text{for}\\: n > -\\infty \\wedge n < \\infty \\wedge n \\neq 0 \\\\0 & \\text{otherwise} \\end{cases}$$"
      ],
      "text/plain": [
       "⎧L⋅g⋅m                            \n",
       "⎪─────  for n > -∞ ∧ n < ∞ ∧ n ≠ 0\n",
       "⎨  2                              \n",
       "⎪                                 \n",
       "⎩  0            otherwise         "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E1(n) # the top line is the one that counts: n neq 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T19:19:48.514864Z",
     "start_time": "2019-01-21T19:19:48.508867Z"
    }
   },
   "outputs": [],
   "source": [
    "# first order state corrections"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T19:20:36.024645Z",
     "start_time": "2019-01-21T19:20:22.672856Z"
    }
   },
   "outputs": [
    {
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\n",
      "text/latex": [
       "$$\\begin{cases} 0 & \\text{for}\\: \\left(\\ell = 0 \\wedge n = 0\\right) \\vee \\left(\\ell = 0 \\wedge \\ell = n \\wedge n = 0\\right) \\vee \\left(\\ell = 0 \\wedge \\ell = - n \\wedge n = 0\\right) \\vee \\left(\\ell = 0 \\wedge \\ell = - n \\wedge \\ell = n \\wedge n = 0\\right) \\\\- \\frac{L g m}{2} & \\text{for}\\: \\left(\\ell = 0 \\wedge \\ell = - n\\right) \\vee \\left(\\ell = - n \\wedge \\ell = n\\right) \\vee \\left(\\ell = - n \\wedge n = 0\\right) \\vee \\left(\\ell = 0 \\wedge \\ell = - n \\wedge \\ell = n\\right) \\vee \\left(\\ell = - n \\wedge \\ell = n \\wedge n = 0\\right) \\vee \\ell = - n \\\\\\frac{L g m}{2} & \\text{for}\\: \\left(\\ell = 0 \\wedge \\ell = n\\right) \\vee \\left(\\ell = n \\wedge n = 0\\right) \\vee \\ell = n \\\\\\frac{4 \\left(-1\\right)^{\\ell} \\left(-1\\right)^{n} L \\ell g m n}{\\pi^{2} \\ell^{4} - 2 \\pi^{2} \\ell^{2} n^{2} + \\pi^{2} n^{4}} - \\frac{4 L \\ell g m n}{\\pi^{2} \\ell^{4} - 2 \\pi^{2} \\ell^{2} n^{2} + \\pi^{2} n^{4}} & \\text{otherwise} \\end{cases}$$"
      ],
      "text/plain": [
       "⎧                               0                                           fo\n",
       "⎪                                                                             \n",
       "⎪                            -L⋅g⋅m                                           \n",
       "⎪                            ───────                              for (ell = 0\n",
       "⎪                               2                                             \n",
       "⎪                                                                             \n",
       "⎪                             L⋅g⋅m                                           \n",
       "⎨                             ─────                                           \n",
       "⎪                               2                                             \n",
       "⎪                                                                             \n",
       "⎪       ell     n                                                             \n",
       "⎪ 4⋅(-1)   ⋅(-1) ⋅L⋅ell⋅g⋅m⋅n             4⋅L⋅ell⋅g⋅m⋅n                       \n",
