{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Crystal Momentum Operator in Lattice Models"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Saumya Biswas (saumyab@uoregon.edu)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bloch's theorem, Born von Karmen Boundary Condition\n",
    "In a periodic crystal with periodic boundary condition, the wave function of a particle assummes the bloch form resolvable into the product of a plain wave($|k \\rangle$) and a cell periodic part($| u_{n}(k) \\rangle$).\n",
    "\\begin{eqnarray}\n",
    "|\\psi_n(k) \\rangle = |k \\rangle \\otimes | u_{n}(k) \\rangle \n",
    "\\end{eqnarray}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The periodicity of the crystal limits the crystal momenta, k to a set of discrete values(as many as the number of cells(L) in the crystal).\n",
    "We name this set $k_{lat}$ and enumerate the crystal momenta of the $k_{lat}$ with the wave-vector associated with them.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In a 1d crystal of L cells, for $k \\in k_{lat}$, k is one of the values,\n",
    "\\begin{eqnarray}\n",
    "k = \\frac{2\\pi}{La}\\times n, \\ \\ \\ \\ \\ \\text{where} \\ \\ \\ n = 0,1,2,...., L-1 \\ \\ \\text{and a is the length of the unit cell}\n",
    "\\end{eqnarray}\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Crystal Momentum Operator\n",
    "Like any operator with a discrete momentum, the crystal momentum operator can be formed as a sum of its projections into all the individual eigen-kets.\n",
    "\\begin{eqnarray}\n",
    "\\hat{\\bf{k}} = \\sum\\limits_{k \\in k_{lat}} k| k \\rangle \\langle k |\n",
    "\\end{eqnarray}\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We enumerate the position operator with the index, j. So a complete set of position eigenkets have corresponding position eigenvalues labeled as, \n",
    "\\begin{eqnarray}\n",
    "x = ja ,\\ \\ \\ \\ \\ \\ \\ \\text{where} j = 0, 1, 2, ..., L-1\n",
    "\\end{eqnarray}\n",
    "Each position eigenket defined as such simply refer to a particle localized at a certain cell, j. We label these position eigenkets as $|j\\rangle$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The momentum eigenfunctions in the position basis are the properly normalized plane waves(continuum state normalization).\n",
    "\\begin{eqnarray}\n",
    "\\langle j|k \\rangle = \\sqrt{\\frac{a}{h}} e^{i \\frac{2\\pi x}{L}j } \\\\\n",
    "\\langle k'| k \\rangle = \\delta(k-k')\n",
    "\\end{eqnarray}\n",
    "The $\\langle j|k \\rangle$ have the dimension of $[\\text{momentum dimension}]^{-\\frac{1}{2}}$\n",
    "\n",
    "The matrix elements of the crystal momentum operator($\\hat{\\bf{k}}$) in the basis of ${|j \\rangle}$ are formed as follows\n",
    "\\begin{eqnarray}\n",
    "\\langle j|k|j' \\rangle = \\sum\\limits_{k \\in k_{lat}} \\langle j | k \\rangle \\langle k |j' \\rangle \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\\\n",
    "=\\frac{1}{h} \\frac{2\\pi}{L}\\sum\\limits_{n=0}^{L-1} n e^{-i \\frac{2\\pi n}{L}(j'-j) } \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\\\n",
    "=\\frac{2\\pi}{h}\\left\\{\n",
    "                \\begin{array}{ll}\n",
    "                  (L-1)/2, \\ \\ \\text{when} \\ \\ \\ j=j'\\\\\n",
    "                  \\frac{1}{e^{i\\frac{2\\pi}{L}(j-j')}-1}, \\ \\text{otherwise}\n",
    "                \\end{array}\n",
    "              \\right.\n",
    "\\end{eqnarray}\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "from qutip import *\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "periodic_atom_chain8 = Lattice1d(num_cell=8, boundary = \"periodic\")\n",
    "k8 = periodic_atom_chain8.k()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "[ks8, pw8] = k8.eigenstates()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([4.4408921e-15, 1.0000000e+00, 2.0000000e+00, 3.0000000e+00,\n",
       "       4.0000000e+00, 5.0000000e+00, 6.0000000e+00, 7.0000000e+00])"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ks8 # In units of 2*pi/(L*a),  if ks[1] = 1, the wavevector/crystal-momentum of the\n",
    "    # corresponding eigen-vector of k is a plane wave with wavelength of L*a, the length\n",
    "    # of the crystal. ks[2] = 2 has a corresponding plane wave eigenvector with\n",
    "    # wavelength of 2*L*a, twice the length of the crystal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[8], [1]], shape = (8, 1), type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}0.354\\\\0.354\\\\0.354\\\\0.354\\\\0.354\\\\0.354\\\\0.354\\\\0.354\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[8], [1]], shape = (8, 1), type = ket\n",
       "Qobj data =\n",
       "[[0.35355339]\n",
       " [0.35355339]\n",
       " [0.35355339]\n",
       " [0.35355339]\n",
       " [0.35355339]\n",
       " [0.35355339]\n",
       " [0.35355339]\n",
       " [0.35355339]]"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pw8[0] # ks[0] = 0 has an eigenvector that has a frequency of 0 and wavelength of\n",
    "       # infinity, the eigen-vector is normalized to unity, meaning the sum of all\n",
