{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A particle moves in a periodic box of length $L=10$.  Consider a position-space wavefunction $\\psi(x)$ which is unity in the interval $x_1=2\\le x \\le x_2=3$ and zero for all other values of $x$.  Approximate this wave function as a Fourier series with $2\\ell+1$ terms with $-\\ell \\le m \\le \\ell$ and plot the result for values of $\\ell$ that you can choose.\n",
    "\n",
    "First, determine the Fourier amplitudes:\n",
    "\\begin{eqnarray}\n",
    "\\widetilde \\psi_m &=& \\frac{1}{\\sqrt{L}}\\int_{x_1}^{x_2}dx e^{-i 2\\pi mx/L} \\psi(x) \\\\\n",
    "                  &=&\\sqrt{L}\\frac{\\Bigl\\{e^{-i2\\pi m x_2/L} - e^{-i2\\pi m x_1/L}\\Bigr\\}}\n",
    "                                                 {-2\\pi i m}\n",
    "\\end{eqnarray}\n",
    "Note the case $m=0$ is special and should be evaluated directly:\n",
    "$$\n",
    "\\widetilde \\psi_0 = \\frac{x_2-x_1}{\\sqrt{L}}\n",
    "$$\n",
    "We can then obtain an approximate expression for $\\psi(x)$ by summing $m$ over the specified range:\n",
    "$$\n",
    "\\psi^{(\\ell)}(x) = \\sum_{m=-\\ell}^{\\ell}e^{i2\\pi m x/L}\\widetilde \\psi_m\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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F4HUAQIERrJCtouwKZPA6AKALBCtkqyjBiooVAKALBCtki2AFACgwghWyxeB1AECBEayQ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\nySXSV74S374+8Yk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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1222c0adc18>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# This is the first version of this example of a simple square wave function \n",
    "# for which the coefficients of the Fourier series are computed analytically \n",
    "# and then used to reconstruct the wave function from a truncated version of\n",
    "# the Fourier series, including only a finite number.  This serves as both \n",
    "# sample code and and easy place to experiment with Fourier series to see how\n",
    "# the description improves as more terms are included and to recognize the \n",
    "# difficulty that the Fourier series has in representing the sharp edges of\n",
    "# the square wave function.\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "# assign constants\n",
    "L=10 # the spatial extent of the system\n",
    "x2=3 # upper limit of psi(x)\n",
    "x1=2 # lower limit of psi(x)\n",
    "l=100 # The largest value of m that is included: |m| \\le l or (-l \\le m \\le +l)\n",
    "Dx = 0.01 # Accuracy for plotting position\n",
    "Nx = int(L/Dx) # Number of positions plotted.\n",
    "# prevent dividing by zero:\n",
    "d=0.000001\n",
    "psi_tilde = np.zeros((2*l+1),dtype=complex) # Fourier coefficients\n",
    "# Calculate psi_tilde from analytic results in Latex at the top\n",
    "# We take advantage of Python allowing negative values for the indices of\n",
    "# arrays but must be aware that negative values are stored starting from \n",
    "# the uppper limit of the array so if the array is insufficiently large \n",
    "# strange things might happen.\n",
    "for m in range(-l,l+1):\n",
    "    psi_tilde[m] = L**0.5*1j/(2*(np.pi)*(m + d))*(np.e**(-2j*np.pi*m*x2/L)-np.e**(-2j*np.pi*m*x1/L))\n",
    "#    print('psi_tilde = ', psi_tilde[m], 'm = ',m)\n",
    "# correct the value of psi_tilde for the singular case m-l=0\n",
    "psi_tilde[0] = (x2-x1)/L**0.5\n",
    "# Reconstruct psi\n",
    "psi = np.zeros((Nx+1),dtype=complex) # Reconstructed wave function, one element for each position. \n",
    "x = np.zeros((Nx+1)) # Values for each of the Nx positions, needed for plotting our results.\n",
    "# compute the value of the position and the wave function for each position:\n",
    "for n in range(0,Nx+1):\n",
    "    x[n]=n*Dx\n",
    "    psi[n] = 0+0j\n",
    "    # sum over the 2l+1 values of m\n",
    "    for m in range(-l,l):\n",
    "        psi[n] = psi[n]+psi_tilde[m]*np.e**(+2j*np.pi*m*x[n]/L)/L**0.5 \n",
    "fig = plt.figure(figsize=(10,4))\n",
    "ax = fig.add_subplot(1,1,1)\n",
    "ax.plot(x,psi.real,'r-')     #Add a curve described by the arrays x and psi.real to Plot, choose a red solid curve.\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}