       "⎪────────────────────────────── - ──────────────────────────────              \n",
       "⎪ 2    4      2    2  2    2  4    2    4      2    2  2    2  4              \n",
       "⎩π ⋅ell  - 2⋅π ⋅ell ⋅n  + π ⋅n    π ⋅ell  - 2⋅π ⋅ell ⋅n  + π ⋅n               \n",
       "\n",
       "r (ell = 0 ∧ n = 0) ∨ (ell = 0 ∧ ell = n ∧ n = 0) ∨ (ell = 0 ∧ ell = -n ∧ n = \n",
       "                                                                              \n",
       "                                                                              \n",
       " ∧ ell = -n) ∨ (ell = -n ∧ ell = n) ∨ (ell = -n ∧ n = 0) ∨ (ell = 0 ∧ ell = -n\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                 for (ell = 0 ∧ ell = n) ∨ (ell = n ∧ n = 0) ∨\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                       otherwise              \n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "0) ∨ (ell = 0 ∧ ell = -n ∧ ell = n ∧ n = 0)          \n",
       "                                                     \n",
       "                                                     \n",
       " ∧ ell = n) ∨ (ell = -n ∧ ell = n ∧ n = 0) ∨ ell = -n\n",
       "                                                     \n",
       "                                                     \n",
       "                                                     \n",
       " ell = n                                             \n",
       "                                                     \n",
       "                                                     \n",
       "                                                     \n",
       "                                                     \n",
       "                                                     \n",
       "                                                     \n",
       "                                                     "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# first, the integrals are\n",
    "integrate(phi0(x,ell)*m*g*x*phi0(x,n),(x,0,L))\n",
    "# The last line is the only good one: ell neq n and both are > 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T19:22:49.757394Z",
     "start_time": "2019-01-21T19:22:48.378242Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\begin{cases} - \\frac{L g m}{2} & \\text{for}\\: \\ell = -1 \\\\\\frac{L g m}{2} & \\text{for}\\: \\ell = 1 \\\\- \\frac{4 \\left(-1\\right)^{\\ell} L \\ell g m}{\\pi^{2} \\ell^{4} - 2 \\pi^{2} \\ell^{2} + \\pi^{2}} - \\frac{4 L \\ell g m}{\\pi^{2} \\ell^{4} - 2 \\pi^{2} \\ell^{2} + \\pi^{2}} & \\text{otherwise} \\end{cases}$$"
      ],
      "text/plain": [
       "⎧                       -L⋅g⋅m                                      \n",
       "⎪                       ───────                         for ell = -1\n",
       "⎪                          2                                        \n",
       "⎪                                                                   \n",
       "⎪                        L⋅g⋅m                                      \n",
       "⎪                        ─────                          for ell = 1 \n",