    "       # the elements sqared is 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# The eigenvectors are better demonstrated with a latice of longer length\n",
    "# Here we plot the second and third eigen-vectors(real and imaginary parts) of the first\n",
    "# 3 eigenvectors of the crystal momentum operator for a 1d lattice with 64 cells\n",
    "periodic_atom_chain64 = Lattice1d(num_cell=64, boundary = \"periodic\")\n",
    "k64 = periodic_atom_chain64.k()\n",
    "[ks64, pw64] = k64.eigenstates()\n",
    "plt.plot(np.real(pw64[1]),'r')\n",
    "plt.plot(np.imag(pw64[1]),'b')\n",
    "plt.plot(np.abs(pw64[1]),'g')\n",
    "plt.xlabel('position')\n",
    "plt.ylabel('crystal momentum eigen vector')\n",
    "plt.show()\n",
    "plt.close()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZQAAAEKCAYAAAA1qaOTAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDMuMC4wLCBodHRwOi8vbWF0cGxvdGxpYi5vcmcvqOYd8AAAIABJREFUeJztnXl8VNX5/z9P2EH2TQQUUERR2cEFhJBhmSiyJO5aFQW7W7W21W9/rXbx29pvW2sX2wou1ErVmiiIZlgmCaAgEGTfNGyyE/adkOT5/fHcK5MwSe7M3HvPmeS8X69LMjd3+Qx3ec45z3KImWEwGAwGQ6KkqBZgMBgMhpqBMSgGg8FgcAVjUAwGg8HgCsagGAwGg8EVjEExGAwGgysYg2IwGAwGVzAGxWAwGAyuYAyKwWAwGFzBGBSDwWAwuEJd1QL8pE2bNtylSxfVMgwGgyGpWL58+QFmblvddrXKoHTp0gUFBQWqZRgMBkNSQUTbnWxnhrwMBoPB4ArGoBgMBoPBFYxBMRgMBoMrGINiMBgMBlcwBsVgMBgMrmAMisFgMBhcwRgUg8FgMLhCrcpDiZfHQ49j5d6VqmUkD8zA4cPAmTNA27ZAvXqqFX1NSQlQVCSSWrcGiFQrMiTEuXPAgQNA/fpAq1ZaXdATJ4AjR0RW48ZqtfS5uA/+FPyT5+cxBsXgHmfPAnv3AHv2yu8AUFgItGkNdLgEaNlSmbSjR4E9e4D9RQCXybp69YGLLwY6XAw0aqRMmiEeDh+WC3rggDRgADEqHTrIRW3YUIms0lJg3z55BE4cl3WbNwPNmst91q4dkFKTx4WYudYs/fv3Z4MHbNnCfMstzESyjBrF/O67zKtWMT/xBHPr1swA82WXMU+d6qu0d99l7tFDTt+sGfO3v828bBnzzJnMY8cy16kjfxs+nHnNGl+lGeJh2jTmbt3korVsyfzYY8wrVjBnZTEHg+XvwS++8E3WqVPM3/wmc+PGIu3aa5lfeol50ybm3/2O+corz9+DTz7JfO6cb9JcAUABO3jHKn/J+7kYg+IBBw/K09K8OfP/+39iXCpy5gzz228z33ij3HJvveWLtI8/Zk5JYe7dm/n115lPnLhwm127mJ9/nrldO+aOHZl37vRFmiEe/vtfuX8GDZJ76PTpC7fZvp352WeZW7USw7N/v+eySkuZ77hDpD38MPNnnzGXlZXfpqyMef585nvuke2++c0Lt9EZY1CMQfGeM2eYhw5lrl+fecGC2LZfuNBTaatWMV90kRiTY8eq337lStm+Tx/m48c9lWaIh88+Y27YUBolp05Vv/3ixee3j2Z4XOTpp+VN+rvfOdv+Jz+R7X//e09luYoxKMageEtZGfM3vsEx9zjsHk3r1p4NSezaxdypE/MllzDv2OF8P7tHc+utzCUlnkgzxMOWLdKFjLXHYfdo7rpLuhEeMGWKnOLRR533OEpLmW+/XUbmsrI8keU6xqAYg+Itv/iF3D6//GXs+375pRiU7t2ZDxxwVdaJE8z9+jE3acL8+eex7//yy/K1vv99V2UZ4uXwYearr2Zu0YJ5w4bY93/hBbmg//M/rkubM0d8cKNGMRcXx7bvqVPM11/P3KgR85IlrktzHWNQjEHxjjfflFvngQfiHwj+5BMZ+rr5ZhkKc4GSEnG0p6Qwf/hh/Md58kn5ei+95IosQ7ycPcuclsZcrx5zXl58xygrY548WS7oq6+6Jm3tWnGwX3st85Ej8R1j3z7mLl2k87V1q2vSPMEYFGNQvGHdOjEEw4bJA58I06fLLfjYY65I+9Wv5HB/+UtixykpYR4/XoYkPv3UFWmGePjRj+SCvvFGYscpLmYeOZK5bl2JCEuQ06dl9O3iiyUGIBHWr5d4lr599R5mNQbFGBRvGD9emmZuRc9885vyoBcWJnSYfftkmCsz0x1ZJ04wd+jAPHhwckXj1Bi2b5eGy0MPuXO8Q4ckzDg9PeFDvfiivDnnzHFBFzP/5z9yvDffdOd4XmAMijEo7rN4McftN6mM3btlIPneexM6zGOPyXj2xo0u6WLmv/9dvu6sWe4d0+CQiRPFoCTaBYjE9qfMnx/3IY4dY27ThjkQcE9Waan0ULp2TbzT7xXGoBiD4i5lZcypqcxt27ofV/v00zK+tHJlXLtv2ybvnkmT3JVVXMx8+eXMvXp5FiRkiMb69eIIe+IJd4978qSE/t10U9zdzueek7fm0qXuSguF5Lh//au7x3ULY1CMQXGX2bPZM0/1oUMSxXPrrXHt/tBDzA0axBYi7BTbzeNTLqaBmTkjQ5KCvEhK/Mc/5ILOnBnzrvv3MzdtKvLcpqxM3JLt20dPwFWNMSjGoLhHaSlz//5SOsWliKwL+M1v5HaMMeFx3TppzP7wh97IKi2V5Mhu3fQdjqhRLF0q98Fzz3lz/OJi5iuukPCsGL3gTzwh99r69d5IW7RIvvrzz3tz/EQwBsUYFPd49112JdqmKk6elLCZIUNiGo6YMEFajUVF3kmbNUu+/ssve3cOg0UgIE4KJ+UN4iUOL/hXX0kveOJE72QxM992m0R9HTzo7XlixRgUY1Dc4dw5qa7Ys6f3cY12VuFHHznafMkSdj1GIBplZWLnOnQQu2fwiHnz5IK++KK35yktlRo7MXjBH3nE/RiBaKxeLe7EH//Y2/PEijEoxqC4w9Spcpu8/7735zp7VsaWevd25AVPS5MYAS8bszYLF8p/w29/6/25aiVlZcwDBzJ37ux57S1mljo7APPf/lbtphs2yFDX4497L4uZ+f77pQyZToVKk8KgAAgC2ASgEMDTUf4+FMDnAEoA3F7hbw8C+NJaHnRyPmNQYuTsWSmKdf31/iVj/Pvfclu+806Vm+XmymZ/+pM/spilQn+LFsxHj/p3zlrDBx+w29nsVVJWJlUa2rev1oDddZd3MQLR2LxZUrO+8x1/zucE7Q0KgDoANgPoBqA+gFUAelbYpguAXgD+FWlQALQCsMX62dL6vWV15zQGJUbee09ukUTqmMRKaanE6g4dWuVmmZky1O5HY9bGTsP5xz/8O2etYeRI6Z34OVHInDlyQadPr3STPXvk5f7UU/7JYpbIxSZN/Ol9O8GpQVE5d9ggAIXMvIWZiwG8DWBc5AbMvI2ZVwMoq7DvaABzmfkQMx8GMBfS2zG4ydSpQMeOQHq6f+dMSQEmTQIWLAC++CLqJvv3AzNmAA884O/EfNdfD1x3nfy3GFxk61Zg7lzgkUeAuj5OIhsIAF27VnlBp02TaaMnT/ZPFgA8+ihw8iTwzjv+njdRVBqUjgB2RHzeaa3zel+DE7ZvB2bPBh5+GKhTx99zP/ignLOSB91+yCdN8lcWkZyzoABYudLfc9doXntN/nMnTvT3vCkpYsRyc2We3gowyy04dChw5ZX+SrvhBqBnT2DKFH/PmyhVGhQiSiGitR6dm6KsY7f3JaJHiaiAiAqKiooci6v1vP66/Hz4Yf/P3aEDcNttwBtvAMXF5f5kP+RDhgBXX+2/tPvvBxo0SL4HXVtKSsSgBIPApZf6f/6JE8WwRGm85OcDhYX+904Asa+TJwNLlwKrV/t//nip0qAwcxmAVUTkxZXeCaBzxOdOAHa7vS8zv8LMA5h5QNu2beMSWusoLZWHfORIoEsXNRomTQKKioAPPyy3euFCGQnzu3di06oVkJkJvPUWcOqUGg01ilAI2L1b3QW95BLg1lul8XLuXLk/TZ0KtGgh11sF998P1K+fXEOsToa8OgBYR0RhIpppLy6cexmA7kTUlYjqA7gbgNPjzgYwiohaElFLAKOsdQY3mDMH2LFDTdPMJhgEOnW6oCswZQrQrBlwxx2KdEH+W44eBd57T52GGsOUKUD79tIjVcXkycDevcBHH3296tAhICtLXuqNGqmR1aYNkJEBvPkmcPq0Gg0xU53XHsCwaIsTj7+DY98C4AtItNdPrXW/BDDW+n0gpDdyEsBBAOsi9n0YEm5cCGCik/OZKC+HTJggCR6qa438/OeS5bVtGzNLya+GDZm//W21ssrKpHrHzTer1ZH07NolJaJ/8hO1Os6dk6KREbXkXnpJAsBWrVKoi5nDYdHx73+r1QE3w4YBtAcwxlraOdlHx8UYFAeoipOMxrZtYlB+/nNmlomzgPim9nWb3/5WtMQzK63B4vnn5T/xiy9UK2H+6U8le3HHDi4rk1JfAweqFiVR9N26SeFIlTg1KNUOeRHRnQCWArgDwJ0AlhDR7Ql2jAy6oiqEKhqXXQaMGgW89hq4pBRTpgD9+gF9+6oWBjz0kES4JtP4tlaUlcl/Xmoq0L27ajUS7VVWBrz2GpYsAdauVTvia2NH0c+fX2kUvVY48aH8FMBAZn6QmR+A5I/8zFtZBiVExkn26KFajTB5MrBzJwr++hlWr9bjIQdk2H/sWLG/FQLRDE7Iy5P8E10uaNeuEoTy6quYOqUMTZoAd9+tWpTw0EMSRf/qq6qVVI8Tg5LCzPsjPh90uJ8h2Zg/X+Ikdeid2Nx2G9C2Lab86SQaNwbuvVe1oPNMmgQcOCBJloYYmTIFaNlSvM66MGkSjn91CG9PL8PddwNNm6oWJHToAIwZEzWKXjucGIYQEc0mooeI6CEAHwHI8VaWQQl2nOTtGo1o1q+PE/c+iv9svxF3jjmFZs1UCzrPqFGSOmFyUmLkwAHg/feBb3zD31IH1TFuHN6+aBJOnqmrTcfJZvJkqRBRIYpeO6o1KMz8IwD/hNTU6g3gFWb+sdfCDD5z6pQ85HffrS5OshI+7PxtnEBTTGz/sWop5ahTR5L6580D9u1TrSaJyMqSprbfmfHV0aAB3mr+XfSk9Rh01THVasoRDEpPZfp01UqqxolT/gVmzmbmJ5n5CWZ+n4he8EOcwUdCITEqKhM8KiFrcUdcXLcIQ1b9TbWUC7j9dnE9mWGvGMjOBq64AujdW7WScuzfDyzcczlu5/+CPv6o+h18pE4dYMIEICdH74RaJ0NeI6Os87FaoMEXsrOB1q3FIa8Rp07JQzShz1akfLJAnnqNuO46eTdmZalWkiQcPiy1szIzpb6IRsyYAZSVETJbz5fnQTMyMyXBMRRSraRyKjUoRPRtIloD4CoiWh2xbAWwxj+JBs85e1YGZ8eN87faqwNmzxajkjGptYR1atYVIBK/cm6uvCsN1fDhhxKWrpMz3iIrC7j8cuC6O64CPv5Yu67A0KHS5tPQ1n1NVT2U6QBuAzDD+mkv/Zn5Ph+0GfwiNxc4dkxd0aIqyM6W+lnDJnaTp13DrkBmprwjdXeYakFWFtC5MzBwoGol5ThyBAiHrY7T7ZliTObMUS2rHHXrSpvvww+lDagjlRoUZj7KzNsAvATgEDNvZ+btAM4R0fV+CTT4QFaWFMgKBFQrKUdx8fmOU736JE97OKxdV2DgQCk7pqGt04vjx6XLmZGh3XBXuY7TsGHSitHwgmZmStsvHFatJDpOfCh/B3Ai4vNJa52hJlBSAnzwgQS6N2igWk05wmEpwvj16EhGhuidNUuprorYw16zZwMnTlS/fa0lJ0ea1poOd3XqZHWcIrsCmiV+BALS9tN12MuJQSGrlguAr0va6zXQboifhQuBgwe1He5q2hQYMcJaoXFXIDNT3pUf6xXZrBdZWUC7dsDgwaqVlOPECWkMTJggpU4AyAU9elSGgzWiQQNp+33wgbStdMOJQdlCRI8RUT1r+QFkDndDTSArS/JORo9WraQcdsfp1lsjct9SUuSp17ArMHiwvCt1bTkq58wZKQ8/frz/M4BWQ06OyCvXphoxQlozGjZeMjKkDbhwoWolF+LEoHwLwE0AdkFKyV8P4FEvRRl8oqxMkhnT04EmTVSrKccnn0hC9QUdp8xMefpz9CrWUKeOvCs/+kjkGSowZ45Mkq5pT7htW5kF9GsiuwKlpcq0RSMYlDaghrbOUab8fma+m5nbMXN7Zr63Qm0vQ7KyZInMlqfhQ56VJT2T9IoZT0OGyNOvYVcgM1M6TnPnqlaiIdnZUtZn+HDVSspx5oy45KJ2nDIzpVWjWVegSRN5Lt5/X9qEOuEkU/5Ka7bGtdbnXkT0/7yXZvCcrCygXj0ZV9KIsjJ5/wSDUTpOdldg1iztugKpqfLO1LHlqJRz54CZM6U8c716qtWUY+5caQREjRPQuCuQkSFtwSVLVCspj5MhrykAngFwDgCYeTVkul5DMsMsb+2RI4HmzVWrKcfSpdV0nOyuwLx5vuqqjvr15Z05c+YF05PXbvLzJdRbw55wdrbc/mlpUf7YpIkYFQ27AmPGiG3WraPuxKA0ZualFdZpGF9giImVK2U+Ck1DOOvVk4cmKsOHy1tA05bj4cPyDjVYZGXJy3lktCpO6jh3TgovjB0rjYGoZGQAu3ZJK0cjmjeXuIGsLGkb6oITg3KAiC4HwABgzda4x1NVBu/Jzpbho3HjVCsph91xCgRk+CgqdldgxgztugKjRsm7U0Nbp4bS0vPheppVsZ4/X4x/lW0quyug4QXNzJQ24cqVqpWcx4lB+S6kfP1VRLQLwOOQyC9DMvPhh+LgbtNGtZJybNgAbNkibpIqmTBB3gaLF/uiyymNGonDdNYsvVqOyigokNr+1V5Q//nwQwn8GDWqio1atJDxMM2SaQFpUxHpJc2JQdnOzCMAtAVwFTMPsUqwGJKV3buBVauAW25RreQC7GjgC6K7KhIISEazZuHDgGjftUvmJa/15ORI/lCVb2015OTI6GnjxtVsmJ4ObNwIbNvmhyzHtG0rub46PQJODMpWInoFwA0oX4LFkKzMni0/g0G1OqIQCgE9e8pMiFXSrJlkE2pYy9vOEdVQmv+EQsCgQVImVyM2bwa+/NLhI2BvpOEFDQYl0uvQIdVKBCcGpQeAeZChr61E9FciGlLNPgadCYWASy6RyTw04uRJYMECB70Tm/R0GUDeo5dLr2NHoFcvvVqOSjhwQJzZji+of9htKkfSrrwS6NpVS4OSni4BaLrkPjlJbDzNzO8ycwaAvgCaAZjvuTKDN5SUSNZyMKhdxde8PKnF57jjZG9ovx00IhiUbP/jx1UrUcjcueJI0rAnnJMDdOsmk6NVC5F8h3BYu2KRAwcCLVvqY+uc9FBARMOI6GUAnwNoCOBOT1UZvGPpUpn8QcOHPBSS8eybb3a4Q69eMtG2Lk9TBMGgBKDl5alWopBQSIa6+vdXraQcZ89Kzcf09BjaVMGg5D59+qmn2mKlTh1xT4VCegSBOMmU3wqJ7FoI4FpmvpOZ9YuhMzgjJ0fuQs1yAphFWlpaDFX07ZbjnDnalV4dPBi46KJaPOxVViZvudGjtSsGuXChzJ8VU5sqLU3ChzW8oOnpwN69EmejGic9lN7MPIGZ/8PMJz1XZPCWUAi44YYqkjzUUFgo4cIxd5yCQQkfXrbME13xUr++BKLp0nL0nZUrgf37te0J168fY1mxiy6SrrOGvWE7gE4HaU58KMf8EGLwgf37JS9A04cciMN/O2KEhKXq8DRVIBiUSNNNm1QrUYDdktcwXDgUkvnZYy6wHQwCa9ZITLhGdOgA9OmjR+fJkQ/FUEOw58jWMOomJwfo3l0cpTHRqpX0uHR4miqgcbSp94RC4jtp3161knLs2AGsWxdnm8p+bjS8oOnpwKJFMieYSoxBqU2EQjILVN++qpWU4/RpqX0Vt50LBqXnVVTkpqyE6dIFuOoqLW2dtxw5IhUMalJPGACuuUZiwjU0KMGguBFVzzXvxCnfgIjuJaL/IaKf24sf4gwuUlYm4bWjR0fMc6oHCxeKUYn7/ZOeLo4KXYLxI0hPl5pRp06pVuIj8+ZJDS8Ne8KhENC5M3D11XHsTCTfae5c7YJAbrxRcn1V2zonb5YZAMZBKgyfjFgShoiCRLSJiAqJ6Okof29ARO9Yf19CRF2s9V2I6DQRrbSWf7ihp0azfLkkmmnYaszJkZpKqalxHqBfP6lJpmFXIBiUMNVaVX04J0fK4V5/vWol5Th3TmxdTOHCFQkGZVzps89c1ZYo9eqJOzEnR20QSF0H23RiZtffQkRUB8DfAIyETC28jIhmMvP6iM0eAXCYma8gorsBvADgLutvm5m5j9u6aiyhkDxFmjpJhw1LoBhtSor0vGbPlp6YRj2woUPle4VCWpZOcx9m+bIjR0qtNY1YvBg4dizBNlUgIGHQoVCFOYPVEwxKpe7162V0TgVOnrxFRORFjY5BAAqZeQszFwN4G9ITimQcgGnW7+8BCBBplt6dLOTkSFqtZtWFt22TunsJd5zS08WH8vnnbshyjYYNJTxV9VCEb6xdK8VHNRzuyskRGxcIJHCQFi2Am27StjcMqL3XnBiUIQCWW0NTq4loDRGtduHcHQHsiPi801oXdRtmLgFwFIBdZa4rEa0govlE5DS3unZy6JBUkNNwuCshJ2kko0ZJD0zDN3cwKIUIN29WrcQH7BetXSFTI0IhSTht1izBAwWD0nDZt88VXW7RubP0TFTaOicGJR1AdwCjANwGYIz1M1Gi9TQqjv5Vts0eAJcyc18ATwKYTkRRbxMiepSICoiooEizKCDfmDdPhoI0NShdukj9vYRo21bCVDU1KICW0twnFJKiox0rtg3VsmeP5Fq68gjYB7HD8DUiGJQglxOK6sI7SWzcDqAzgDTr91NO9nPATuu4Np0A7K5sGyKqC6A5gEPMfJaZD1r6lgPYDCDqK4mZX2HmAcw8oG3bti7ITkLmzJGu+qBBqpWUo6REaiqNHu1SncpgUAbKVQfjV8DOr9GwhqW7nDwpFTE17J3Mmyc/XZHWp4+E32t4QYNBqV85X1H5Xidhw88C+AmAZ6xV9QD824VzLwPQnYi6ElF9AHcDmFlhm5kAHrR+vx1ALjMzEbW1nPogom6QHtQWFzTVTMJhGcjXrKbSsmVSjXfECJcOGAhIT2zBApcO6B6BgDzkmkWbussnn0golWsX1D3CYXEf9u7twsFSUqS2V26udnV1Bg+WWni5uWrO76SnMQHAWFihwsy8G0DTRE9s+US+B2A2gA0A3mXmdUT0SyIaa232KoDWRFQIGdqyQ4uHAlhNRKsgzvpvMbMmU8xoxpYt4vlOyBPpDXYSVkw1larixhvFC646uysKgYBEGC1frlqJh4TDEr+qWfQT8/k2lWsBgIGAjKNt3OjSAd2hUSOJGVD1CDiJ6yu2egUMAEQUawWcSmHmjwF8XGHdzyN+PwPgjij7ZQEwFY+dYDdV0tLU6ohCbq6MHrg2mV+DBvIyU9U8qwLbaObmapee4R65uVIGJ+YiWd5SWAjs3OnyI2AfLDc3zixJ70hLA372M0k78zuo04m9fpeI/gmgBRFNhszeOMVbWQbXCIeletxVV6lWUo7Tp6X2kOsdp0BACvjt3+/ygROjXTvxVWvYeXKHQ4ck8knjnrCr0rp1k2gSDS+o/T1VzMXjxCn/e8iwUhZkOuCfM/NfvBZmcAFmaUEFAtrNzvjpp5JB7olBAbTspQQC8r3PnFGtxAPy8+V+09SgdO7scHbGWAgE5K1dWurygRNj4ECgaVM1ts7RiCIzz2XmHzHzU8ysX8EkQ3TWrpWWuobDXeGwJJk5np3RKf36SdkPDVuOaWliTBYvVq3EA3JzZbpNzSIJy8rknZ+W5kGbKi1NCmGuWOHygROjbl2pPKGiTeUkyus4ER2rsOwgovetCCuDrth3lIatRtuXcNFFLh+4Th0pCqZhD2XYMJGnoa1LnHBY6szUr69aSTlWrwYOHvToEYj0o2hGWpok0+7YUf22buKkh/JHAD+CZK13AvAUxIfyNoDXvJNmSJhwWPr5l16qWkk5jhyRavOe2blA4Hx0m0Y0aybDETXOoOzaJdFOGjZcPPGf2Fx8saSma3hB7e/rtzQnBiXIzP9k5uPMfIyZXwFwCzO/A6Clx/oM8VJSIokPGj7k8+fLUISnBgXQ9kFftkxCiGsMtvdXw3stHJZ4lEsu8egEgYCkpp8969EJ4uPaa6V4hN+dJycGpYyI7iSiFGu5M+JvemX1GM5TUCBvLU39J40aeRg+e/XV0nrU0KCkpYkPV8Pcy/gJh2XmTFeyBt2juFj+nz19BNLSJGRRs3L2du5lOOxv7qUTg3IfgG8A2A9gn/X7/UTUCJKYaNARu2niWtage+TmijO+QQOPTkCkbSbzTTdpm3sZH55kDbrDsmVSDcbTjtOwYfK9NfWj7N4NfPGFf+d0Eja8hZlvY+Y2zNzW+r2QmU8z8yd+iDTEQTgsLUbN6pft3Stzens+OhIISDXY9eur39ZHGjaU8hgavn/iY/Nm8fxqOtxFlMDEbU5o0QIYMEDLFoKKkd9KDQoR/dj6+Rci+nPFxT+Jhpg5fVoSHjQc7vItcd8+gYYPelqaRB9plnsZH/b/r4b3WjgM9O0ro3GekpYm00OoKvFbCd26AZddpolBgdTXAoACAMujLAZdWbzYo6zBxMnNlUZd374en6hLF3miNOwKqMxkdp1wWErVJzz/gLucOiWPgS+PQCAgQTALF/pwMufYI795eRIE4weVGhRm/tD6OY2ZpwF4z/7d+mzQlXBYEh6GDlWt5AJ8LXwcCEgGt2Ylfvv3lxBiDW1dbNhZgxpWYrALH/tiUAYPlvwbDXvDgQBw+LDMBeMHThIbbySi9bB6LETUm4he9lyZIX7CYclYbppwUWhXsVNDfBsdSUuTuVE0mxbYzmTW8P0TG2vWSAVCTYe7fCt8rLrEbxX4PfLrJCzjTwBGA7AntFoFKR9v0JGjRyW8RcPhLk+TzKKhsR8lEBB/9vbtqpUkgO8X1DnhsM+FjwMB6QYcOODTCZ3RoQPQs6d/vWGntbwqJvDrVQ3NcJ4FCzzOGoyf3FyfCx9rXOJX49xL5+Tmiu+kUyfVSspx+LCCwscaO8bS0uS1UFzs/bmcGJQdRHQTACai+kT0FM477A26kZ8vsak33KBaSTmYRZonRfqqIi1N6uT78TTFwDXXSES3qqlaE8Z2Qms43LVwodxvvkobMEAK02l4Qe+4A3jiCQn+9BonBuVbAL4LqeW1E0Af67NBR/Lyzs9cqBGbNkkOiqc5AdFITZUnaelSn09cNXZ+RF6edrmXzlixQiox+H6MqzUAAAAgAElEQVRBqycvT25/Xwsf2w4bDXsoQ4cCv/61FOH2GieJjQeY+T5mbs/M7Zj5fmY+6L00Q8zY4RwaPuT5+fLTd2lDh8rb2xagEampkhO4datqJXFg/38OG6ZURjTy88VH7lklhspITZVE2hqRYBQfetVKMCSG3dfX1KB07AhcfrnPJ27VCujVS1uDAmgprXry88UZdvHFqpWU49AhYNUqRY+AfVINh738whiUmoTtP9FskiPbf5KaqihdITVV/CiaVYS9+mrxoySdQbH9Jxo2XJS2qfr1Ez9K0l1Q9zAGpSaRn6+t/2TfPoXvH9uPsmyZIgHRsf0o9uy5ScOKFcDx41oaFKVtKtuPYgxK5RBRCyJ6jIj+aGp5aYzxn1SO8aO4i/GfVE4t96M46aF8DKALgDUwtbz0ZcEC4z+pDONHcZf8/PNzzmiEUv+JTS33ozgxKA2Z+Ulmft3U8tIYu6/v2axV8aHcf2IzfLhUYNbQj9KunZbRptHR2H+iRZuqf/9a7UdxYlDeJKLJRNSBiFrZi+fKDLGhvK8fnY0bxX+ifJ6v1FTgzBlt81GSxo/y+eda+08aNVIck1K3rswelzQtBHdxYlCKAfwfgMU4P9xV4KUoQ4xo0dePjnL/ic3NN2vtR9m5U4pnao/xn1RPaiqwYYO0pGoZTgzKkwCuYOYuzNzVWrp5LcwQA5rnn3TqJFOTKMWe81xTgwJoKe1CbP9J+/aqlZTj0CGZtEyLR6AW+1GcGJR1AE55LcSQACb/xBma5qNcdZX4UbQ3KMZ/4oxanI/ixKCUAlhJRP80YcOaok1fvzwbN0r0pBYPOWD8KIny+ecyza1yh9iF2P6TgQNVK8F5P4oxKFH5AMDzABbBhA3rh/GfOMf4URLDdjQb/0n11FI/ipPikNOiLW6cnIiCRLSJiAqJ6Okof29ARO9Yf19CRF0i/vaMtX4TEY12Q09SolVfvzx5eZr4T2w09qPYjX4NpZ0nP19ma2rXTrWSchw8qGGbqpb6UZxkym8loi0Vl0RPTER1APwNQDqAngDuIaKeFTZ7BMBhZr4CwIsAXrD27QngbgDXAAgCeNk6Xu3D+E9iY/hwLf0oPXqIn1vbaNNz52Sidq3e2sKCBfJTK2n9+skU3NpeUG9wMuQ1AMBAa7kZwJ8B/NuFcw8CUMjMW5i5GMDbAMZV2GYcALs39B6AABGRtf5tZj7LzFsBFFrHq31o19cXNmwAioo0e8iB836UJUtUKymH9n4U23+i3QXVzH9iU0v9KE6GvA5GLLuY+U8A3JgLrSOAyKmFd1rrom7DzCUAjgJo7XBf13j8ceCPf/Tq6Algx0pq6iQFNJRm+1E0bDmmpgK7dslc89qhef7J4MHatankgm7cKDPL1RKcDHn1i1gGENG3ADR14dzRBkIqts0q28bJvnIAokeJqICICoqKimKUKKxYAUyfHteu3mL7TzR9yDt1Arp2Va2kAi1bAn36aDm2rXU+isb+k9WrtXwEaqUfxcmQ1x8ilt8A6AfgThfOvRNA54jPnQDsrmwbIqoLoDmAQw73BQAw8yvMPICZB7Rt2zYuoampYlSOHIlrd+/Iy9Og1sSF2P6T4cM185/YpKYCixfL0JdG2H4U7QyK7T/Rrrt53n+ioTSgb1/xo2h3Qb3DiUF5hJmHW8tIZn4UUo4lUZYB6E5EXYmoPsTJPrPCNjMBPGj9fjuAXGZma/3dVhRYVwDdAXiWXDB8OFBWJs+UVhj/SXxono+i3Tzzy5cb/0k81EI/ihOD8p7DdTFh+US+B2A2gA0A3mXmdUT0SyIaa232KoDWRFQIKQHztLXvOgDvAlgPIATgu8xcmqimyrjhBnlna3Vf2H19TR9yQEtpgsZ+lOHDgd27gcJC1UoisC/o0KFKZUTD9p/Ur69aSSXUMj9K3cr+QERXQcJymxNRRsSfmgFwZUpAZv4YMt9K5LqfR/x+BsAdlez7PCTh0nMaNhSjopVB0TJWUsjLAzp31tB/YtOypQxH5OcDzz6rWk05Iv0o3burVBJBfj5wzTXa+U8OHJA21a9/rVpJFUT6Ue66S6kUP6iqh9IDwBgALQDcFrH0AzDZe2l6oZ0fRdO+vrb5JxXR1I9y5ZUyb5U2jReTf5IYffsCzZpp2Rv2gkoNCjPPYOaJAMYw88SI5TFmXuSjRi1ITRU/ysKFqpVYaBoruX69tBy1dJJGkpoqyY0mH6Vqli8HTp7U8oLm5wONG2vXpipPLfOjOPGhFBLR/xDRK0T0mr14rkwztPKj2H19DZtm2vtPbDSv66WNH8X4TxInNRXYtAnYs0e1Es9xYlBmQMJ15wH4KGKpVTRsCNx4oybvH437+vn5wKWXAl26qFZSDS1anPejaIZ9WbUYJcnLA669Fogz5N4rDhwA1qzR8hG4kFqUj+LEoDRm5p8w87vMnGUvnivTEG38KMZ/4g7Dh2vrR+nQQQNbZ/wn7tCnj/hRlF9Q73FiUGYR0S2eK0kCUlPlpancj6JpX9/2nyTFQw6c96N89plqJeXQxo9SUACcOqXlBbX9JwMGqFbigFrkR3FiUH4AMSpniOgYER0nomNeC9OR66/XwI+icV8/afwnNkOGACkpWj7oqaky5P7llwpFGP+Je9QSP4qT4pBNmTmFmRsyczPrczM/xOmGFn4Ujfv6eXlJ4j+xSQI/ilJp+fla+k+KirRtU1WOHSVXw/0oTopDEhHdT0Q/sz53JiK9ikf5iO1HOXxYkQBNYyXLyuRZ0bZ+V2WkpsqQl2Z+lO7dFftRTP0ud7H9KFpEWniHkyGvlwHcCOBe6/MJyMRYtZLhwxX7UfLytOzrJ53/xMb2oyxerFpJOYjkXlNW12vZMuM/cZM6dWToUMPesJs4MSjXM/N3AZwBAGY+DECvt5mPDBokQ19K7ouiImDtWm0fckBLaVVz881a+1H27gW++ELByTX3nwwZAtSrp1pJjKSmysXcHbUweo3AiUE5Z02vywBARG0BlHmqSmOU+lE09p/k5wOXXZZE/hOb5s1lulZNDQqgSFp+PnDddUCbNgpOXjkat6mqpxbkozgxKH8G8D6AdkT0PIBPAPyvp6o0JzUVWLlSgR9F076+7T9JyoccOO9HOX1atZJyXHEFcMklCgxKcTHw6adaXlCN21TVUwvyUZxEeb0F4MeQybX2ABjPzP/1WpjOKMtH0TRWMmn9JzapqfISNfkogsk/8YZa4Edx0kMBgH0AFgJYBKAREfXzTpL+XH+9DH3l5vp40v37te3r24ErGkpzhp2P4usFdYbtR9m40ceT2hdUQ/9JXl6S+k9sbD/Krl2qlXiCk7DhXwFYDRn6sqcC/r3HurSmQQO5qX19/9gPeSDg40mdEQ4D3bolof/EpnlzCcPW0KDYl9tXaeGwDM9o5j/ZuxdYt07LR8A5Si6ofzjpodwJ4HJmTo2YCjjNa2G6k5YmyVX79/t0wnBYxl/79/fphM4oKZEefFqy3xFpaTIl8PHjqpWUo2tXCXYIh3064enTwKJFWl5Q+x2soTTn9OoFtG7t4wX1FycGZS1kki1DBL43NMJhYNgwqQukEZ9/Dhw9muStRkC+QEnJea+vJhCJtLw8oNSzSa4j+PRTycvR8IKGw+eLGyQtKSmSYBQOazLhjbs4MSi/AbCCiGYT0Ux78VqY7vTrJyMlvhiUbduALVu0fMhrRKsRAG