Time_evolution.ipynb

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    "colab": {
      "name": "Time_evolution.ipynb",
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  "cells": [
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      "cell_type": "markdown",
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      "source": [
        "<a href=\"https://colab.research.google.com/github/iashyam/Computation-QM/blob/main/Time_evolution.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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      "cell_type": "markdown",
      "metadata": {
        "id": "BR3FPjUcnKzN"
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      "source": [
        ""
      ]
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    {
      "cell_type": "code",
      "metadata": {
        "id": "nbEG5ytJnKzh"
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      "source": [
        "import numpy as np \n",
        "import matplotlib.pyplot as plt \n",
        "import numpy.linalg as lin \n",
        "import math\n",
        "from matplotlib.animation import FuncAnimation\n",
        "plt.style.use('seaborn-whitegrid')"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "tCmXZ6HgygYx"
      },
      "source": [
        "# __Problem__:\n",
        "A beam of electrons with energy $5 eV$ approaches a potential barrier with height $10eV$ and width $1nm$ Calculate the transmission cofficient. Conpare it with the theoritical value."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "4WzMjOTjzQtb"
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      "source": [
        "## Step1: Nondimessionlisation\n",
        "\n",
        "We the width of the potential barrier as the natural length scale so:\n",
        "\n",
        "$l_0 = 1\\times 10^{-9} m$\n",
        "\n",
        "Then we have the natural energy salce as:\n",
        "\n",
        "$$\\varepsilon  = \\frac{\\hbar^2}{2ml_0^2}$$\n",
        "\n",
        "So the scrodinger equation becomes \n",
        "\\begin{equation}\n",
        "\\iota \\frac{\\partial \\Phi}{\\partial \\tau} = -\\frac{\\hbar^2}{2m}\\frac{\\partial^2 \\Phi}{\\partial y^2} + V(x)\\Phi\n",
        "\\end{equation}\n",
        "\n",
        "With potential as \n",
        "\\begin{equation}\n",
        "V(x) =\n",
        "\\begin{cases}\n",