       "⎨                          2                                        \n",
       "⎪                                                                   \n",
       "⎪          ell                                                      \n",
       "⎪    4⋅(-1)   ⋅L⋅ell⋅g⋅m            4⋅L⋅ell⋅g⋅m                     \n",
       "⎪- ──────────────────────── - ────────────────────────   otherwise  \n",
       "⎪   2    4      2    2    2    2    4      2    2    2              \n",
       "⎩  π ⋅ell  - 2⋅π ⋅ell  + π    π ⋅ell  - 2⋅π ⋅ell  + π               "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_15.subs(n,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T19:25:01.579317Z",
     "start_time": "2019-01-21T19:25:00.643896Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sum_{\\ell=2}^{\\infty} \\begin{cases} - \\frac{L g m}{2} & \\text{for}\\: \\ell = -1 \\\\\\frac{L g m}{2} & \\text{for}\\: \\ell = 1 \\\\- \\frac{4 \\left(-1\\right)^{\\ell} L \\ell g m}{\\pi^{2} \\ell^{4} - 2 \\pi^{2} \\ell^{2} + \\pi^{2}} - \\frac{4 L \\ell g m}{\\pi^{2} \\ell^{4} - 2 \\pi^{2} \\ell^{2} + \\pi^{2}} & \\text{otherwise} \\end{cases}$$"
      ],
      "text/plain": [
       "    ∞                                                                         \n",
       "_________                                                                     \n",
       "╲                                                                             \n",
       " ╲        ⎧                       -L⋅g⋅m                                      \n",
       "  ╲       ⎪                       ───────                         for ell = -1\n",
       "   ╲      ⎪                          2                                        \n",
       "    ╲     ⎪                                                                   \n",
       "     ╲    ⎪                        L⋅g⋅m                                      \n",
       "      ╲   ⎪                        ─────                          for ell = 1 \n",
       "       ╲  ⎨                          2                                        \n",
       "       ╱  ⎪                                                                   \n",
       "      ╱   ⎪          ell                                                      \n",
       "     ╱    ⎪    4⋅(-1)   ⋅L⋅ell⋅g⋅m            4⋅L⋅ell⋅g⋅m                     \n",
       "    ╱     ⎪- ──────────────────────── - ────────────────────────   otherwise  \n",
       "   ╱      ⎪   2    4      2    2    2    2    4      2    2    2              \n",
       "  ╱       ⎩  π ⋅ell  - 2⋅π ⋅ell  + π    π ⋅ell  - 2⋅π ⋅ell  + π               \n",
       " ╱                                                                            \n",
       "╱                                                                             \n",
       "‾‾‾‾‾‾‾‾‾                                                                     \n",
       " ell = 2                                                                      "
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "summation(_16,(ell,2,oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T19:31:27.495967Z",