66ScYyNRyKCASkwvXKlT6cLDf3fDFDzcjNlXdxnTqqlSRIIADs3Km4UJs3OGnuTgPwAoA1qMX5JxWpW1c6DL70XO2XnIYGJRyWdAXNphuPnUaNxKhoOBRhlxgJh30Y8QyHJXu3aVOPTxQbW7ZIu+qpp1QrcYHI4Y0rr1SrxWWc9FAOMPOfmTmPmefbi+fKkoBA4PyN7inhsLyxr7nG4xPFxpkzUu4p6XsnNoEAsGqVZM9pRIcOQM+ePti6I0ckZFjThgtQQ+61K64AOnXSsvGSKE4MynIi+g0R3UhE/ezFc2VJgC9+FGY5QVqadlUX7ZqKGr5/4sP+IhrmCQQCkvdUXOzhSRYskCxVDS9obq4Y1quuUq3EBSIdY2U1a9DHiUHpC+AGSHa8CRuOoGdPoH17jxsaGzZIvKSGD3k4LOPZw4apVuISAwbIUI+GLce0NAnA8jT3MhyWob8bbvDwJLFjt6kCAe3aVPETCAAHD0qPuAZRrQ+FmZOpSLSvEMmDnpsrN70nN7v9ctPUoAwcKNHMNQJfHWOxkZoqAULhsIf5huGwJFg1aODRCeJj7VoJz9fwEYgfe+wuNzfJw9bK4ySxsTkR/ZGICqzlD0TU3A9xyUAgIB2IDRs8OkFurmQMdu3q0Qni4/hxSduoEWPakQQCQGEh8NVXqpWUo0ULcch7Nry6b5+2WYM1JpIwko4dgR49tGy8JIKTIa/XAByHJDjeCeAYgNe9FJVM2De5J/dFaamM52v4kC9YIPI0lJYYkS1HzUhLkyGvEyc8OLjGb+1wWPzYl16qWonLBALyIHnqGPMXJwblcmZ+lpm3WMsvAHTzWliy0LWrLJ4YlM8/l8gbDd/a4bCMjNx0k2olLmNPeathy9HOvfSkKKmdNdhPr3ibkhKpZK3hI5A4gQBw8qR09WsITgzKaSIaYn8gosEA9KrzrZhAQDoSJSUuH9h+qWk412k4LIWPGzZUrcRlUlKkla5hJrNdaNoTWxcOi6NGs6zBggLg2DEtO06Jk5oqjlcNe8Px4sSgfBvA34hoGxFtB/BXAN/yVlZykZYm5UdWrHD5wLm5knty8cUuHzgxioqA1atraKsRkAu6Zw+waZNqJeVo3Fgmd3P9/bN1qyRTafjWtr+rhm2qxGnVShzyGvaG48XJfCgrmbk3gF4ArmPmvsxcs2LdEsQTP8rZs5I1qOFbW+PCx+5gfzENH/RAQEqwHDzo4kE1jyTs3VtGIWskgQCweLEMfdUAnER5tSCixwA8B+DXRPRnIvqz58qSiPbtZejd1ffP4sWSeKDpQ65h4WP36NbN5xK/zgkEZCTONuquEA5LL/jqq108aOKcPi21KjV8BNwjEADOnZPGYw3AyZDXxwC6QGp5LY9Y4oaIWhHRXCL60vrZspLtHrS2+ZKIHoxYn09Em4hopbUorySVlib3xNmzLh0wN1fG8zWc5Cg3V8vCx+5hJxjl5/tU4tc5AwcCF13k4rCXxpUYFi+W50nDkTj3sGcLqyF+FCcGpSEzP8nMrzPzNHtJ8LxPAwgzc3cAYetzOYioFYBnAVwPYBCAZysYnvuYuY+1+DUrSaUEAlKGZNEilw4YDkvmdgu9Zg746itJ06jRrUZAvuDhwx44xhKjXj1pY8yb59IB163TNmswHJZGi4ZtKvdo0kQqE7h2QdXixKC8SUSTiaiD1bNoZb3sE2EcpIoxrJ/jo2wzGsBcZj7EzIcBzAUQTPC8npGaKjf/nDkuHOzwYUk4GDnShYO5y+zZ8lNDae4yYoT8dOWCusvIkVL5fOtWFw5mX1D7+2rE7NnyrtWs8LH7jBwpKQK+zdbnHU4MSjGA/wOwGOeHuwoSPG97Zt4DANbPaENWHQHsiPi801pn87o13PUzIvV99WbNJKwzFHLhYOGwFI1LT3fhYO6SkyMJZpoNt7tP+/aSk+HKBXUX+7awbUFChEISSahZ1uC+fcDy5Vo+Au5jf8m5c9XqcAEnBuVJAFcwcxdm7mot1SY2EtE8IlobZRnnUFs0I2EnBtzHzNcBuNlavlGFjkftsjFFHpclDwYlAmfPngQPlJMjs3ddf70rutzi3DnpmQeD2g23e0MwKGOYR46oVlKOK6+Uajw5OQke6MQJydQO6tfxtzuGGkpzn379gDZtXLig6nFiUNYBOBXrgZl5BDNfG2WZAWAfEXUAAOtntL7eTgCdIz53ArDbOvYu6+dxANMhPpbKdLzCzAOYeUBbj2MP7Zs/oZYjs7QaR47Uzuu9eLHU8KoVDzkgX7S0VLtoLyKRFg4nWLUjP18OoOEFDYVkCqA+fVQr8YGUFGD0aHlxJHk5eycGpRTASiL6px0y7ELY8EwAdtTWgwBmRNlmNoBRRNTScsaPAjCbiOoSURsAIKJ6AMZA5r1XTu/eEn2ZUENjzRpg924t+/o5OWLjNPTfesONN0pPUcOWY3q6pC4kFG2akyPZkppN91taKu/WYFDetbWC9HTgwAEZ50tinFyuDwA8D2ARXAobBvBbACOJ6EsAI63PIKIBRDQVAJj5EIBfAVhmLb+01jWAGJbVAFYC2AVgSoJ6XMFuOc6dm0AZFnvMfvRo13S5RSgkfqIaU66+OurWFWd1KKRdGZbhwyXiK24XD7MYlLQ07crVL18uiZsadpy8Y9QoeYFo6LOLBSeZ8tOiLYmclJkPMnOAmbtbPw9Z6wuYeVLEdq8x8xXW8rq17iQz92fmXsx8DTP/gJm1SRZIT5cgrWXL4jxAKCSTtHfsWP22PrJnj/iHatVDDsgX3rVLwms1omlT6VjE/f4pLJQwMQ0vaE6OvFtrfCRhJG3bSqZwTTcohtgYMUK66XGNkhw/LmMYGg532X4hDaV5i/3C1XDYKxiUEdKdO+PY2f4+Gl7QUAgYNEj81LWK9HRJFzh0SLWSuDEGxWVatZLY+bgaGrm5EkqlaauxQwegVy/VSnymUyepq6NhyzGh8OFQCOjeXcrMaMTBg8CSJVo+At4TDIpTPomTHI1B8YBgUMpuxxylHApJXY3Bgz3RFS8lJeIXqjXhwhVJT5dJSDyZ2Sp+rrlGRkZjtnWnT0uEl4a9k7lzxb2joTTvGTQIaNlSy8aLUyo1KET0IRHNrGzxU2SyEQzKQxFTkrXtJA0EZNILjVi2TPxCtbLVCMgXP3dOu3pLcQeBLFggRkXDCxoKSS9/wADVShRQt644jjQMAnFKVT2U3wP4QxWLoRL695fx35gaGps2Adu3a/mQ5+SIX0jD6hz+MHiw1FzSsOWYni5z8Xz2WQw7hUIS2TVsmGe64qGsTKSNGqXdPF/+EQxKBMzq1aqVxEWlmXPMPN9PITWJinlKjmLp7ZeVhgYlFBK/UKtEK7glKw0aSM8xJ0dajhqN+wUC8vINhaRwrSNCISk+17ixl9JiZtUqKblSK4e7bOznPxSSxLYkw8l8KN2J6D0iWk9EW+zFD3HJTHq6+FA+/9zhDjk5wFVXSU0NjSgqEn+QhnbOX4JBmdXwiy9UKylHixbATTfFEIS2bRuwcaOWF9T+DqNGqdWhlA4dxJBoGFXoBCdt59cB/B1ACYDhAP4F4E0vRdUEYspTOnUKmD9fy4d8zhxplGsozV8iW46aEQxKw2XfPgcba94T7ttXuxmv/ScYlJnFjh1TrSRmnBiURswcBkDMvJ2ZnwNQk6e8cYW2bcWx6KihkZ8vMwlp2NcPhcQfVGNnZ3RK165Ajx5athxjCh/OyZFecI8eXkqKmaNHpQ6nho+A/6SnS5SFZjXknODEoJwhohQAXxLR94hoAqKXmzdUIBh0mKcUCgGNGmk3k1BZmbykRo+uRTWVqiIYFON/KuZaqZ7Su7dU26/W1hUXy0tKw/jvefOkhpeGHSf/uekmKYWgYeOlOpy8Jh4H0BjAYwD6A7gfwANeiqopjBkjL+WPPqpiI2ZgxgzxrjZs6Js2JyxaJD6UW29VrUQTxoyRnqRmk26lpMg1ysmppvpwOCwVJTW8oB98ICkYN9ygWokG1KsnrbgPP0y66sNODEoXZj7BzDuZeSIzZwLQazYeTRkwQBKts7Or2Gj5cplXNzPTN11Oyc6WlBgN3z9qGDZM3npVXlA1ZGTIsFGVqTLZ2dLy1axIVnGxvDvHjZN3qQFyQffulTkjkggnBuUZh+sMFUhJASZMkBGtSpOss7Ik7nPsWF+1VQezvH9GjapF1YWro149eevNnJngRCTuM2KE2IqsrEo2KCmRbsCYMdpVF87NFWOoYZtKHbfeKq25Si+onlSVKZ9ORH8B0DFyHhQiegMS8WVwQGYmcOZMJcOhzHLDDB+uXZLH559LnmVGhmolmuGoK+A/DRqIrfjgg0qy5hculPk2NLygWVlScajWJs5Go1kz6UlmZydV1nxVPZTdkLnjz6D8PCgzAeg3WYemDBkiEV9RR0nWrQO+/FLLppmmHSf1jBwpbz8Nh70yM8VmRJ10KztbfHSahVGVlp7vOGnmQlRPZqa06hwns6mnUoPCzKuseU+uiJgDZSaAQmY+7JvCJKdOHWD8eGDWLOmplCMrS6Jtxo9Xoq0y7I5TairQurVqNZrRsKEMR3zwgbwNNSIYlGDBC0ZJysrEoASDUkJGIzTuOKln7Fh5gSTRsJcTH8pcImpGRK0ArALwOhH90WNdNYrMTPGhXFCVOjtb6kRplsm1fr0khGvYcdKDzEwJf0to/l33adJEbMb771cIDlq6VKaV1vCCatpx0oPWraVVl5WVNMNeTgxKc2Y+BiADwOvM3B+AGe2MgeHDZWrycg2NwkIpAKfpQ65hx0kf0tPlLahhyzEjQyaYXLo0YmVWlgQUjBmjTFc0IjtOF12kWo2mZGZK6279etVKHOHEoNQlog4A7gQwy2M9NZL69aX3OmOGVEEHcP5lNGGCMl2VkZUluVUdOqhWoikXXSR5AtnZ2uUJjBkjtuNrW2ePXwYCUvhLI5YuFeNnhruqYPx4ad1p2HiJhhOD8ksAsyG+k2VE1A3Al97KqnlkZMicIvPtGs7Z2ZKoctllSnVVZPNmqfqqYcdJLzIz5W24bJlqJeVo0UJsx9fBQatWydzxGl7Q7GyZAuS221Qr0ZgOHaR1p2EQSDScGJQwM/di5u8AADNvsZIbDTEwerRUC8/KArBjhzTPNGya2fethh0nvRgzRt6GGrYcMzOBLVvEliArSxKixo1TLascGnec9CMjQy7m5s2qlVSLE4OyhIj+S0S3EGlWACiJaNQIuOUWcZiWZn0gKzVtNfbvr10Vff1o2bJCV0Afxo0TG5KdDfln6FCJXdeI1avF6Gn4COiH3fBMgl6KE4NyJYBXAHwDQCER/S8RXemtrJpJZqaUGF/8xibg2muBK/X6b9y5U4pZmofcIZmZ0mrUbHa9tm3FhmRNPyvOXA0vqKYdJz3p0kVaeRr2hitSrUFhYS4z3wNgEoAHASwlovlEdKPnCmsQt9wC1K/PyF51uZbDXe+/Lz81lKYn5boCepGRAazf3AAb0UPL8cvsbODmm4F2pm65MzIygCVLpNWnMU5mbGxNRD8gogIATwH4PoA2AH4IYLrH+moUzZoBo3p8hSxkoGyCnq3Ga67RbqoMfWnXTt6K772n3bCXbUOyOj8OdOyoVkwFNm6UIhEadpz0xf7P0rDxEomTIa/FAJoBGM/MtzJzNjOXMHMBgH94K6/mcffp1/AVLkP+wetUSynHli0SgXbXXaqVJBl33y3DSgUFqpWUo9Oh1bgZCzDtzF262Tq88YYkgBuDEgM9egB9+sh/nsZUaVCIqA6AWcz8K2a+oK/FzC94pqwmsnYtMgp/hxaNzmDKVL3iG159VUZvJk5UrSTJuOceCd+bOlW1kvJMnYpJdd7Al0Utz4eqa8C5c/JOvPVW4JJLVKtJMiZNAlas0Lq2V5UGhZlLAfT2SUvNZ+pUNKpXim/cx8jOBg4eVC1IKCkBXn9dfDydOqlWk2Q0bw7ceScwfXoVcxT4zOnTwJtv4vaMMjRvrpetmzVLAlMmT1atJAm57z6p0DBlimolleJkyGslEc0kom8QUYa9eK6spnHmDPDmm8CECZj0/UYoLpaPOvDxx8CePdIAMsTBpEliTN59V7USITsbOHIEjb/1AO67T1w81U5D7RNTp0rPxEz1GwctWgB33CGNl5MnVauJihOD0grAQQBpAG6zFr2KAiUD778vT/XkyejVCxg0SB4uHca3p06VhFwzM2Oc3HQTcPXV+rQcp0wBLr8cSE3F5Mkya/Fbb6kWJfm8oRDw8MOSE2qIg8mTgWPHgP/+V7WS6DCz7wvESM2FlHCZC6BlJduFAByB+HEi13cFsMTa/x0A9Z2ct3///qyM4cOZu3ZlLi1lZuZXXmEGmBctUieJmXnnTuaUFOZnnlGrI+n5wx/kgq5Zo1bHpk2i43//9+tV/fszX3cdc1mZQl3M/NxzIm3LFrU6kpqyMuYePZhvusnX0wIoYAfvWCdhw9OIqEXE55ZE9FqCduxpSEmX7gDC1udo/B8kobIiLwB40dr/MIBHEtTjLYWFQF4e8Mgj4vmGBAc1aaK+Ufv661Lf8BG9/wf154EHpAqoaofFq69KCNVDD329avJkYM2aChWIfaa0FHjtNZmfrGtXdTqSHiIZYl20SMsKxE6GvHox8xH7A8vkWn0TPO84ANOs36cBiFoonZnDAI5HrrPKv6QBeK+6/bUhSghV06YSIPTOO9KDVUFZmUgLBGSExJAAbdpI8sebb0aZSc0nioslhOq228qVitYhEG3uXOCrr4wz3hUeeEBKSqtuvETBiUFJIaKW9gdroq1ER0DbM/MeALB+xpIv2xrAEWa2Z87eCUCvzK1IqoiTnDQJOHUK+M9/1EgLh4Ft24wz3jUmTRI/mV1ywG9mzQL277/ggjZrJvlF//kPcPx4Jft6zNSpYnPNlNIu0K6dVGn417/EQaYRTgzKHwAsIqJfEdEvASwC8LvqdiKieUS0NsqSaPWeaAkclbq2iehRIiogooKioqIETx0HH30E7N0btWk2aBBw3XXqhr2mTAFatdKyMkdykpYm4zkqL2jHjlFDqCZPlsCgt9/2X9a+fTIX0IMPAg0a+H/+GsnkyZJ38MEHqpWUw0ktr38ByASwD0ARgAxmrjbglZlHMPO1UZYZAPZZk3bB+rk/Bs0HALQgIruX1AnA7ip0vMLMA5h5QFsVFVftOMkoc5zaw6HLl0u+kp8UFcm9+MAD5iF3jZQUcUbl5YnfzE+++gqYPVtCqOrUuZubmxcAAA36SURBVODPN9wA9OypZpTkX/+SXCfjp3ORESNkLiXNhr2c9FDAzOuZ+a/M/BdmdsMTNBNSZBLWzxlOd7QiDvIA3B7P/r6ycyeQkyO+k0riJO+/X17oft8X//qXjMaZMW2XmThRXuivvurveV+z4mQqeWsTybVeutSaJ8UnmOXeHjJEIqsNLmE3XubNk7pJuuAkFMztBeIHCUPCfsMAWlnrBwCYGrHdQkiv6DTEVzLaWt8NwFIAhQD+C6CBk/P6Hjb82GPMdepUGyf5wAPMjRsz79njj6zTp5k7dWK++WZ/zlfrmDCBuUUL5kOH/DnfsWPMbdow33JLlZsdOMDcqBHzPff4I4uZOStLQoX//W//zllr2LmTuV495m9+0/NTwWHYsBKDomrx1aBs3cpcvz7z5MnVbvrll8x16zJ/97vey2I+nzKRl+fP+Wodq1YxE/mX3GMneCxdWu2mzzwjm65Y4b2sc+eYr75alnPnvD9freR735NG6xdfeHoaY1BUG5QHH2Ru0IB5xw5Hm3/zm2JUNm/2VtbRo8ytWzOPGuXteWo9994r3YHdu709z/79zBddxJyZ6WjzQ4ek81RNZ8YVXntN3jBZWd6fq9ayd68Mb9x9t6enMQZFpUFZu1bSz596yvEuu3YxN2zIfP/9Hupi5p//XK56QYG356n1FBZKC+E73/H2PE88Iffahg2Od/ntb+UeWLjQO1mnTzNfeinzwIHqM/RrPD/9KXvd7TQGRaVBGT+euVkzGbSOgR//WEZKVq/2Rta+fcxNmjDfcYc3xzdU4NvfFqNSWOjN8bdvl2HVhx+OabeTJ5k7dGAeMsS7l/2LL8rbZd48b45viODwYeaWLZnT0z07hTEoqgzKZ5/Jf+uvfhXzrgcPMjdvzjx2rAe6mPkHP5Dh1o0bvTm+oQK7d8uw1333eXP8hx+WYdXt22Pe9e9/l9v0o4/cl2XHCAQC7h/bUAm/+51c0AULPDm8MSgqDEpZmRSBbNeO+fjxuA7x/PNyVT791F1p27ZJY/aRR9w9rqEann5aup2rVrl73A0bZKjriSfi2r24mPnyy5l79/66Xqlr/OIXcg8vWeLucQ1VcOoU8yWXMA8e7Em30xgUFQZlzhz5L/3zn+M+xIkTzO3bMw8b5u598dBD0pj96iv3jmlwgO0FHzPG3eNmZjI3bSpO+TiZPl1u1+nT3ZNVVCSyMjLcO6bBIf/8p1zQDz90/dDGoPhtUMrKmAcMYL7sMuYzZxI61F//KlcmJ8cdaevWSWP2ySfdOZ4hRn7zG7mgn3zizvGWLZPjPfdcQocpLZUeyuWXS4/FDZ58Uu619evdOZ4hBoqLma+4QuYqKClx9dDGoPhtUJ59Vv47p01L+FBnz8pD3qmTRH8lwvHjzH36SIxAUVHC0gzxcPKkDEdccUXiF+HQIearrmJu21acFQny8cdy237nO4n3iGfPFh9djDECBjd55x25oE8/7ephjUHx06BMmyb/lRMnujZOtWKFRGT16yfDYPFQUsJ8663ykLvV2zHEyaJFMuY4eLDE1MbD2bPio6tXj3n+fNek/ehHcvu++GL8x1izRoa6evVyxc4Z4qWsjPlb35ILOmWKa4c1BsUvg5KfLw94Wpo88C4ya5YMH4wdG18P9vvflyv88suuyjLEi916vOee2BseZWXiCAOY33zTVVmlpeKSIWL+4IPY99+9W3JOLrnEcR6vwUvOnWMOBqUlOWeOK4c0BsUPg7Jxo8R/X321xIJ7gO1Pefzx2PZ76SXZz/hNNMP2p/zsZ7Ht9+tfsxt+k8o4eZJ50CBJul62zPl+J06I67BxY+blyz2RZoiHo0fFl9KsmSRaJ4gxKF4blP37mbt1k7FsjyfJfvxxuVJ/+Yuz7WfMkNbm+PGu++YMiVJWJrHbAPMbbzjbxw7Huv9+T9PO9+6VmJKLL3aW2lJSwjxunNxrM2d6JssQL199JRmsl12WcOVZY1C8MijnzsnT07ev1EpZvDjxY1ZDSYkMe6WkSOmUyoYVioqY//hHaS0OGBC/78XgMcXFkvVXt64kHlVW72vvXqmTUr8+89ChCUcPOmHdOkmu7dGD+d13o4/ilpVJHcrbb5c3yEsveS7LEC8FBfJCGDhQuqFxYgyK2wZlyxapmXPJJfLf1r59fAPOcXLihBgVQAzLLbcwv/++PPDz5jHfdZe8dwDx+3pdk9CQIIcPS4VOQMa6x48Xp1lxsURQZGSIwQHEPxdjGZ9EyMtj7txZTt2mDfMPfyh5lIcPyxBs797yt0aNJLjRoDkzZjDfeGNCEYbGoLhlUEpLJVSq4pvcrcD9GNm8ubxds41Iy5Yy/YpXdcAMHrFpkxRxa9eu/AWNfJMroKTkQrtmS+vXTwI9jhxRIs0QDwmWQ3BqUEi2rR0MGDCACwoKYt/xiSeAFi1ketXOnd0XFgclJTIZ5EcfATffDGRkAI0aqVZliJviYmDWLGDOHCAQAMaO1WZu5n37ZIbPvXuB++4D+vVTrcjgN0S0nJkHVLudMSgGg8FgqAqnBsXRnPIGg8FgMFSHMSgGg8FgcAVjUAwGg8HgCsagGAwGg8EVjEExGAwGgysYg2IwGAwGVzAGxWAwGAyuYAyKwWAwGFyhViU2ElERgO1x7t4GwAEX5fhNsusHkv87GP1qSXb9gLrvcBkzt61uo1plUBKBiAqcZIrqSrLrB5L/Oxj9akl2/YD+38EMeRkMBoPBFYxBMRgMBoMrGIPinFdUC0iQZNcPJP93MPrVkuz6Ac2/g/GhGAwGg8EVTA/FYDAYDK5gDEo1EFGQiDYRUSERPa1ajxOI6DUi2k9EayPWtSKiuUT0pfWzpUqNVUFEnYkoj4g2ENE6IvqBtT4pvgMRNSSipUS0ytL/C2t9VyJaYul/h4jqq9ZaFURUh4hWENEs63Oy6d9GRGuIaCURFVjrkuIeAgAiakFE7xHRRutZuFF3/cagVAER1QHwNwDpAHoCuIeIeqpV5Yg3AAQrrHsaQJiZuwMIW591pQTAD5n5agA3APiu9f+eLN/hLIA0Zu4NoA+AIBHdAOAFAC9a+g8DeEShRif8AMCGiM/Jph8AhjNzn4hQ22S5hwDgJQAhZr4KQG/ItdBbv5N5gmvrAuBGALMjPj8D4BnVuhxq7wJgbcTnTQA6WL93ALBJtcYYvssMACOT8TsAaAzgcwDXQxLS6lrry91bui0AOkFeWGkAZgGgZNJvadwGoE2FdUlxDwFoBmArLD93sug3PZSq6QhgR8Tnnda6ZKQ9M+8BAOtnO8V6HEFEXQD0BbAESfQdrOGilQD2A5gLYDOAI8xcYm2i+730JwA/BlBmfW6N5NIPAAxgDhEtJ6JHrXXJcg91A1AE4HVr2HEqETWB5vqNQakairLOhMX5BBFdBCALwOPMfEy1nlhg5lJm7gNp6Q8CcHW0zfxV5QwiGgNgPzMvj1wdZVMt9UcwmJn7QYasv0tEQ1ULioG6APoB+Dsz9wVwEroNb0XBGJSq2Qmgc8TnTgB2K9KSKPuIqAMAWD/3K9ZTJURUD2JM3mLmbGt1Un0HAGDmIwDyIb6gFkRU1/qTzvfSYABjiWgbgLchw15/QvLoBwAw827r534A70MMe7LcQzsB7GTmJdbn9yAGRmv9xqBUzTIA3a3olvoA7gYwU7GmeJkJ4EHr9wchfgktISIC8CqADcz8x4g/JcV3IKK2RNTC+r0RgBEQh2oegNutzbTVz8zPMHMnZu4Cuedzmfk+JIl+ACCiJkTU1P4dwCgAa5Ek9xAz7wWwg4h6WKsCANZDc/0msbEaiOgWSOusDoDXmPl5xZKqhYj+AyAVUpl0H4BnAXwA4F0AlwL4CsAdzHxIlcaqIKIhABYCWIPzY/j/A/GjaP8diKgXgGmQeyYFwLvM/Esi6gZp8bcCsALA/cx8Vp3S6iGiVABPMfOYZNJvaX3f+lgXwHRmfp6IWiMJ7iEAIKI+AKYCqA9gC4CJsO4naKrfGBSDwWAwuIIZ8jIYDAaDKxiDYjAYDAZXMAbFYDAYDK5gDIrBYDAYXMEYFIPBYDC4gjEoBoNCiOhbRPSA9ftDRHRJxN+mJkkxUoMBgAkbNhi0gYjyITkfBaq1GAzxYHooBkOcEFEXa66KaUS02pq7ojERBayCfmusuWkaWNv/lojWW9v+3lr3HBE9RUS3AxgA4C1r/o5GRJRPRAOs7e6xjreWiF6I0HCCiJ635l75jIjaq/i/MBgAY1AMhkTpAeAVZu4F4BiAJyHz0dzFzNdBsrS/TUStAEwAcI217a8jD8LM7wEoAHAfy/wdp+2/WcNgL0BqavUBMJCIxlt/bgLgM5a5VxYAmOzZNzUYqsEYFIMhMXYw86fW7/+G1FzaysxfWOumARgKMTZnAEwlogwAp2I4x0AA+cxcZJWPf8s6JgAUQ+YrAYDlkHlwDAYlGINiMCSGIyekZQgGQSoojwcQiuEc0UrH25zj847QUkiPyGBQgjEoBkNiXEpEN1q/3wNgHoAuRHSFte4bAOZbc7s0Z+aPATwOGbqqyHEATaOsXwJgGBG1saalvgfAfDe/hMHgBqY1YzAkxgYADxLRPwF8CZmH/TMA/7XmDlkG4B+QCr0ziKghpMfxRJRjvQHgH0R0GjLFLgCZmY+InoGUjycAHzOzVmXLDQbAhA0bDHFjTU88i5mvVSzFYNACM+RlMBgMBlcwPRSDwWAwuILpoRgMBoPBFYxBMRgMBoMrGINiMBgMBlcwBsVgMBgMrmAMisFgMBhcwRgUg8FgMLjC/wf20cHUHMj+FgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "k64 = periodic_atom_chain64.k()\n",
    "[ks64, pw64] = k64.eigenstates()\n",
    "plt.plot(np.real(pw64[2]),'r')\n",
    "plt.plot(np.imag(pw64[2]),'b')\n",
    "plt.plot(np.abs(pw64[2]),'g')\n",
    "plt.xlabel('position')\n",
    "plt.ylabel('crystal momentum eigen vector')\n",
    "plt.show()\n",
    "plt.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Relationship of crystal momentum eigen-vectors with the Hamiltonian"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [],
   "source": [
    "Hamt8 = periodic_atom_chain8.Hamiltonian()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "pw8_M = np.array([pw8[0].full(),pw8[1].full(),pw8[2].full(),pw8[3].full(),\n",
    "       pw8[4].full(),pw8[5].full(),pw8[6].full(),pw8[7].full()])\n",
    "pw8_M = np.squeeze(pw8_M, axis=2)  # changing shape from (8,8,1) to (8,8)\n",
    "     # pw8_M is a matrix of 8 columns, each of which are the eigenvectors\n",
    "     # of the crystal momentum operator\n",
    "pw8_M = Qobj(pw8_M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[8], [8]], shape = (8, 8), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}-2.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & -1.414 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 1.414 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 2.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.414 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & -1.414\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[8], [8]], shape = (8, 8), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[-2.          0.          0.          0.          0.          0.\n",
       "   0.          0.        ]\n",
       " [ 0.         -1.41421356  0.          0.          0.          0.\n",
       "   0.          0.        ]\n",
       " [ 0.          0.          0.          0.          0.          0.\n",
       "   0.          0.        ]\n",
       " [ 0.          0.          0.          1.41421356  0.          0.\n",
       "   0.          0.        ]\n",
       " [ 0.          0.          0.          0.          2.          0.\n",
       "   0.          0.        ]\n",
       " [ 0.          0.          0.          0.          0.          1.41421356\n",
       "   0.          0.        ]\n",
       " [ 0.          0.          0.          0.          0.          0.\n",
       "   0.          0.        ]\n",
       " [ 0.          0.          0.          0.          0.          0.\n",
       "   0.         -1.41421356]]"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pw8_M * Hamt8 * pw8_M.dag()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So, eigenvectors of the crystal momentum are eigenvectors of the Hamiltonian as \n",
    "well. This is a consequence of the translational symmetry of the lattice. The generator of the lattice translational operator is the crystal momentum operator. Due to the translational symmetry, the lattice translational operator, the crystal momentum and the Hamiltonian all commute with each other. So they can have simultaneous eigenstates."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[8], [8]], shape = (8, 8), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[8], [8]], shape = (8, 8), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[0. 0. 0. 0. 0. 0. 0. 0.]\n",
       " [0. 0. 0. 0. 0. 0. 0. 0.]\n",
       " [0. 0. 0. 0. 0. 0. 0. 0.]\n",
       " [0. 0. 0. 0. 0. 0. 0. 0.]\n",
