1D_eiogenvalue.ipynb

{
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  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "1D-eiogenvalue.ipynb",
      "provenance": [],
      "collapsed_sections": [],
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/iashyam/Computation-QM/blob/main/1D_eiogenvalue.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Y8m07-X2iRkX"
      },
      "source": [
        "import numpy as np\n",
        "import math\n",
        "import matplotlib.pyplot as plt\n",
        "import numpy.linalg as lin\n",
        "\n",
        "plt.style.use('seaborn-whitegrid')\n",
        "\n",
        "%matplotlib inline"
      ],
      "execution_count": 2,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "l-ZkDxNNlEAe"
      },
      "source": [
        "#### Square Well Potential\n",
        "\\begin{equation}\n",
        "V(x) = \n",
        "\\begin{cases}\n",
        "-10 && if && |x|<a \\\\\n",
        "0 && elsewhere\n",
        "\\end{cases}\n",
        "\\end{equation}\n",
        "\n",
        "The natural length scale here is the width of well $a$.\n",
        "And natural energy scale turn out to be $\\dfrac{\\hbar^2}{2ma^2}$\n",
        "\n",
        "#### Harmonic Osicalltor potential\n",
        "\n",
        "$$V(y) = y^2$$\n",
        "\n",
        "Natural length scale here is $\\hbar\\omega/2$ and natural length scale here is $l_0 = \\sqrt{\\frac{\\hbar}{m\\omega}}$\n",
        "\n",
        "#### Hygrogen Atom Radial Equation\n",
        "\n",
        "\\begin{equation}\n",
        "\\frac{d^2 u }{dr^2} = \\frac{2\\mu}{\\hbar}\\left(V(r) - E\\right)u\n",
        "\\end{equation}\n",
        "\n",
        "Taking ground state energy $E_0 = -13.6eV$ as natural energy scale and $l_o = \\sqrt{\\frac{\\hbar^2}{2mE_0}}$ as natural length scale we get \n",
        "\n",
        "$\\mathcal{U}(y) = - \\dfrac{ke^2}{l_0 E_0}\\dfrac{1}{y}$\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "P2LJuUmTe76s"
      },
      "source": [
        "$$H_{i,j} = -\\frac{1}{\\Delta^2}(\\delta_{i+1,j} + \\delta_{i-1,j} - \\delta_{1,j}) + V(\\Delta i)*\\delta_{i,j}$$\n",
        "\n",
