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    "<center> <h2>A Quantum Bead on a Circle </h2> </center>\n",
    "\n",
    "A bead represented as the wave packet\n",
    "$$\n",
    " \\psi(\\phi,0) =  e^{-ik_0\\phi} e^{-8*(\\phi-\\pi)^2}\n",
    "$$\n",
    "slides frictionlessly on circle. \n",
    "With $\\phi$ as the only spatial coordinate, and with periodic boundary conditions, the time-dependent Schrodinger equation becomes\n",
    "$$\n",
    "\\frac{1}{b^2} \\frac{\\partial^2\\psi}{\\partial\\phi^2}  =  i \\frac{\\partial \\psi}{\\partial t}.\n",
    "$$\n",
    "Here the wave function  is complex $\\psi(\\phi,t) =  R(\\phi,t) + i I(\\phi,t)$."
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    "\"\" \"From COMPUTATIONAL PHYSICS\" & \"COMPUTER PROBLEMS in PHYSICS\",\n",
    "    \"    by RH Landau, MJ Paez, and CC Bordeianu (deceased).\n",
    "    \"    Copyright R Landau, Oregon State Unv, MJ Paez, Univ Antioquia,\n",
    "    \"    C Bordeianu, Univ Bucharest, 2020.\n",
    "    \"    Please respect copyright & acknowledge our work. \"\"\n",
    "\n",
    "# QMbeadOnCircle.ipynb: a quantum bead slides on a frictionless circle\n",
    "\n",
    "%matplotlib notebook\n",
    "\n",
    "from numpy import *\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.animation as animation\n",
    "\n",
    "dt = 0.0005                  \n",
    "k0  = 0.1                   # wave packet wave vector\n",
    "xmax = 2*pi                # maximum phi value 2pi\n",
    "b = 4                        # radius of circle\n",
    "b2 = b**2.\n",
    "nmax = 201                   # for arrays\n",
    "dx  =  2*pi/(nmax-1)\n",
    "dx2  =  dx*dx\n",
    "fc = dt/dx2/b2               # factor\n",
    "R  =  np.zeros((nmax,2), float) # Real part wave function\n",
    "I  =  np.zeros((nmax,2), float) # Imaginary part wave function\n",
    "xx  =  np.arange(0,xmax+dx,dx)  # values for the x axis\n",
    "\n",
    "def initial():             # Gaussian wave packet \n",
    "    R[ : ,0] = exp(-8*(xx-pi)**2) * cos(k0*xx)    # Re I\n",
    "    I[ : ,0] = exp(-8*(xx-pi)**2) * sin(k0*xx)    # Im I     \n",
    "    \n",
    "initial()\n",
    "fig = plt.figure()              \n",
    "ax  =  fig.add_subplot(111, autoscale_on = False,xlim = (0,2*pi),ylim = (0,1.5))\n",
    "ax.grid()   \n",
    "plt.title(\"Bead on circle with periodic boundary conds.: $\\psi(0,t) = \\psi(2\\pi,t)$\")\n",
    "plt.xlabel(\"angle $0\\leq \\phi\\leq 2\\pi $\")\n",
    "plt.ylabel(\"$\\psi^2 = R^2+I^2$\")\n",
    "line,  =  ax.plot(xx, R[ :,0]**2+ I[ :,0]**2)  # plot first lin\n",
    "\n",
    "def animate(dum):                   # this is the animation\n",
    "   xx  =  arange(0,xmax+dx,dx)\n",
    "   for i in range (1,nmax-1): R[i+1,1] = R[i,0] -fc*(I[i+1,0]+I[i-1,0]-2*I[i,0])  \n",
    "   for i in range (1,nmax-1): I[i+1,1] = I[i,0] + fc*(R[i+1,1]+ R[i-1,1]-2*R[i,1])\n",
    "   R[0,1] = R[200,0] -fc*(I[0,0]+I[199,0]-2*I[200,0])    # for boundary Cond              \n",
    "   R[1,1] = R[0,0] -fc*(I[1,0]+I[200,0]-2*I[0,0])  \n",
    "   I[0,1] = I[200,0] + fc*(R[0,1]+ R[199,1]-2*R[200,1])  # for boundary cond               \n",
    "   line.set_data(xx,R[ : ,-1]**2+I[ : ,-1]**2)           #plot new line \n",
    "   for i in range (0,nmax):             # new R is now old R\n",
    "       R[i,0] = R[i,1]\n",
    "       I[i,0] = I[i,1]\n",
    "   '''\n",
    "   R[200,1] = R[0,0]                     # for periodic boundary conditions\n",
    "   I[200,1] = I[0,0]\n",
    "   R[0,1] = R[200,0]\n",
    "   I[0,1] = I[200,0]  \n",
    "   '''\n",
    "   return line,\n",
    "ani  =  animation.FuncAnimation(fig, animate)  # call the animation   \n",
    "plt.show() "
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