     "start_time": "2019-01-21T19:31:27.479980Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'Piecewise((0, (Eq(ell, 0) & Eq(n, 0)) | (Eq(ell, 0) & Eq(ell, n) & Eq(n, 0)) | (Eq(ell, 0) & Eq(n, 0) & Eq(ell, -n)) | (Eq(ell, 0) & Eq(ell, n) & Eq(n, 0) & Eq(ell, -n))), (-L*g*m/2, Eq(ell, -n) | (Eq(ell, 0) & Eq(ell, -n)) | (Eq(ell, n) & Eq(ell, -n)) | (Eq(n, 0) & Eq(ell, -n)) | (Eq(ell, 0) & Eq(ell, n) & Eq(ell, -n)) | (Eq(ell, n) & Eq(n, 0) & Eq(ell, -n))), (L*g*m/2, Eq(ell, n) | (Eq(ell, 0) & Eq(ell, n)) | (Eq(ell, n) & Eq(n, 0))), (4*(-1)**ell*(-1)**n*L*ell*g*m*n/(pi**2*ell**4 - 2*pi**2*ell**2*n**2 + pi**2*n**4) - 4*L*ell*g*m*n/(pi**2*ell**4 - 2*pi**2*ell**2*n**2 + pi**2*n**4), True))'"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "str(_15)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T19:33:27.401221Z",
     "start_time": "2019-01-21T19:33:27.385231Z"
    }
   },
   "outputs": [],
   "source": [
    "Hprimeelln = 4*(-1)**ell*(-1)**n*L*ell*g*m*n/(pi**2*ell**4 - 2*pi**2*ell**2*n**2 + pi**2*n**4) - 4*L*ell*g*m*n/(pi**2*ell**4 - 2*pi**2*ell**2*n**2 + pi**2*n**4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T19:38:48.482747Z",
     "start_time": "2019-01-21T19:38:48.443772Z"
    }
   },
   "outputs": [],
   "source": [
    "sumterm = Hprimeelln/(E0(n)-E0(ell))*phi0(ell,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T19:38:56.256966Z",
     "start_time": "2019-01-21T19:38:55.299553Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAeYAAABGBAMAAADr1iqmAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMom7q0QizZkQVN1mdu8ilFydAAAACXBIWXMAAA7EAAAOxAGVKw4bAAALk0lEQVR4Ae0bf4wUZ/Xt3u3c7t7+qkRtQuTGC6FH1XgEbWyCdgwYUiphhf7VP2ASMARpuxdrmhhR1iotRJpOKhWjbTixJf2jyJpUoyWxq4jaEHRFFJvewRHBGK3tgkBLW1zfe9+P+WaZWW5huV1iX7Lf9773a96b+Wbm+968BWgT9rptKtwg4slqpKPWBwuRvG4xdnodOPI3o2303xfN6xYn9uVrP3JmLNrG4BdCeM8+tzuEOnOkA9d+qDnGXEm7vr0sou+vYbPHpwER947mDMrMo33k1LXBF5U6Xu+DCn8SB2WAH+I4U2wi9hdmIWXVOTr0sKuYV9MPACTuWI6a8XltqFsX2hAOFU1flORsHVKOxGOTFYCVYD2K4x2SBpIIgzCBpyE7hfT4Hn1ClFQ7/QkUPk4Kt36+HbVvtCPsyw41CGg8aEvqPXVI6gjSGPOAB9nnALZIPoAk7oDd4wB9DtI30ly/akjsQ9WP4c+q1NoxUnLDpZ9RZDVt4orA/QI9GvIEmnq5jldQAYUXH+cRYhIMIkC+itRhtHp4opbZ/oGj2V8X4He/nCxL2el0T+1CqXfwF7dtyEweJn1n4tOA5v7aSr/PDuWmy4qspk1sVFGwNwaLJTmbqsN6LULhCRvWWAgRSUMuNqfnQ9bpczKrqzkn68DBV+HvWvrKSA1jjtHNmVvpwVG84qh/ChEy57ZQz0yFMncqqj9tDisS9v1FPaDzTLAHY14lUGwp5sQoDUXLDJ+IQ7oPrUsFKHkD1Q2vQL6YtK2nKqCeDqzQukm7GDM/Fkp/BOsibCL98/AQkDmvha71VihzEVI34h3r0rSZfHjExRvXMLMerK0PFEiTzzP2VhVjfhpg1kIEl2MWkyHthBBRAe/DJxKnAQ5B3oXlMAcGx6G/lphC1jThXsCY8bGQKq4uQqYCF1A/NgUvAJpb3dLGzxTXelVhGMEU4tvuxbsNp03WXTwwjmfUQZqELbAO4udpkKkLUmx45Ow4vYME+HPbuA18Igrh/LCTDsBcDBdOwp+gFINcIWnzmZRWWnfvGX7sVnosxAFXgnEHw80VsqNWvYDmHoq10n1EMRNvKgwgaQOkYN53AXDaZOCTZCHlT7vUGNwJ8BrJ949RS4CzK/gMy9aYXuGWGopZEml+pJ3PjSZwbiwBq46X5kAcSjB4T0GLXxkpefRYeDoxC4ppp99B/f5aZtRGcz/5QSvtE5rpaQzvLsLHbgbAaQNwhjmbuKUmOQ6ve7DCRTTnYMNw4Nx4n8sYxn/LmzWQo9skLUCEY2dHNhdh8jeQPPxVSDjwIKx7Ge6G7IeV9DT6zB3LspuHHzkP2QkXth+rof6Aay330NyRaiv9fWEnll9AidG7gaYNxOoWWfBf5d8HuAtgBWnmbWJJSNgKo/7bYrDSpCmiSTtpDq4ez4UFEm5uaDyE/grRktU5QNPm/vgY3thyxUMIiIvxIqH8miWE4WGFUD9fDJKu6EUriZoUq2TH9OAakNlwbPrau6ooG5uYHFt/Cmb/476jNVJdSs2gN4grqQl3x8GPsKdrPCSmi9hUiB27RG1pnFoFcRKRkCFJAjNMTRQsvAsWjRg6itp+v/ZX26avlK+hLE7DCjwPcKom3j0Lw/SHCkg9PqaeQ4NlEmqaJob/KWITaKQJZ2ZXGr4hV3tg0xvzU2ItB18Kc6VUxMvyOD7PxON5K8usQSIAr7471pDFnfsJfnx9jOfLaHfwP3/hVQJeX56x9NLOB2JAwi4KL3UOV8jYQ6ZMbfN1ZlrPNxxz6pZGma6zijn8OhcomMdc+Dn1u8FysTPmdvrIZ4iBayNP9Ea7dy7PBy2iESkj+eHKJKMVOmF8Vw0tboAnz1wxZnz9I5TKaRu7mA1ZCq1UxUZAjtaClAU4pGJODS+R6FSqjmuleUIEpTQidZnPuFaWHNl11Dg7vRfgs2bMarOk0hl8YH5uQ99F3BfjYnzr1ruIyqeM2WDRKhKzAOmNKuYkJCuC5+LKEHf2LEIUiZQJJ2A+Ib4yjXzoqHGenD/y4Daa2A/I+/m4PhjvW8To99wlLj1L/eZG4w3qc2VqJXycswDvxVecgHzREks4nO9FsbNHEQGMjKoR8Rl8Zc2SSAeN84N35eEjxVmv3T7rvwsOvO3gMcQlpYP10VAALckQvvag6EU7wG5bLzYaFyBZoCxAqqpjznjWGcGCr+MUsG0SETt6kgUgZd6+EZ8yBoaysI9t543/wdPGfSRXVHi+qjCQ6+19jqYgkpyiUfI7VRsXH7hcOz3fmjwxj2gM8YpgxT2bd/YoInb0JCti5u0b8SljEFAWJq6D8W9Jy4EuaauheHDRSO2rBl3Foz7Duy0vvqEMA423OQuQMmKe7TELb4UKbdFIhHf0hCDgdRbbN+SLjIGpTBIInTf+vDAcbOmJI8C/szOOopl96i0eDcTK3FMWgECmDqzbMRUoWWKLhjy1o08uXPi3hYvE9o10IjMGykKnjKfeocNdBssUBV/aTwg8eH0VHz7B2CrcvxEkRcc4NuvgfaBYvEUjhrGjp/uZt29Ej8wYKAudMp6mHNrloPNheEpswebV1+WS4mm+Xc4MygL4YC24aRsoFm3RiGUZO3qKmbZvDJEZA2WhU8aTY+KATW26LAiczmDUIv9CgFekuDTjxRlQFsCHXKNR91m0s0cwd/Rkk7dvxIjMGHTaeG6cDnc5PCNIIp1BuLlNNMXj4SfNFInGI85jtEI7nEjj/jOqHXOm7Elz0CZeblO+LfFI4/9sy0yY8G/DiL1M60BiZqDWywGG+Ha/G0Jsj5T6aXvy3ZZOvdABD9Z7HTAycyb6qh04llXogJGZMyGXWDN3wHePdGOcgUBe8P9jcGNcmHe9vIHOQHN22M8GB4PQSWAtoJGgYG+MhuQDgb2RmxrtmZkdPoRLPT8brEUY4SRwQEBoBqV6ZrTA8ERtXjXJzA4fX/5nPxusJQTCSeCAAGsa5Q1NCl0dBtzSSQrtUp+fHS4REcfhgEngoABLRqQ4wk3MGLXfDALzYhp09ldmh+HA93Cph9ngJvCTwEpAFJixpE44Nyl1d7gev+queEMEa00ZvujsL9V9UVJ48UuPYs7CNkQY9TPMSoDSxVIy6zSL98J4CzqRrwhP/OSykf1V2WH4N4xwtrjJa50EVgIiXUx5ZT813aTT3SEWH+FnTVs4kS+azlDxDgEX8HDW0Jz6iSozsVFJYKsOQsBMF29SUj3UJ8fRmRMyVi7aUc7p7K/8WpQZTV10FRM/49UUrpLAWsBMF/slP0q8+z0WH+GnPukHF+0on1T2V2WHuehLMc2YVRJYC5jp4mvP9PmH7BTGxUf8bRMtLjWt6uyvzA5T0ZfrC/jXWSWBtYCZLva/M/qqXcR08VHirPQCvwNPG/yYW6pwyU9LiRllHsfHV5buSlEzikhoMUeET9OMuSQfFRFWZpjsFx8N1OWhsWgn9RUuFSo0Fe+Ym3QUTu7f/4t/7d/vkJ7JasKRKz6wkFxPgC4+yuMFz7jo0/W4zoWeiFU7oYqP1tTwEyxRo2LWO0Wtaj63TaIqWtK0IVejPYHI4iP4KF6LCfLIL9ph/x6fe6co32muLyKueT+3EOyx57YqPgIsk41zoULTy7QMWP6QndL1RXwiVGPG3EJQfCRWSt3vRfFRamljydKGQ+40XZTzUCrI8h3cKTaBGXMLQbmIa1Lu4jBYfIQ1BsWAMy9RzFy+Q/VF8i9ESsKMuYVgz6239zkqAtGb+yqmfAj3Hy5itJXEHSL9hUiBv8cgSpSgKvlRWt3vm4tT/KId4VsMyxboHudaJPkXolCvIwUzTqh8LxGXBZ1Z64l/IzFV/YUoKCJGkYLNZzVMucu0YD7McmIu/xuJvVJ/IQpxMVqwN/NhgRCCec+1N93s+eU78i9EAXk5iBSMKvkJM9I1WiC//Tr+59Yv35F/IQpzLVIwquQnzEhr2v8AxG11upbOfN8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{\\sqrt{2} \\left(\\frac{4 \\left(-1\\right)^{\\ell} \\left(-1\\right)^{n} L \\ell g m n}{\\pi^{2} \\ell^{4} - 2 \\pi^{2} \\ell^{2} n^{2} + \\pi^{2} n^{4}} - \\frac{4 L \\ell g m n}{\\pi^{2} \\ell^{4} - 2 \\pi^{2} \\ell^{2} n^{2} + \\pi^{2} n^{4}}\\right) \\sin{\\left (\\frac{\\pi \\ell x}{L} \\right )}}{\\sqrt{L} \\left(- \\frac{\\pi^{2} \\ell^{2} \\hbar^{2}}{2 L^{2} m} + \\frac{\\pi^{2} \\hbar^{2} n^{2}}{2 L^{2} m}\\right)}$$"
      ],
      "text/plain": [