       " [0. 0. 0. 0. 0. 0. 0. 0.]\n",
       " [0. 0. 0. 0. 0. 0. 0. 0.]\n",
       " [0. 0. 0. 0. 0. 0. 0. 0.]\n",
       " [0. 0. 0. 0. 0. 0. 0. 0.]]"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# A check that the Hamiltonian and the crystal momentum operator do indeed commute\n",
    "Hamt8 * k8 - k8 * Hamt8"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Relationship of Expansion in momentum eigenkets to fft"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We form a Gaussian state in the position basis and evaluate its Discrete Fourier Transform(DFT) for evaluating its frequency components."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {},
   "outputs": [],
   "source": [
    "xs8 = np.arange(0,8)\n",
    "Gaussian8_sum_to_1 = 1/np.sqrt(2*np.pi* 0.5 **2) * np.exp(-(xs8 - 3)**2/2/0.5**2)\n",
    "Gaussian_state8 = np.sqrt(Gaussian8_sum_to_1) # A normalized Gaussian state"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(Gaussian_state8,'r')\n",
    "plt.xlabel('position')\n",
    "plt.ylabel('Probability Amplitude')\n",
    "plt.show()\n",
    "plt.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Inner Product of crystal momentum eigenket and a state"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {},
   "outputs": [],
   "source": [
    "k_exps = pw8_M.dag() * Gaussian_state  # ket vector of inner products with crystal\n",
    "                                       # momentum eigenkets\n",
    "k_exps = k_exps/ np.sqrt( np.sum(np.multiply(k_exps, np.conj(k_exps)))) # normalization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### fft spectrum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "dft_Gaussian_state = np.fft.fft(Gaussian_state)\n",
    "shf_dft_Gaussian_state = np.roll(dft_Gaussian_state,4)\n",
    "sq_dft_Gaussian_state = np.multiply(shf_dft_Gaussian_state, np.conj(shf_dft_Gaussian_state))\n",
    "norml_dft_Gaussian_state = shf_dft_Gaussian_state / np.sqrt(np.sum(sq_dft_Gaussian_state))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Comparison"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(np.abs(norml_dft_Gaussian_state),'b')\n",
    "plt.plot(np.abs(k_exps),'ro')\n",
    "plt.xlabel('frequency number')\n",
    "plt.ylabel('frequency Amplitude')\n",
    "plt.show()\n",
    "plt.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Therefore, expanding a state in momentum eigenkets is equivalent of a DFT for a periodic lattice."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "QuTiP: Quantum Toolbox in Python\n",
      "================================\n",
      "Copyright (c) QuTiP team 2011 and later.\n",
      "Original developers: R. J. Johansson & P. D. Nation.\n",
      "Current admin team: Alexander Pitchford, Paul D. Nation, Nathan Shammah, Shahnawaz Ahmed, Neill Lambert, and Eric Giguère.\n",
      "Project Manager: Franco Nori.\n",
      "Currently developed through wide collaboration. See https://github.com/qutip for details.\n",
      "\n",
      "QuTiP Version:      4.5.0.dev0+c8074e5\n",
      "Numpy Version:      1.15.2\n",
      "Scipy Version:      1.1.0\n",
      "Cython Version:     0.28.5\n",
      "Matplotlib Version: 3.0.0\n",
      "Python Version:     3.5.4\n",
      "Number of CPUs:     2\n",
      "BLAS Info:          INTEL MKL\n",
      "OPENMP Installed:   True\n",
      "INTEL MKL Ext:      True\n",
      "Platform Info:      Linux (x86_64)\n",
      "Installation path:  /home/ssaumya7/Desktop/GSOC/Aug5/qutip/qutip\n",
      "==============================================================================\n",
      "Please cite QuTiP in your publication.\n",
      "==============================================================================\n",
      "For your convenience a bibtex reference can be easily generated using `qutip.cite()`\n"
     ]
    }
   ],
   "source": [
    "qutip.about()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "@article{qutip2,\n",
      "doi = {10.1016/j.cpc.2012.11.019},\n",
      "url = {https://doi.org/10.1016/j.cpc.2012.11.019},\n",
      "year  = {2013},\n",
      "month = {apr},\n",
      "publisher = {Elsevier {BV}},\n",
      "volume = {184},\n",
      "number = {4},\n",
      "pages = {1234--1240},\n",
      "author = {J.R. Johansson and P.D. Nation and F. Nori},\n",
      "title = {{QuTiP} 2: A {P}ython framework for the dynamics of open quantum systems},\n",
      "journal = {Computer Physics Communications}\n",
      "}\n",
      "@article{qutip1,\n",
      "doi = {10.1016/j.cpc.2012.02.021},\n",
      "url = {https://doi.org/10.1016/j.cpc.2012.02.021},\n",
      "year  = {2012},\n",
      "month = {aug},\n",
      "publisher = {Elsevier {BV}},\n",
      "volume = {183},\n",
      "number = {8},\n",
      "pages = {1760--1772},\n",
      "author = {J.R. Johansson and P.D. Nation and F. Nori},\n",
      "title = {{QuTiP}: An open-source {P}ython framework for the dynamics of open quantum systems},\n",
      "journal = {Computer Physics Communications}\n",
      "}\n"
     ]
    }
   ],
   "source": [
    "qutip.cite()"
   ]
  }
 ],
 "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": "3.5.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}