        "$y$  $\\epsilon$ $[0,1]$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "qkyrMXX8Hicp"
      },
      "source": [
        "n = 400 #number of states we want to know\n",
        "y = np.linspace(-6,6,2*n)\n",
        "Delta = y[1]-y[0]"
      ],
      "execution_count": 27,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "hUZhxaPBgLrW"
      },
      "source": [
        "def k(i,j): #kronekar delta function\n",
        "    return 1 if i==j else 0\n",
        "\n",
        "def V(x):\n",
        "    # return x**2 #for harmonic potential\n",
        "    return -50 if abs(x)<1 else 0 #for square well\n",
        "    # return (-k*e**2/l0) *(1/x) # for hygrogen atom "
      ],
      "execution_count": 39,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "tjtCfiDBgT5A"
      },
      "source": [
        "def Hemiltonian(i,j):\n",
        "    return (-1/Delta**2)*(k(i+1,j)+k(i-1,j)-2*k(i,j)) + V(Delta*i)*k(i,j)"
      ],
      "execution_count": 68,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "WTbfHA8liYCB",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "d3ee93fe-9744-4d13-9df6-adf19541a378"
      },
      "source": [
        "H = np.array([[Hemiltonian(i,j) for i in range(-n,n)] for j in range(-n,n)])\n",
        "len(range(1,n))"
      ],
      "execution_count": 69,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "99"
            ]
          },
          "metadata": {},
          "execution_count": 69
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "5IdqwV0UgVpu"
      },
      "source": [
        "eval, evac = lin.eig(H)"
      ],
      "execution_count": 80,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "LPhAswWShDFg"
      },
      "source": [
        "sort_eval = np.sort(eval)"
      ],
      "execution_count": 71,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "zwpd_gT0hGxv"
      },
      "source": [
        "idx = eval.argsort()\n",
        "eval = eval[idx]\n",
        "evac = evac[:,idx]\n",
        "psi1 = evac[:,1]\n",
        "psi2 = evac[:,2]\n",
        "psi0 = evac[:,0]\n",
        "\n",
        "potential = 0.006*np.array([V(Delta*i) for i in range(1,n)])"
      ],
      "execution_count": 72,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Yim4cT940qPA",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 291
        },
        "outputId": "bb5758e8-fd7f-45a5-f591-3a24b84984c7"
      },
      "source": [
        "plt.plot(y,psi1, label=\"n=1\")\n",
        "# plt.plot(y,psi2, label=\"n=2\")\n",
        "# plt.plot(y,psi0, label=\"n=0\")\n",
        "# plt.plot(y, potential, label=\"Potential\")\n",
        "plt.ylabel(r'$\\Psi$')\n",
        "plt.xlabel('x')\n",
        "plt.title('Wave function for Particle in a box')\n",
        "plt.legend();"
      ],
      "execution_count": 75,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "UsMIgAThR3J8",
        "outputId": "9418c4e9-ed98-4385-a94e-14a34ad3ee54"
      },
      "source": [
        "print(eval[0])"
      ],
      "execution_count": 13,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "2.9993638658266395\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "0rinEhJWo6FY"
      },
      "source": [
        ""
      ],
      "execution_count": 13,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "540lD2uK2P_f"
      },
      "source": [
        "## Double Potential well\n",
        "\n",
        "Let's say we have double potential well as\n",
        "\n",
        "\\begin{equation}\n",
        "V(x) = \\begin{cases}\n",
        "V_1 & \\frac{b}{2} < x < \\frac{a}{2} \\\\\n",
        "V_2 &  0< x <\\frac{b}{2} \\\\\n",
        "V(-x) & x<0\n",
        "\\end{cases}\n",
        "\\end{equation}"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "e85e8PnI3iBV"
      },
      "source": [
        "a = 2\n",
        "b = 1\n",
        "def V(x):\n",
        "    V1 = -10\n",
        "    V2 = -5\n",
        "    if abs(x) > b/2 and abs(x) < a/2:\n",
        "        return V1\n",
        "    elif abs(x) < b/2:\n",
        "        return V2\n",
        "    else:\n",
        "        return 0"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "agzR8EJl4BP-"
      },
      "source": [
        "n = 1000 #number of states we want to know\n",
        "y = np.linspace(-2,2,2*n)\n",
        "Delta = y[1]-y[0]"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "TFVfOQl55CVT"
