{
 "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 \\ell=5, 20 and 100.\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": 121,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1222a0ff0f0>"
      ]
     },
     "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 # thespatial extent of the system\n",
    "x2=3 # upper limit of psi(x)\n",
    "x1=2 # lower limit of psi(x)\n",
    "l=400 # 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",
    "# Since Python indices might best begin with zero, our index \"m\" will \n",
    "# run from 0 to 2l.  Thus, where ever m would appear, we will instead\n",
    "# use m-l so that the variable m-l will go from -l to +l as needed.\n",
    "for m in range(0,2*l+1):\n",
    "    psi_tilde[m] = L**0.5*1j/(2*(np.pi)*(m-l + d))*(np.e**(-2j*np.pi*(m-l)*x2/L)-np.e**(-2j*np.pi*(m-l)*x1/L))\n",
    "#    print('psi_tilde = ', psi_tilde[m], 'm = ',m-l)\n",
    "# correct the value of psi_tilde for the singular case m-l=0\n",
    "psi_tilde[l] = (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(0,2*l+1):\n",
    "        psi[n] = psi[n]+psi_tilde[m]*np.e**(+2j*np.pi*(m-l)*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()"
   ]
  },
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   "outputs": [],
   "source": []
  }
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