        " V_o & |x| < a \\\\\n",
        " 0 & \\text{elsewhere}\n",
        "\\end{cases}\n",
        "\\end{equation}"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "g67JSdJtmdvf"
      },
      "source": [
        "## Time Evolution\n",
        "\n",
        "For a given state $\\Phi_0(x,0)$ we can calculate the wave funciton at any time\n",
        "\n",
        "\\begin{equation}\n",
        "\\Phi(x,t) = \\sum_{n = 0}^{\\infty}C_n \\phi_n e^{-\\iota E_n t/\\hbar}\n",
        "\\end{equation}\n",
        "\n",
        "Where \n",
        "\n",
        "\\begin{equation}\n",
        "C_n = \\sum_{n = 0}^{\\infty}\\Phi(x,0) \\phi_n \n",
        "\\end{equation}"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "VQZhPOsNnKzy",
        "outputId": "6cc53706-3b3d-4448-d763-6873c2d00eb7"
      },
      "source": [
        "h = 1.05e-34\n",
        "m = 9.1e-31\n",
        "l0 = 1e-9\n",
        "E = h**2/(2*m*(l0**2))\n",
        "eV = 1.6e-19\n",
        "V0 =  5*eV/E\n",
        "p0 = h/l0\n",
        "E0 = 10*eV/E\n",
        "P = np.sqrt(2*m*E0)\n",
        "# po = P/p0\n",
        "po"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "14.536216402543934"
            ]
          },
          "metadata": {},
          "execution_count": 138
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "VFClTorRnKz9"
      },
      "source": [
        "N = 500 #number of lattice points\n",
        "l = 30\n",
        "ys = np.linspace(-l,l,2*N) #descretising the space\n",
        "delta = ys[1]-ys[0]"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "dJcd62m6nKz_"
      },
      "source": [
        "#potential function\n",
        "def V(x):\n",
        "    return  V0 if  x>-0.5 and x<0.5 else 0\n",
        "\n",
        "#kroneker delta function\n",
        "def k(i,j):\n",
        "    return 1 if i==j else 0\n",
        "\n",
        "#hemiltonian function\n",
        "def Hemiltonian(i,j):\n",
        "    return -(1/delta**2)*(k(i+1,j)+k(i-1,j)-2*k(i,j)) + V(i*delta)*k(i,j)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "meh1DMT2nK0C"
      },
      "source": [
        "#hamiltonian matrix\n",
        "H = np.array([[Hemiltonian(i,j) for i in range(-N,N)] for j in range(-N,N)])\n",
        "eig_values, eig_functions = lin.eig(H)\n",
        "idx = eig_values.argsort()\n",