       "   ⎛       ell     n                                               ⎞          \n",
       "   ⎜ 4⋅(-1)   ⋅(-1) ⋅L⋅ell⋅g⋅m⋅n             4⋅L⋅ell⋅g⋅m⋅n         ⎟    ⎛π⋅ell\n",
       "√2⋅⎜────────────────────────────── - ──────────────────────────────⎟⋅sin⎜─────\n",
       "   ⎜ 2    4      2    2  2    2  4    2    4      2    2  2    2  4⎟    ⎝   L \n",
       "   ⎝π ⋅ell  - 2⋅π ⋅ell ⋅n  + π ⋅n    π ⋅ell  - 2⋅π ⋅ell ⋅n  + π ⋅n ⎠          \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                             ⎛   2    2   2    2   2  2⎞                      \n",
       "                             ⎜  π ⋅ell ⋅h̅    π ⋅h̅ ⋅n ⎟                      \n",
       "                          √L⋅⎜- ─────────── + ─────────⎟                      \n",
       "                             ⎜        2            2   ⎟                      \n",
       "                             ⎝     2⋅L ⋅m       2⋅L ⋅m ⎠                      \n",
       "\n",
       "   \n",
       "⋅x⎞\n",
       "──⎟\n",
       "  ⎠\n",
       "   \n",
       "───\n",
       "   \n",
       "   \n",
       "   \n",
       "   \n",
       "   "
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sumterm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T19:51:27.993629Z",
     "start_time": "2019-01-21T19:46:34.171332Z"
    }
   },
   "outputs": [],
   "source": [
    "with assuming((Q.nonzero(n-ell)),Q.positive(n),Q.positive(ell)):\n",
    "    summation(sumterm,(ell,1,n-1))+summation(sumterm,(ell,n+1,oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T20:52:18.172638Z",
     "start_time": "2019-01-21T20:52:18.168641Z"
    }
   },
   "outputs": [],
   "source": [
    "theta = symbols('theta', real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T20:53:35.342177Z",
     "start_time": "2019-01-21T20:53:35.326187Z"
    }
   },
   "outputs": [],
   "source": [
    "Hmatrix = Matrix([[cos(theta),sin(theta)/sqrt(2),0],[sin(theta)/sqrt(2),0,sin(theta)/sqrt(2)],[0,sin(theta)/sqrt(2),-cos(theta)]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T20:53:39.628539Z",
     "start_time": "2019-01-21T20:53:37.934581Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\cos{\\left (\\theta \\right )} & \\frac{\\sqrt{2} \\sin{\\left (\\theta \\right )}}{2} & 0\\\\\\frac{\\sqrt{2} \\sin{\\left (\\theta \\right )}}{2} & 0 & \\frac{\\sqrt{2} \\sin{\\left (\\theta \\right )}}{2}\\\\0 & \\frac{\\sqrt{2} \\sin{\\left (\\theta \\right )}}{2} & - \\cos{\\left (\\theta \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡           √2⋅sin(θ)           ⎤\n",
       "⎢ cos(θ)    ─────────      0    ⎥\n",
       "⎢               2               ⎥\n",
       "⎢                               ⎥\n",
       "⎢√2⋅sin(θ)             √2⋅sin(θ)⎥\n",
       "⎢─────────      0      ─────────⎥\n",
       "⎢    2                     2    ⎥\n",
       "⎢                               ⎥\n",
       "⎢           √2⋅sin(θ)           ⎥\n",
       "⎢    0      ─────────   -cos(θ) ⎥\n",
       "⎣               2               ⎦"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Hmatrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T20:54:52.823523Z",
