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    "\"\"\" From: \"A SURVEY OF COMPUTATIONAL PHYSICS\" \n",
    "   by RH Landau, MJ Paez, and CC BORDEIANU \n",
    "   Copyright Princeton University Press, Princeton, 2008.\n",
    "   Electronic Materials copyright: R Landau, Oregon State Univ, 2008;\n",
    "   MJ Paez, Univ Antioquia, 2008; and CC BORDEIANU, Univ Bucharest, 2008.\n",
    "   Support by National Science Foundation\n",
    "\"\"\"\n",
    "# Version in which the real part is updated, than the imaginary part\n",
    "#is updated using the new values\n",
    "# Harmonic oscillator using MatPlotLib\n",
    "# runs very slowly\n",
    "from numpy import *\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.animation as animation\n",
    "%matplotlib notebook\n",
    "#psr=np.zeros(())\n",
    "#initialize wave function, probability, potential\n",
    "dx = 0.02;    dx2 = dx*dx;  k0 = 3.0*pi;  dt = dx2/4.0;  xmax = 7.5\n",
    "xs = np.arange(-xmax,xmax+dx/2,dx)             # array of x positions\n",
    "nmax=751\n",
    "R=np.zeros((nmax,2),float)\n",
    "I=np.zeros((nmax,2),float)\n",
    "v=np.zeros((nmax),float)\n",
    "#psr = exp(-0.5*(xs/0.5)**2) * cos(k0*xs)    # Re wave function Psi\n",
    "#psi = exp(-0.5*(xs/0.5)**2) * sin(k0*xs)    # Im wave function Psi\n",
    "R[:,0]= exp(-0.5*(xs/0.5)**2) * cos(k0*xs) \n",
    "I[:,0]= exp(-0.5*(xs/0.5)**2) * sin(k0*xs) \n",
    "  \n",
    "#print(R[750,0],I[750,0])\n",
    "v[:] = 5.0*xs**2\n",
    "fig=plt.figure()                           # figure to plot (a changing line)\n",
    "# select axis; 111: only one plot, x,y, scales given\n",
    "ax = fig.add_subplot(111, autoscale_on=False, xlim=(-xmax,xmax), ylim=(0, 1.1))\n",
    "ax.grid()                                                       # plot a grid\n",
    "plt.title(\"Harmonic Oscillator\")\n",
    "line, = ax.plot(xs, R[:,0]**2+I[:,0]**2, lw=2)             # x axis, y values, linewidth=2\n",
    "\n",
    "def animate(dum):\n",
    "   R[1:-1,1] = R[1:-1,0] - (dt)*(I[2:,0]+I[:-2,0]-2*I[1:-1,0])/dx2 +dt*v[1:-1]*I[1:-1,0]\n",
    "   line.set_data(xs,R[:,1]*R[:,0]+I[:,0]**2+0.001*v[:])\n",
    "   I[1:-1,1] = I[1:-1,0] + (dt)*(R[2:,1]+R[:-2,1]-2*R[1:-1,1])/dx2 -dt*v[1:-1]*R[1:-1,1]\n",
    " \n",
    "   #R[:,0]=R[:,1]\n",
    "   #I[:,0]=I[:,1]\n",
    "   for i in range(0,nmax):\n",
    "     R[i,0]=R[i,1]\n",
    "     I[i,0]=I[i,1]\n",
    "   \n",
    "   return line,\n",
    " \n",
    "ani = animation.FuncAnimation(fig, animate,frames=1000,interval=1)   \n",
    "\n",
    "plt.show()\n"
   ]
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