        "eig_values = eig_values[idx]\n",
        "eig_functions = eig_functions[:,idx]"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
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        "outputId": "abe19961-c787-4cba-d541-5eb727ae7af3"
      },
      "source": [
        "psi0 = eig_functions[:,1]\n",
        "plt.plot(ys,psi0)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f0bd69aec10>]"
            ]
          },
          "metadata": {},
          "execution_count": 142
        },
        {
          "output_type": "display_data",
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IRVH467E6libHEh81Tu1y/J5Go+GW+ZM5WNE4ZlZglEAXIkBY6lqptLXx1Xn+fwMHf/HVeZNRFNhXOjbOdpFAFyJAvH5M2i1DNStBR3L8BP5xfGz00SXQhQgAiqLw+tE6rk+JI1Yn7ZbB0mg03Dx3Eocqm2hq61K7HJ+TQBciAJSeu8iZxnZptwzDzXMTcfcovHnivNql+JwEuhAB4PVjdYRoNaySdsuQpU+OZpohkr8dk0AXQqhMURT+fqyOZcmxGCbIrRyH6lLb5UC5PegvMpJAF8LPnW5wcKaxnVVz5dv5cN18bSKuHoW3LMF9tosEuhB+bt/x3lbBTXLf0GGbn6Rnsj6Cvwf52S4S6EL4uTdO1LNgagwJsnbLsGk0Gr4yN5H3TttpdQZv20UCXQg/Vnuhg2O1LXIw1AtuvnYSXa4e3jnZoHYpPjOoQN+yZQt5eXnk5+dz9OjRfmMHDx5kzZo15OXlsW3btn5jTqeTnJwc9uzZ472KhRhD3ijtbbesSpd2y0gtmjaRON043jwRvH10j4FeXFxMdXU1BQUFbN68mc2bN/cb37RpE1u3bmXXrl0cOHCA8vLyvrHf/va36PV671ctxBjxRmk9M4w6kuN1apcS8LRaDTlpRt49ZaPLFZw3kPYY6GazmZycHABSUlJoaWnB4XAAYLVa0ev1JCYmotVqyc7Oxmw2A1BRUUF5eTlf+tKXfFe9EEGsua2L4jNN8u3ci3LSEmjtdHGoslHtUnwi1NML7HY76enpfY8NBgM2mw2dTofNZsNgMPQbs1qtADz55JP89Kc/5bXXXrvqe1sslmEX7nQ6R7S9vwiWeYDMxdveLG/F3aMwM7Jj2LX4wzy8xRtziXP3MC5Ew6sHThLvtnupsqHz1efiMdC/SFEUj6957bXXWLBgAVOnTh3wdWlpaUPdfR+LxTKi7f1FsMwDZC7e9qsPD5Ooj+Abyxeg0WiG9R7+MA9v8dZcVnzcQUltC6mpqcP+7zpSI5lLSUnJVcc8BrrRaMRu/+z/ZA0NDcTHx19xrL6+HqPRyP79+7Farezfv5/z588THh7OpEmTuP7664c1ASHGmvYuF++V2cjPnKpa6ASr3DkJvHmintJzF5k7JbiO8XkM9KysLLZu3Up+fj6lpaUYjUZ0ut4DNElJSTgcDmpqapg0aRKFhYU8/fTTrF27tm/7rVu3MmXKFAlzIYbgvTI7na4ebpLTFb3uxlQjGg28eaJ+7AV6RkYG6enp5Ofno9Fo2LBhA3v27CEqKorc3Fw2btzIunXrAFi9ejUmk8nnRQsR7N45WU9URCiLTQbPLxZDEqsbx6JpE3nLUs//yZ2ldjleNage+vr16/s9Tk1N7fs5MzOTgoKCq257//33D7M0Icamnh6Fd07ayJ4VT1iIXPvnC7lzEnji7yepvdDBlJjxapfjNfKvRQg/c6y2BbujkxvTjGqXErRyPl0X560gu8hIAl0IP/P2yQY0GsieJYHuKynxvbemC7bVFyXQhfAzhScbyJg2UdY+97HctATMFY1cDKLFuiTQhfAj9RedHKtt4cup8u3c125MS8DVo3DgtHoXGHmbBLoQfqTw05UApX/uexnTYoiOCKXwVPCsviiBLoQfeftkA1NixjM7IUrtUoJeaIiWFbPiKTxlG9QV8IFAAl0IP+HsdnOg3M6XU41ydegoWTnbiK21k9JzF9UuxSsk0IXwE0VVTbR3uaV/PoqyZ/cuY1IYJDe9kEAXwk+8Y6knIkzLspRYtUsZM+J045ifpA+aProEuhB+QFEU3j7ZwPIZcUSEhahdzpjypdlGPrZeoKmtS+1SRkwCXQg/cLrBQU1zB19OlZtZjLYvpxpRFHj/tE3tUkZMAl0IP7D/0z/5V6bGq1zJ2HPtFD2xE8KDoo8ugS6EH3i3zMbshCgS9cGzUFSg0Go1ZM+O590yG+6ewD59UQJdCJW1d7n4sKqZFbPi1C5lzFo520hzezefWC+oXcqISKALobJDlY10uXtkMS4VrZgZT4hW09f6ClQS6EKo7N1TNsaHhXDd9IlqlzJm6SPDWDRtYsCfviiBLoTK3jttZ2myQU5XVNmXUuM5XnuRhlan2qUMmwS6ECqqbmyjyt5G9iw5u0VtK2b2fgYfBPDqixLoQqjovbLec5+zZ0v/XG1zEqOJnRDO+xLoQojheLfMzlTDeKbHRqpdypin1WpYPjOO90/b6AnQ0xcl0IVQSZerh4MVdrJnxcvqin5ixcx47I4uLOcDc/VFCXQhVHK4und1RTld0X/cMLP3WoBAbbsMKtC3bNlCXl4e+fn5HD16tN/YwYMHWbNmDXl5eWzbtg2Ajo4OHnjgAdauXcttt91GYWGh9ysXIsC9V2YnVKuR1RX9iDE6gtRJUX3HNgJNqKcXFBcXU11dTUFBARUVFTzyyCMUFBT0jW/atIkdO3aQkJDA2rVrWbVqFWVlZcydO5e7776b2tpa7rrrLlauXOnTiQgRaN4ts3Hd9Inoxnn8NRSjaMWseP7rwBnau1xEhgfWZ+PxG7rZbCYnJweAlJQUWlpacDgcAFitVvR6PYmJiWi1WrKzszGbzaxevZq7774bgLq6OhISZAU5IT6v4aITS91Fabf4oRtmxtHl7qGoskntUobM4/9+7HY76enpfY8NBgM2mw2dTofNZsNgMPQbs1qtfY/z8/M5f/4827dvv+J7WyyWYRfudDpHtL2/CJZ5gMxlKN4sbwVgWlirT/cjn8nQRbl6CA/R8FrRKSYpjT7Zh6/mMuS/J4ZyM9VXXnkFi8XCj3/8Y/bu3XvZkfy0tLSh7r6PxWIZ0fb+IljmATKXofjNJx8THzWO1dfP9+kZLvKZDM/SD9sobezw2f5GMpeSkpKrjnlsuRiNRuz2z474NjQ0EB8ff8Wx+vp6jEYjx48fp66uDugNbbfbTVNT4P35IoQv9PQofHDaxg0z4+R0RT+1YmYc5Q0Ozl3oULuUIfEY6FlZWezbtw+A0tJSjEYjOp0OgKSkJBwOBzU1NbhcLgoLC8nKyuLw4cPs3LkT6G3ZtLe3M3GiLDwkBMCJuos0t3f3XWou/M8Nn342gXYXI48tl4yMDNLT08nPz0ej0bBhwwb27NlDVFQUubm5bNy4kXXr1gGwevVqTCYTiYmJPProo3znO9/B6XTy+OOPo9XKKe9CABwo7/2r9no5XdFvzUrQkRA9jvdO28nLnKZ2OYM2qB76+vXr+z1OTU3t+zkzM7PfaYwAERERPPPMM14oT4jg80G5nVkJOozREWqXIq5Co9Fww8x43jxRj7tHIUQbGK0x+dosxChydrv58EwTWTPk7kT+7oaZcbR0dHOstkXtUgZNAl2IUfTR2Wac3T0sl0D3ezfMjEejIaCuGpVAF2IUHSi3E6LVsCRZ+uf+zjAhnPTJ0XxQHjjrukigCzGKPihvZOHUGLncP0BkpcTx8dlm2rtcapcyKBLoQoySlvZujtVckP55AMmaEUe3W6G4KjCuo5FAF2KUmCsb6VFg+UwJ9ECROd1AeIiWgxW+WQLA2yTQhRglB8rtTAgPYcHUGLVLEYM0PjyEhdNi+q4d8HcS6EKMkgPldpYkxxIWIr92gSRrRlzv1b1tXWqX4pH8yxJiFNRe6KDS3ib98wCUNSMWReltmfk7CXQhRsGlP9nl/PPAMy8phgnhIQHRdpFAF2IUHCi3E6cbx6wEndqliCEKC9GyJDk2IA6MSqAL4WOKonCg3M7yGbGyXG6Auj4llip7m98vpyuBLoSPnapvxe7okv55ALv02fl720UCXQgf++B0bwhIoAeu2QlRxE4I9/u2iwS6ED52oNxOcvwEJseMV7sUMUxarYZlKbF8UG4f0m04R5sEuhA+1OXqoaiqSc5uCQLLZ8Rha+2kvMGhdilXJYEuhA8dqblAe5db7k4UBAKhjy6BLoQPmSsa0WhgiUkCPdBNNUQy1TCeA37cR5dAF8KHzBWNpE6KZuKEcLVLEV6QlRLHocpGXO4etUu5Igl0IXzE2e2m5Gwzy+RmFkHj+hlxtDpdHD93Ue1SrkgCXQgf+fjsBbpcPSyT/nnQuHQsxF/76BLoQviIubIRrQYWmwxqlyK8JE43jtkJURzy04