     "start_time": "2019-01-21T20:54:50.505948Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$$\\left [ \\left ( -1, \\quad 1, \\quad \\left [ \\left[\\begin{matrix}\\frac{\\sin^{2}{\\left (\\theta \\right )}}{\\left(\\cos{\\left (\\theta \\right )} + 1\\right)^{2}}\\\\- \\frac{\\sqrt{2} \\sin{\\left (\\theta \\right )}}{\\cos{\\left (\\theta \\right )} + 1}\\\\1\\end{matrix}\\right]\\right ]\\right ), \\quad \\left ( 0, \\quad 1, \\quad \\left [ \\left[\\begin{matrix}-1\\\\\\frac{\\sqrt{2}}{\\tan{\\left (\\theta \\right )}}\\\\1\\end{matrix}\\right]\\right ]\\right ), \\quad \\left ( 1, \\quad 1, \\quad \\left [ \\left[\\begin{matrix}- \\frac{\\sin^{2}{\\left (\\theta \\right )}}{\\sin^{2}{\\left (\\theta \\right )} + 2 \\cos{\\left (\\theta \\right )} - 2}\\\\\\frac{\\sqrt{2} \\left(\\cos{\\left (\\theta \\right )} - 1\\right) \\sin{\\left (\\theta \\right )}}{\\sin^{2}{\\left (\\theta \\right )} + 2 \\cos{\\left (\\theta \\right )} - 2}\\\\1\\end{matrix}\\right]\\right ]\\right )\\right ]$$"
      ],
      "text/plain": [
       "⎡                                                ⎛      ⎡⎡          2         \n",
       "⎢⎛       ⎡⎡      2      ⎤⎤⎞                      ⎜      ⎢⎢      -sin (θ)      \n",
       "⎢⎜       ⎢⎢   sin (θ)   ⎥⎥⎟                      ⎜      ⎢⎢────────────────────\n",
       "⎢⎜       ⎢⎢─────────────⎥⎥⎟  ⎛      ⎡⎡  -1  ⎤⎤⎞  ⎜      ⎢⎢   2                \n",
       "⎢⎜       ⎢⎢            2⎥⎥⎟  ⎜      ⎢⎢      ⎥⎥⎟  ⎜      ⎢⎢sin (θ) + 2⋅cos(θ) -\n",
       "⎢⎜       ⎢⎢(cos(θ) + 1) ⎥⎥⎟  ⎜      ⎢⎢  √2  ⎥⎥⎟  ⎜      ⎢⎢                    \n",
       "⎢⎜-1, 1, ⎢⎢             ⎥⎥⎟, ⎜0, 1, ⎢⎢──────⎥⎥⎟, ⎜1, 1, ⎢⎢√2⋅(cos(θ) - 1)⋅sin(\n",
       "⎢⎜       ⎢⎢ -√2⋅sin(θ)  ⎥⎥⎟  ⎜      ⎢⎢tan(θ)⎥⎥⎟  ⎜      ⎢⎢────────────────────\n",
       "⎢⎜       ⎢⎢ ─────────── ⎥⎥⎟  ⎜      ⎢⎢      ⎥⎥⎟  ⎜      ⎢⎢   2                \n",
       "⎢⎜       ⎢⎢  cos(θ) + 1 ⎥⎥⎟  ⎝      ⎣⎣  1   ⎦⎦⎠  ⎜      ⎢⎢sin (θ) + 2⋅cos(θ) -\n",
       "⎢⎜       ⎢⎢             ⎥⎥⎟                      ⎜      ⎢⎢                    \n",
       "⎣⎝       ⎣⎣      1      ⎦⎦⎠                      ⎝      ⎣⎣          1         \n",
       "\n",
       "  ⎤⎤⎞⎤\n",
       "  ⎥⎥⎟⎥\n",
       "──⎥⎥⎟⎥\n",
       "  ⎥⎥⎟⎥\n",
       " 2⎥⎥⎟⎥\n",
       "  ⎥⎥⎟⎥\n",
       "θ)⎥⎥⎟⎥\n",
       "──⎥⎥⎟⎥\n",
       "  ⎥⎥⎟⎥\n",
       " 2⎥⎥⎟⎥\n",
       "  ⎥⎥⎟⎥\n",
       "  ⎦⎦⎠⎦"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Hmatrix.eigenvects(simplify=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-21T20:54:31.728496Z",
     "start_time": "2019-01-21T20:54:31.706511Z"
    }
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'Series' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-39-34e76326a0af>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mSeries\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0m_37\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m: name 'Series' is not defined"
     ]
    }
   ],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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