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    "### <span style=\"color:blue\">Exploration of discrete states</span>  \n",
    "#### J Wang, _UMass Dartmouth_, <u>www.faculty.umassd.edu/j.wang/</u>\n"
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  {
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   "source": [
    "### <span style=\"color:blue\">1 Discrete energies (<em>not as weird as it seems</em>)</span> \n",
    "-  Visually manipulate possible wave function of Schrodiner eqn in a 1D box.\n",
    "-  Wave function ($A,B,k = $ adjustable parameters): $ \\frac{d^2\\psi}{dx}+ k^2 \\psi=0,  \\quad \\psi = A \\cos(kx )+ B \\sin(k x ), \\quad k = \\frac{\\sqrt{2mE}}{\\hbar}$\n",
    "-  Activity: Interactively inspect the wave function to observe that discrete energies are a natural result of finding continuous wave functions. "
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  {
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   "source": [
    "#### <span style=\"color:blue\">Load libraries, initialize</span>, actual templates for my class to use in HW"
   ]
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   "execution_count": 1,
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   "source": [
    "from ipywidgets import interact\n",
    "import matplotlib.pyplot as plt\n",
    "from numpy import *\n",
    "\n",
    "a = 1.0            # width\n",
    "x = linspace(0, a, 100)  # grid"
   ]
  },
  {
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   "source": [
    "#### <span style=\"color:blue\">Main visualization</span>  $\\quad \\psi = A \\cos(kx )+ B \\sin(k x ), \\quad k=n\\pi/a$"
   ]
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   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
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   "source": [
    "def psi(A, B, k):\n",
    "    wf = A*cos(k*x) + B*sin(k*x)\n",
    "    return wf\n",
    "\n",
    "def plotwf(A=1, B=1, n=1.2):\n",
    "    plt.plot([0, 0, a, a],[1, 0, 0, 1], 'k-') # the well\n",
    "    wf = psi(A, B, n*pi/a)\n",
    "    plt.plot(x, wf, 'b-', lw=2)         # plot w.f.\n",
    "    plt.xlabel('x'), plt.ylabel('Wave function')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "<span style=\"color:blue\"><b>Interactive plotting</b></span>, $ \\psi = A \\cos(kx )+ B \\sin(k x ), \\quad  k=n\\frac{\\pi}{a}$"
   ]
  },
  {
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   "execution_count": 3,
   "metadata": {
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   "outputs": [
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       "model_id": "57f2cb12b5794da0b1a0c4d4ae62e13d",
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      "text/plain": [
       "interactive(children=(FloatSlider(value=1.0, description='A', max=1.0), FloatSlider(value=1.0, description='B'…"
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   "source": [
    "interact(plotwf, A = (0,1,0.1), B=(0,1,0.1), n=(0,5,.2));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "### <span style=\"color:blue\">Observations</span>,  $ \\psi = A \\cos(kx )+ B \\sin(k x ), \\quad k=n\\pi/a$\n",
    "1.  \"$A$\" must be 0 for $\\psi(x=0)$ to be zero for arbitrary $B$ and $n$.\n",
    "3.  \"$B$\" controls scale only (normalization), $\\psi = B \\sin(k x )$.\n",
    "3.  \"$n$\" must be an integer for $\\psi(x=a)$ to be zero.\n",
    "4.  Only when $n=$ <u> an integer</u> will $\\psi$ be continuous across the well to the outside where $\\psi=0$.\n",
    "5.  Discrete $n$, hence discrete energies, are a <em>natural</em> result of obtaining physically acceptable, continuous solutions from an ODE.\n",
    "6.  Discrete energies are as natural as discrete harmonics on a guitar string from classical mechanics."
   ]
  }
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