W6BhXoW7ZsIS8vj/z8fI4ePdpv7ODBg6xZs4a8vDy2bdvW9/xTTz1FXl4et956K2+88YZ3qxYiAByqbCR9sh79+DC1SxFetDTZwOEzzXS5/K+P7jHQi4uLqa6upqCggM2bN7N58+Z+45s2bWLr1q3s2rWLAwcOUF5ezqFDhzh9+jQFBQW88MILbNmyxWcTEMIfObvdfHL2grRbgtCylFg6ut0crbmgdimX8RjoZrOZnJwcAFJSUmhpacHh6D2x3mq1otfrSUxMRKvVkp2djdlsJjMzk2effRaA6OhoOjo6cLvdPpyGEP6lpLqZLnePHBANQos/PQXVH9suHm89brfbSU9P73tsMBiw2WzodDpsNhsGg6HfmNVqJSQkhMjISAB2797NihUrCAkJuey9LRbLsAt3Op0j2t5fBMs8QObyeX/5qAmtBqK7bFgs6v3iy2fiG6aJ4bx1zMqNia5hbe+ruXgM9C8ayjoGb731Frt372bnzp1XHE9LSxvq7vtYLJYRbe8vgmUeIHP5vNP7DzIvKYZF89I9v9iH5DPxjS+V97Cr+CzJM2cxLvTyL6uejGQuJSUlVx3z2HIxGo3Y7Z+dotPQ0EB8fPwVx+rr6zEajQC8//77bN++neeff56oqKhhFS5EIGrrdHHEKv3zYLY0ORZndw9HrC1ql9KPx0DPyspi3759AJSWlmI0GtHpeu+6kpSUhMPhoKamBpfLRWFhIVlZWbS2tvLUU0/xu9/9jpgYucO5GFsOVzfj6lGkfx7ElpgMaDS9VwL7E48tl4yMDNLT08nPz0ej0bBhwwb27NlDVFQUubm5bNy4kXXr1gGwevVqTCYTBQUFNDc388Mf/rDvfZ588kkmT57su5kI4SfMFY2EhWi4bvpEtUsRPhITGU7apGgOVTbyADPVLqfPoHro69ev7/c4NTW17+fMzEwKCgr6jefl5ZGXl+eF8oQIPObKRuYnxRAZPuRDVCKALEuJ5aVD1Ti73USEDb2P7gtypagQXtTq7OZ4bYv0z8eApcmxdLl6+MTqP+ejS6AL4UUfnmnCLf3zMWGxyYDWz/roEuhCeJG5opHwEC0Z10j/PNjpx4eRPlnvVxcYSaAL4UXmykYWTovxm56q8K2lyQY+PnsBZ7d/XAkvgS6El7S0d1N67qL0z8eQZSmxdLl7+Ohss9qlABLoQnhNUVUjitJ7sEyMDddN7+2jH/KTProEuhBeYq5sZFyoloXT5GK6sSI6Ioxrp+gx+0kfXQJdCC8xVzSy6JqJw1rbQwSupcmxfGK9QEeX+n10CXQhvKC5rYuT51vldMUxaGlKLN1uhZJq9fvoEuhCeEFRVRPQ+8stxpbM6QZCtBq/OH1RAl0ILyiq6u2fz0vSq12KGGW6caF+00eXQBfCC4qrmsiYJv3zsWpZSixHrBdo7xreDS+8RQJdiBFq6ejmRN1FFpsMnl8sgtLS5FhcPQqHz6jbR5dAF2KESqqbUBRYkiyBPlZdd81EQrQaiqrUbbtIoAsxQkWVTYSFaMiYJuu3jFUTxoUyd4qe4k8PjqtFAl2IESqqamJ+kqzfMtYtNRk4Ym1RdV0XCXQhRqCt08Wx2hZptwgWmwx0uXv4+Kx666NLoAsxAh+dbcbdo7DYJOefj3XXTe+9z6iafXQJdCFGoKiyiRCthkWy/vmYpx8fxpzEaIoq1eujS6ALMQLFVU3MnRyNbpzcP1T0tl0+OttMl6tHlf1LoAsxTM5uN59YL7BE1m8Rn1piiqXT1cPRGnX66BLoQgzTx2cv0OXuYfF0OSAqel26uKxIpdMXBxXoW7ZsIS8vj/z8fI4ePdpv7ODBg6xZs4a8vDy2bdvW93xZWRk5OTn84Q9/8G7FQviJ4qomNBrIlCtExacME8KZlaDz30AvLi6murqagoICNm/ezObNm/uNb9q0ia1bt7Jr1y4OHDhAeXk57e3t/PznP2fZsmU+K1wItRVVNZI2KRr9+DC1SxF+ZIkplpIzTbjco99H90atqjoAAAwoSURBVBjoZrOZnJwcAFJSUmhpacHhcABgtVrR6/UkJiai1WrJzs7GbDYTHh7O888/j9Fo9G31Qqiky9V7H0lZv0V80WKTgbYuN6XnLo76vj0emrfb7aSnp/c9NhgM2Gw2dDodNpsNg8HQb8xqtRIaGkpoqOej/haLZZhlg9PpHNH2/iJY5gFjay4nGpw4u3tICu/w6zmPpc/EXxhcvSsu7j1kIdxx5dsR+mouQz7XSlEUr+08LS1t2NtaLJYRbe8vgmUeMLbmUlhfDsA3l19LrG7caJU1ZGPpM/EnpsJGqttDr1rvSOZSUlJy1TGPLRej0Yjdbu973NDQQHx8/BXH6uvrpc0ixoSiyiZmGnV+HeZCPUtMBoqrmnD3eO8L8GB4DPSsrCz27dsHQGlpKUajEZ1OB0BSUhIOh4OamhpcLheFhYVkZWX5tmIhVOZy91BSLf1zcXVLkg1cdLo4db51VPfrseWSkZFBeno6+fn5aDQaNmzYwJ49e4iKiiI3N5eNGzeybt06AFavXo3JZOL48eM8+eST1NbWEhoayr59+9i6dSsxMVfuJwkRSE7UXcTR6ZILisRVXVrbp6iqkTmTo0dtv4Pqoa9fv77f49TU1L6fMzMzKSgo6Dc+d+5cXnrpJS+UJ4T/ubTm9RL5hi6uYkrMeJImjqeosok7s0yjtl+5UlSIITpU2cT02EgSoiPULkX4scUmA8Vnmrx6IoknEuhCDEFPj8KHZ5pYIsvlCg+WmmJpauuivMExavuUQBdiCE7Vt9LS0S0HRIVHaqzrIoEuxBAUVfbevEDuUCQ8uSY2koTocRLoQvir4jNNnx7wilS7FOHnNBoNS0yxFFU2jlofXQJdiEFSFIXiqiY5u0UM2mKTgYbWTqob20dlfxLoQgxSha0Nu6NL+udi0JYmX+qjj859RiXQhRikS7+UckGRGKyUeB2xE8JHrY8ugS7EIBVXNREfNY7psdI/F4Oj0WhYbDKM2o2jJdCFGARFUSiq7O2fazQatcsRAWSJyUDthQ5qmn3fR5dAF2IQrE0dnL/olHaLGLJL67oUj0LbRQJdiEE4dKl/LgdExRClTooiOiJ0VNouEuhCDEJRZROGCeHMNOrULkUEGK1W07eui8/35fM9CBEEis80kjl9ovTPxbAsMcVSZW+j4aLTp/uRQBfCg3MXOrA2dciCXGLYliSPzrouEuhCeHDpYJZcUCSGa05iNLpxoT6/wEgCXQgPiqoaiYoIJS1x9O48I4JLaIiWjGsm+vxMFwl0ITwoqmpi8XQDIVrpn4vhW2IyUFbvoLmty2f7kEAXYgANrU4qbW3SbhEjdunf0Ic+PNtFAl2IAXxY1QzI+i1i5OYl6QkP1fq07SKBLsQAiqoaiQwPIX0U79wugtO40BAWTo3x6fnoEuhCDKC4qolF10wkLER+VcTILTEZOF7bQnt3j0/ef1D/Srds2UJeXh75+fkcPXq039jBgwdZs2YNeXl5bNu2bVDbCBEImtu6OHm+VS73F16z2BRLjwKWBt9cYBTq6QXFxcVUV1dTUFBARUUFjzzyCAUFBX3jmzZtYseOHSQkJLB27VpWrVpFU1PTgNsIEQgu/Wks/XPhLRnXxBCq1XCsXqVAN5vN5OTkAJCSkkJLSwsOhwOdTofVakWv15OYmAhAdnY2ZrOZpqamq24zUoqiYK5o5EiVg9Nd5644Prz3HWCMqw8OuN2A79nr3LlWSttrvrDdAPu7+lsOODjsOQy0uy8M1p2/yMcXz/psfwNtOJQ6+49defB8fQvvWe1MjAxjXpJ+oKqEGLTI8FC+s2QaE2nzyft7DHS73U56enrfY4PBgM1mQ6fTYbPZMBgM/casVivNzc1X3ebzLBbLkAu+0OHmX16txq0ANAx5e/9kU7sAL7KrXYDX6MK1PHB9HJWny9QuZdicTuewfs/8UbDM5duzQnA6w30yF4+B/kXD+QZ8tW3S0tKG/F4A5hkz+OTEKZKTU644PtD6SQNdGjLQwksDbzfQ/gZ4Tw2Ul5czY8aMAd59iPtTYQ6XnD59mpkzZw5qfwMNDnZ/Q3jLIf93KSsrY+G1cwgPDeyDoRaLZdi/Z/5G5tKrpKTkqmMeA91oNGK3f/atq6Ghgfj4+CuO1dfXYzQaCQsLu+o23mCMjmCqPpwZQbCUqSMqjKmG4LilWVNkKAnREWqX4RXRESEBH+Zi7PH4LzYrK4t9+/YBUFpaitFo7GudJCUl4XA4qKmpweVyUVhYSFZW1oDbCCGE8A2P39AzMjJIT08nPz8fjUbDhg0b2LNnD1FRUeTm5rJx40bWrVsHwOrVqzGZTJhMpsu2EUII4VuD6qGvX7++3+PU1NS+nzMzM694SuIXtxFCCOFb0iQUQoggIYEuhBBBQgJdCCGChAS6EEIECY0y3GvlR2igk+OFEEJc3aJFi674vGqBLoQQwruk5SKEEEFCAl0IIYLEkBfnUktjYyM/+clP6OzspLu7m4cffpj58+dz8uRJNm7cCMDs2bP52c9+pm6hg+ByuXj00Uc5e/YsbrebBx98kOuuuy4g5wK9a+Y/8MADbNmyhZUrVwIE7Fy2bNnCkSNH0Gg0PPLII8ybN0/tkoasrKyMe++9l+9973usXbuWuro6HnzwQdxuN/Hx8fzyl78kPDxc7TI9euqppygpKcHlcvGv//qvXHvttQE5j46ODh566CEaGxvp7Ozk3nvvJTU11TdzUQLEzp07lb179yqKoihFRUXKnXfeqSiKoqxdu1Y5cuSIoiiK8qMf/UjZv3+/ajUO1u7du5UNGzYoiqIoZWVlyq233qooSmDOpbq6WrnnnnuUe++9V3nnnXf6ng/EuRQVFSk/+MEPFEVRlPLycuX2229XuaKha2trU9auXas89thjyksvvaQoiqI89NBDyt/+9jdFURTlmWeeUV5++WU1SxwUs9msfP/731cURVGampqU7OzsgJyHoijK66+/rvz+979XFEVRampqlJtuuslncwmYlsudd97J17/+dQDq6upISEigq6uL2travm9RK1euxGw2q1nmoNxyyy08/PDDQO9a8RcuXAjYucTHx/Pcc88RFRXV91ygzuVqN3MJJOHh4Tz//PMYjca+54qKirjxxhuBwPksMjMzefbZZwGIjo6mo6MjIOcBvWtc3X333cBn2eWruQRMywXAZrNxzz330NbWxosvvkhzczPR0Z/djT02Nhabzf9vFhEWFtb384svvsjXvva1gJ3L+PHjL3suUOcy0M1cAkVoaCihof1/rTs6Ovr+nA+UzyIkJITIyN5lpXfv3s2KFSv44IMPAm4en5efn8/58+fZvn07d955p0/m4peB/uqrr/Lqq6/2e+7+++/nhhtu4E9/+hPvvvsuDz/8ME888US/1yh+eAbmQHN5+eWXKS0tZfv27TQ1NfV7TaDNZSD+OJfBCNS6BxJoc3rrrbfYvXs3O3fu5Kabbup7PtDmAfDKK69gsVj48Y9/3K9+b87FLwP9tttu47bbbuv3XHFxMS0tLej1erKzs3nwwQf72hWXXLrBhj+50lygNxzfeecdfvOb3xAWFhbQc/miQJjLlQx0M5dAFhkZidPpJCIiImA+C4D333+f7du388ILLxAVFRWw8zh+/DixsbEkJiaSlpaG2+1mwoQJPplLwPTQ33jjDf785z8DcOrUKRITEwkLCyM5OZnDhw/3vcbTt0V/YLVaeeWVV3juuecYN24cQMDO5UoCdS7BemOW66+/vm9egfJZtLa28tRTT/G73/2OmJgYIDDnAXD48GF27twJ9Lb12tvbfTaXgLlStKmpiYceeoi2tja6urp49NFHWbBgAeXl5Tz++OP09PQwf/78voON/uxXv/oVr7/+OpMnT+57bseOHZw9ezbg5rJ//3527NhBZWUlBoOB+Ph4du7cGZCfC8DTTz/N4cOH+27M8vm1/wPB8ePHefLJJ6mtrSU0NJSEhASefvppHnroITo7O5k8eTJPPPFEv+M4/qigoICtW7diMpn6nvvFL37BY489FlDzgN6bWz/66KPU1dXhdDq57777mDt3bt9p2N6cS8AEuhBCiIEFTMtFCCHEwCTQhRAiSEigCyFEkJBAF0KIICGBLoQQQUICXQghgoQEuhBCBAkJdCGECBL/H+J83F4MfozmAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "z0XpBBDPnK0J"
      },
      "source": [
        "a = 1 #intial spread(uncertainily) in natural lenght scale\n",
        "b = -10 #intial <x> center of the gaussian\n",
        "#intial wave function\n",
        "def Psi0(y):\n",
        "    return 1/pow(np.pi*(a**2),0.25)*np.exp(-((y-b)**2)/(2.0*a**2) - 1j*po*y)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        },
        "id": "y6piuNR8nK0N",
        "outputId": "51076ef3-1241-4835-e557-81d4d6d614c2"
      },
      "source": [
        "#time evolution\n",
        "def Psi(t):\n",
        "    sum = np.zeros((2*N), dtype='complex')\n",
        "    for i in range(len(eig_values)):\n",
        "        cn = np.vdot(Psi0(ys),eig_functions[:,i])\n",
        "        sum += cn*np.exp(eig_values[i]*1j*t)*eig_functions[:,i]\n",
        "    return sum\n",
        "\n",
        "def prob(t):\n",
        "    return abs(Psi(t))**2\n",
        "\n",
        "plt.plot(ys, prob(2.8))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f0bd6f1b510>]"
            ]
          },
          "metadata": {},
          "execution_count": 150
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 266
        },
        "id": "mv-vyGt-nK0S",
        "outputId": "fd771521-ced2-4213-d20f-95f567caab9f"
      },
      "source": [
        "fig = plt.figure()\n",
        "ax = plt.axes(ylim=(0,0.6), xlim=(-20,20))\n",
        "\n",
        "plot, = ax.plot([],[])\n",
        "\n",
        "def init():\n",
        "    plot.set_data(ys, prob(0))\n",
        "    return plot,\n",
        "\n",
        "def animate(i):\n",
        "    plot.set_data(ys,prob(i*0.005))\n",
        "    return plot,\n",
        "\n",
        "anim = FuncAnimation(fig, animate, init_func = init,frames = 150, interval = 60, blit = True)\n",
        "anim.save('time_evoluatoin.mp4', writer = 'ffmpeg', fps = 60)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "N5P8eOJonK0V",
        "outputId": "6fe38bbd-7602-4b6f-e316-dade1d64d706"
      },
      "source": [
        "T = 0\n",
        "R = 0\n",
        "Probs = prob(2.8)\n",
        "Probs_left = Probs[:len(ys)//2]\n",
        "Probs_right = Probs[len(ys)//2:]\n",
        "T = 2*sum(Probs_right)/l\n",
        "R = 2*sum(Probs_left)/l\n",
        "print(f\"T = {T}\")\n",
        "print(f\"R = {R} \")\n",
        "print(f\"{T + R }\")"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "T = 0.9117204360546164\n",
            "R = 0.19827956394539106 \n",
            "1.1100000000000074\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "eQnygLifpMVv"
      },
      "source": [
        ""
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "74hKD0LSt6bn"
      },
      "source": [
        "def Tra(E,V):\n",
        "    t = (1+V**2/4*E*(E+V))*(math.sin((2*0.5/h)*math.sqrt(2*m*(E+V))))**2\n",
        "    return 1/t"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "eDk5UuYfk_pt",
        "outputId": "c520e702-b026-4c51-bc41-a52c0da7d735"
      },
      "source": [
        "Tra(10,5)+T"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "1.0316656515457947"
            ]
          },
          "metadata": {},
          "execution_count": 157
        }
      ]
    }
  ]
}