      },
      "source": [
        "H = np.array([[Hemiltonian(i,j) for i in range(-n,n)] for j in range(-n,n)])"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ob_kXLwI4Zsp"
      },
      "source": [
        "eval, evac = lin.eig(H)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "m3jkzseN4nyN"
      },
      "source": [
        "idx = eval.argsort()\n",
        "eval = eval[idx]\n",
        "evac = evac[:,idx]\n",
        "psi1 = evac[:,1]\n",
        "psi2 = evac[:,2]\n",
        "psi0 = evac[:,0]\n",
        "\n",
        "potential = 0.006*np.array([V(Delta*i) for i in range(-n,n)])"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        },
        "id": "CNhbfiyd4sjX",
        "outputId": "f61f4f04-5e27-436f-9bdd-52ee2f02e3db"
      },
      "source": [
        "plt.plot(y,psi0, label=\"n=0\")\n",
        "plt.plot(y, potential, label=\"Potential\")\n",
        "plt.legend()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7f31a5047cd0>"
            ]
          },
          "metadata": {},
          "execution_count": 16
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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WK9zOa2M7c/BMJi99Y73buYWorA37zzB33QEmXN2Ma9vWNzscvyGJ3sf1bl2XKX1bMn/DYVbtPlX+ApeRO2OFl1TyztgLuQU8+elWmsVE8swI6z2ewJdJovcDTwxRtK4fxbOLt1W8C0duOhHVoRL72avf7uJoejZv3NaZyFAZ/M6TJNH7gXB7MC+PvorUs9ky5qzwSUmHzjJ/4yHuuqYFic3lUkpPk0TvJ3q2jOGOns2Yu/YA21JlXBfhO/ILHTzz+TZia4XzhAyf6RWS6P3I1OHtqFsjjKmf/yo3UgmfMWfNAfSJ80y7qQM1wqTLxhsk0fuR6Ag7027qwI6j5/hgw6GKLSSPKRZeU/6+lZKWxd++283QDg0Z0iG23PmFMZLo/czwjrH0aVOXN5fv5syF3CvMKSdjRXW48n72l//uJMhm44WbOlRTPIFJEr2fsdls/OnGDmTmFfKGnJgVFrZ+32mW7jjBAwNaExct4756kyR6P9S2YU0mXt2cj346zM6j58wOR4jLFDqc/PmrnTSuHcEUeZaN10mi91OPDW5LdISdF77agVP64YXFLNyUwq7j53lmRALh9mCzw/F7kuj9VHSknSeGKH46kMaynSeuMKf8CAgvKaOBcS4nn9eXaXq2iGFEJzkBWx0k0fuxcT2a0qpeFK8t1RR6YpxZITzg3dX7OZOZx/M3tpdhAauJJHo/FhIcxJNDFXtOXuDzLamXFsofmKgGzhJX3Zy+kMvctQcYeVUcHRtHmxRV4JFE7+eGd4zlqibRvLViDzn5hWaHIwLcrJV7yS1w8Pj1bc0OJaBIovdzNpuN/xvWjiPp2cyv6E1UQnjBkfRsFmw4zJhuTWhVv4bZ4QQUSfQBoE+bevRtU49ZK/dyIffSp1tKB47wnkvPC81YsQeA3w+ONyOYgCaJPkA8MaQtZ7PypVUvTHHwdCaLtqQy4ermNKotN0dVN0NPEFJK2YF5QHOgEJiktd5fYp7xwKOAA5ittZ6rlAoB5gKt3et+Umu91nj4oqK6NqtDv/h6zFmzn7t6tyBC2vKiOrhP+s9es5/gIBv3DZCbo8xgtEV/J5Cute4LvAi8XLxQKRUFPA8MBgYAjymlYoCJQKZ7ucnAGwbXLwx4eGA8py/k8dFPh80ORQSQk+dyWLQ5lTGJTWhQM9zscAKS0UQ/CFjsfr0C6FOivBewSWudobXOBta555kPPO6e5xRQ1+D6hQE9W8bQq2UM/1y9j9wCuQJHVI/31h2kwOHgt/KoA9MYTfSxuBI1WmsH4FRKhZZW7nYSiNNa52utc9zTHgU+NLh+YdAjg+I5cS6XJTvcd8vK4xGEF+UXOlmw4RAjOsXRol6U2eEErHL76JVSU4ApJSb3KvG+vA7fS8qVUg8C3YAbS5s5OTm5vLBKlZOTY3hZb7JSXHWcThLqhzF//X5GAfn5+ZaJrTgr1VlxElfFJQD6ZBbncwsY2izIUvFZsb7Ae3GVm+i11nOAOcWnKaXm4Wq1b3WfmLVprfOKzXLUXV6kMbDBvexkXAn+Zq11fmnrTEgwNgJ8cnKy4WW9yWpxPWmry4vvH4AwsNvttLBQbEWsVmdFJK7K0WkF9Iuvx419u5gdyiWsWl9ViSspKanMMqNdN8uAse7XNwIrS5RvBHoopWorpWrg6p9fo5RqBdwHjC7WhSOq2cB2DWjlPox2yEPNhBflFDi4f0Brs8MIeEYHaFwIXK+UWgvkAncDKKWmAqu01j+6Xy/FddfENK11hlLq/3CdgP1GqYuDAA8pcTQgvMxms3FHz2bwHew5nUdTswMSfqfQ4SQYqBsRQu9Wcs2F2Qwleq11ITCplOmvFHu9CFhUovwZ4Bkj6xSe1b9tffgONqZkcp3TKU8RFB717bajjAQ6NAiXfcsC5M7YABUS7Nr0x87ns27vGZOjEf7E6XTy9x/2AdAk2m5yNAIk0QcwVyurRmgQ7/yw1+RYhD9Zvec0O465hrCU1rw1SKIPcD2aRLJ+3xl+PnzW7FCEn5i1ci9xteQOWCuRRB/gusRFEB1h5x33obYQVbH5YBo/HUjjXrkL1lIk0Qe40CC4+5oWLN95An38vNnhCB/3zg/7iIkKZVzPJmaHIoqRRC+4+5oWRIYG849V0qoXxu04msH3u05yT58WRNqNXrktvEESfaAqdpKsTlQo43s148utR0lJyzIxKOHL/v7DPmqEhTCxd4v/TZSTsZYgiV4AMKVfK4JtNv65Wlr1ovL2n7rAN9uOMbF3c6Ij5JJKq5FELwBoWCucWxOb8MnmVE6el6dTiMp5a8UewkKCmdy3pdmhiFJIog94/3vWzX39W1FQ6GDu2gMmxiN8za7j5/jq16NM6tOCejXC3FPlGUpWIoleXNS8bhQjr2rE/B8PkZ4ljx8SFfPGst3UCAvhd9fKw8usShK9uMQD17UmK7+Qv8sVOKICtqaks2znCe7t14royNL65uVkrBVIoheXaBdbi1u6NuZf6w5yJD3b7HCExb22TFMn0s490jdvaZLoxWWeGOJ6hPTry7TJkQgrW737FGv2nOaBAW2oESbXzVuZJHpxmca1I5jUpwWLfz7CjqMZZocjLKig0MFf/ruT5nUj+c01zS+fQcYithRJ9KJUDwxoQ3SEnVe+3YVT/mhFCR9vSmH3iQs8PbwdYSHBZocjyiGJPlCVc8didISdhwfGs2bPaZbvPFFNQQlfcC4nnzeX76ZnyxiGdoi94rzSRLAGSfSiTL/p3Zy2DWsw7audZOcVmh2OsIgZK/aQlpXHcze0l+fN+whJ9KJM9uAgpo/qyJH0bGatlMFJBGw/ksG/1h9kXI+mdGoSbXY4ooIk0Qc4Wzn9771a1eWWro2ZvXo/+09dqKaohBUVOpw8u3gbdSLtTB2WUM7c0mljJZLoRbmeHtGOsJAgnlm8DYdD/oAD1fwNh9iamsFzI9uXcXOUsCpJ9AGr4n2rDWqG89zI9mzYn8b7Px70WkTCulLSsvjrUk2/+Hrc1LlRxReUPnxLkEQvKmRs9yYMUPV5ZckuDp7ONDscUY0KHU6e+HQrAC/d0klOwPogSfSiQmw2G6+Mvgp7cBBPfrqVgkKH2SGJajJ37X5+OpDGn25sT9OYSLPDEQYYum9ZKWUH5gHNgUJgktZ6f4l5xgOPAg5gttZ6rlKqAfBvIBwIBR7XWm80Hr6ouor3ucdGhzN9VEceXfgLb63Yw5NDlRfjElaw42gGry3dzZD2DRmTWIlxYOUmO0sx2qK/E0jXWvcFXgReLl6olIoCngcGAwOAx5RSMcAE4AOt9XXAM8B0g+sXJrm5a2Nu696EWT/sZdXuU2aH41UOh5PM3AJOns/haHo2xzKyOZNVwMlzOaRl5pFX4N9HNRlZ+dw3P4mYqFBeHi1dNr7M6JOIBgHvu1+vAN4rUd4L2KS1zgBQSq0D+mit3yg2T1Mg1eD6RVVV4Y922k0d2ZqSwWMLf+Hrh/vSqHaEBwMDMs8Qlr4PTnivVZidX0jq2WxS07M4cjabMxfyOJOZR1pmLmkX8sjMKyCnwFFGw3TdxVehwUHUCA8hKjSYujXCqF8jjPo1w6hXM5QmtSNoWjeSmMhQbN58XK893OMf6XA4eeyTXziekcPHv+1N3YsDilSW/DhYgdFEHwucAtBaO5RSTqVUqNY6r2S520kgDkApFQt8BdQEBhpcvzBRRGgw70zoxs1vr+OeeZv49L7e1Az34OV2s3rSKuu05z6vFBFAvPtfqYJwdS5WRIH7XxauPd0EUf3/BgnlXdteca8v13y/6yTTbupAYvM6HvtcYY5yE71SagowpcTkXiXel/ezfbFca30c6KGUGoGrn39IyZmTk5PLC6tUOTk5hpf1JivGZb9wlDZAfn6+4dimXluP51Yc5+7Za5g2KJbgIM+03tplp5Eedy2ZLYcZWj4r38Ghs3kcTHf9S8tyPb7BBtSNDCa2pp26kcHUjQihbmQIdSKCsAdXrBczPz8Pu/3KvwAOp5MLuQ7OZBdyOrOA01kFnM4s4PiFAvIKXYcIEXYbjWuF0rJOKK3qhFI3MthQ10hI1klif3mLwswzHtvH/qvPMWvDaYbF16RH7Sxjn1uYTwJQUFBguX0frPk3Cd6Lq9xEr7WeA8wpPk0pNQ9Xq32r+8SsrVhrHuCou7xIY2CDUqo/8KvW+qzW+hul1PuUIsFgyyQ5Odnwst5kybjOurpb7PYQw7ElJEBQzcM8/fk25mzL5fXbungs2RfUaUOTIQ9WaF6n04k+cZ6l20+wPPk4O46ew+mEGmEh9GoZQ4+WMXRuUpuOjWtV+cgjOTmZlgbrq9DhZO/JC/x8+Cw/H07n64NpHNjtulQ1LjqcfvH1uL59LP3i6xFur+ATIU9p+OUt7Ha74biKW7L9OO9s3M/Adg14e2IiIRX8AbxMQS4AISHG9y9vsuTfJFWLKykpqcwyo103y4CxwFLgRmBlifKNwBylVG1cB7V9cF2B82egK/CWUqoTkGJw/cIi7ujZjLTMPP66VGOz2XhtbOeqJ/sKXrGx42gGX/5ylCU7jnPoTBY2G3RrVofHB7flmjb1uKpJdIVb6tUhOMiGiq2Jiq3JuJ7NANeNSGv3nmbtntMs2X6cTzanEhUazKCEhozoFMsA1aBCSd8TP6/fbDvGIx/9TOemtXn7zq7Gk7ywHKOJfiFwvVJqLZAL3A2glJoKrNJa/+h+vRTX9XvTtNYZSqnpwL+VUqOBMOD+qn4BYZTnTpI9eF0bnE4nry3bTW5BIW/c1qXiLdKylNGNcfJ8Dv/5+SifbUll1/Hz2INtXNO6Hr+7tjWD2zegQU3Pn5j0pqYxkdzRsxl39GxGfqGDH/ed4dvtx1i64wRfbj1KzfAQRnVpxG3dm9KpcXQp3Tue2Y6fbk5h6ufb6Nq0Nv+a1IPIUE+NGCUnY63A0NbUWhcCk0qZ/kqx14uARSXKTwM3GFmnsLaHBsYTbg/mxW+SOZK+gXcnJtKgltGke2mL3ul0knToLHPWHGB58gkKHU46N63N9FEduLFzI2pHVvSsqbXZg4O4tm19rm1bn+mjHGzYn8ZnW1L5dHMq8zccpl1sTcb1aMqY7k0vH7rP4HXrhQ4nr3ybzLtrDtCnTV1mT+xOlAwL6HdkiwqPmdKvFc1iIvn9x78w/G9reOXWq7i+fUPDn1dQ6ODb7ceZs/YAW1PSiY6wM6VfS8YmNqFNg5oejNx6QoKD6Btfj77x9Zg2qgNfbT3KJ5tSeOGrnby+bDe392jKXde0oGkV1nHoTCZPfforPx1M4ze9m/PcyPaW6uoSniOJPtB5+FL1IR1i+fKhPjy68BfufX8zwzrEMnV4O1rUi6rU5ySfzOHpv/7AkfRsWtaLYvrNHbm1W2MPdin4jlrhdsb3as74Xs35JSWd99Ye4F/rD/LeugP8Jj6PFwBnJTZkZm4Bc9ce4B+r9hEcZOP1sZ25tTJ3vVaE3BlrKYH3VyO8Lr5hTRY/0Id/rtrH31ftY0XyCYZ2iOXOXs3o0SKG0JDLW40Oh5Nfj2Tw6eYUXgS2HMumabMIpt3UgYHtGhDkoat5fF2XprWZcUdXnh7Rjn+vP8T6jT8C8P7PZ0lomMrghIalXlnkdDrZdyqTz7eksnBTCmcy8xjaoSF/urGD5294E5YjiT5Qefl29tCQIB4eFM/tPZoye/V+Pk1K5b/bjhEVGkynJtE0j4kiIjSY3AIHR9Kz+TU1nfSsfMLtQbwYDMPja/HoXb29GqMvi4uOYOrwdmR3Af4J+QUOHlu4leAgGx0b1aJFvSiiI+wUOpwcy8gh+dg5jmXkEGSDge0acP+ANtVzI5Q8NsESJNELr2pQK5w/jmzPE0MUq3afYs2eUyQfO8d3u06SV1CIPTiIRrUjGJzQkGta12VQuwbw/yAmUnbNioiwu+ppcvcYhqqrWbPnFL+kpJN06CyZuQXYbDZia4WT2LwOvVvXZWC7BsRFSws+0Mhfk6gWEaHBDOsYy7COsVeeUfp2DbEBV7eqy9Wt6podirAgOcUe8CyaWOWQv2IsW08W3a8ClCR6YS3SojdI6k2UTRJ9wLJqS1AI4WmS6IXFFLVM5YdICE+RRC+EEH5OEn3As2bfrjWjsiCrnoyVcy2WIoleWLTITsQAABGWSURBVIskCENsUm/iCiTRByqrtgRFJVl7OzotHl+gkEQvLMbdMpUfIiE8RhK9EH5Bum5E2STRBzjr9u1Ki75CLHvkY9X9KjBJohfWYtkfHquTehNlk0QfsKzaEhSVY/HtaNkjjsAiiV5YjNwZK4SnSaIXwh9Iz424Akn0AU8yhE+zateInGuxFEn0wlqcch29MZJYRdkk0QcqSaR+wurb0erxBQZDQwkqpezAPKA5UAhM0lrvLzHPeOBRwAHM1lrPLVbWENgF3KK1/sFQ5MJPSctUCE8z2qK/E0jXWvcFXgReLl6olIoCngcGAwOAx5RSMcVm+StwyQ+DEKIq5AdSlM1ooh8ELHa/XgH0KVHeC9iktc7QWmcD64rmUUoNBM4D2wyuWwhRRLrgRAUYTfSxwCkArbUDcCqlQksrdzsJxLnn+RPwrMH1Cn/nPhkrTz30dXKEYSXl9tErpaYAU0pM7lXifXl/lUXlU4F3tdbpSqkyZ05OTi4vrFLl5OQYXtabrBhXSPYp4oH8/HxLxWYryKEdUFBQYKm4ilhtW4ZknSAeKMjPs1RcQfmZKGQ7Vpa34io30Wut5wBzik9TSs3D1Wrf6j4xa9Na5xWb5ai7vEhjYANwFxCslHoIaA30VEqN1VrvKP75CQkJBr6K6wfC6LLeZMm4ztUGwG63Wyu2vEwAQqwWl5vltmVGLQDsIRarr9zzAISEhFgrLjfLbUe3qsSVlJRUZpmhq26AZcBYYClwI7CyRPlGYI5SqjZQgKt//lGt9ddFM7h/LOaVTPJCCCOkq0SUzWiiXwhcr5RaC+QCdwMopaYCq7TWP7pfL8W1B07TWmd4IF4hRHFyMlZUgKFEr7UuBCaVMv2VYq8XAYuu8Bl3G1m38DCr3arulIea+QWr7VcBTu6MDVTSEhTVQfYzS5BELyxGWoKVI4lUlE8SvRD+QLpKxBVIohfCl0nXiKgASfQBz2ItQXlMsZ+w2H4V4CTRByxJpP7FqolV9jMrkEQvLMaqCcuqJJGK8kmiFxYlCUwIT5FEL4Qvu3guQ46ERNkk0Qc8iyUIeUyxf5DLPS1FEn2gkqta/IrNoolVfrCtQRK9sBhrJizrkkQqyieJXliTHHEI4TGS6IXwZfKDKCpAEn2As1zfrtXiEQbJdrQSSfQBS1qC/sWiiVWOOCxBEr0QPk0SqSifJHphUZLAhPAUSfRC+LKirhE5tyGuQBK9sBZJWP5BtqOlSKIX1iQn8SpJEqsomyT6QGXZRCoJSwhPk0QvLMqqP0RC+B5J9EL4MssemQkrCTGykFLKDswDmgOFwCSt9f4S84wHHgUcwGyt9Vyl1N3AdGCfe7blWusXjYUuPMNiXSVyEk8IjzOU6IE7gXSt9Xil1BDgZeD2okKlVBTwPNATyAM2KaUWu4sXaq2frELMQgghKsFo180goChxrwD6lCjvBWzSWmdorbOBdaXMI0xl1UN+adFXjtWvo7fqfhZYjLboY4FTAFprh1LKqZQK1VrnlSx3OwnE4Wrd91dKLQHswJNa659LfnhycrKhoHJycgwv601WjCs4N522QH5+vqViC84544qroNBScRWx2rYMyjuPAgoKLLYdczPc27HAUnEVsdp2LOKtuMpN9EqpKcCUEpN7lXhf3s92UfkG4JTW+r9Kqd7A+0CnkjMnJCSUF1apkpOTDS/rTZaMK/MMAHa73VqxXTgJQIg9xFpxuVluW+ZkAGAPsVh9ZaUBFozLzXLb0a0qcSUlJZVZVm6i11rPAeYUn6aUmoer1b7VfWLWVqw1D3DUXV6kMbBBa70L2OX+3B+VUvWVUsFa68IKfhfhaVY75LdaPMIY2Y6WYrSPfhkw1v36RmBlifKNQA+lVG2lVA1c/fNrlFJ/UErdAaCU6oirdS9JXpRC+naF8BSjffQLgeuVUmuBXOBuAKXUVGCVu7U+FViK6+zaNK11hlLqQ+ADpdR97nVPruoXEAbJ9dd+omg7WrQFLfuZJRhK9O5W+KRSpr9S7PUiYFGJ8lTgOiPrFIHCoglLCB8md8YKi5KWYIXIY4pFBUiiD3gWSxCSsPyEbEcrkUQvrEn6doXwGEn0Qvg0q/8gWj2+wGD0qhtxBevXr+eNN94gODiYa6+9loEDB5odkg+RQ34hPE1a9F7wl7/8hZkzZ/LRRx+xbt06UlJSzA5J+Cvp4hIVIC36Cvr8889JSkoiLS2NAwcOMHnyZL788stL5omLi+Phhx8mOjqauLg4APr378/WrVsZMmSIGWH7HjkZa5DF6k22o6X4XKL/LCmVTzaX3kLOysoicnV6pT/ztu5NuTWxSbnz7d69m48//piDBw/y+OOP85///OeyebZs2UJMTMzF9zExMRw6dKjSMQlpqQrhKdJ1UwldunQhODiY2NhYzp8/b3Y4VSOH/H5CrqMX5fO5Fv2tiU3KbH17+4l0ISH/q67CwkImTpx4SXlcXByPPPIIp0+fvjjtxIkTl7TwRXkkYQnhaT6X6K0iODiYDz74oNSyCxcukJqaSmxsLCtXruT++++v5uh8n1OOOCpG6klUgCR6L3jhhRd44oknABgxYgSNGzc2OaIrsNohv9Xi8RE2yx0JWS2ewCaJvoJGjx598XVUVBTff/99mfP26NGDhQsXXnxvxZFsrE9aqkJ4ipyMDViSSP2DtbejdMFZgyR6YTFyyC+Ep0miFxYlLcEKkRazqABJ9AHOcifx5GSsMVarN6vFE+Ak0QshhJ+TRB+orH7Ib/HwrMPiY8bKhrQESfQVlJqaSteuXZk4cSITJkzgtttuY/ny5WXOv2TJEkPrKVouOTmZGTNmlDnfzJkzmT9/vqF1WJtVE5YQvkuuo6+Eli1bXrwbNj09nVtuuYV+/foRHh5+2byzZ89m2LBhlV5H0XIJCQlefZyD9UlLsEKsfmQmLEESvUG1a9emfv36bN++nXfeeYf8/HxsNhsvvvgiS5cuRWvNQw89xNtvv838+fM5ePAghYWFTJgwgZEjRzJ16lQaNGjAjh07OHr0KK+99ho//vjjxeUmTpzIggULmDFjBu+99x5Lly7F4XDQv39/HnroIQ9+E4u1oOUknkFWqzerxRPYfC/R//IR/Fx6l0WzrEzYEFX5z+w6AbrcUalFUlNTSU9P57PPPmPMmDGMGDGCJUuW8Pbbb/Pqq6/y7rvv8vbbb7N582ZOnTrFggULyMvL45ZbbmHw4MEA5OXlMXfuXD766CO++OILnn322YvLbdy48ZL1ffjhhwQFBTFo0CDuvvvuyn9HIUTA8r1Eb6IDBw4wceJEnE4nYWFhvPrqqzz//PMXn2vTq1cvZs2adckyW7ZsQWt98UmXDoeDU6dOAdC9e3cAYmNj+fXXX8tcb3h4OBMmTCAkJISzZ8+Snl75Z+5fzuqH/FaPzyqsXk9Wjy8wGEr0Sik7MA9oDhQCk7TW+0vMMx54FHAAs7XWc93TnwQmAPnAA1rrTZVaeZc7ymx9H/byY4qL99EXsdlsON3dDfn5+QQFXXp+OzQ0lMGDB/Pcc89d9nnBwcEXXzvL6LI4cuQI8+bNY/HixURFRTFy5Miqfg2Lk0N+Q6TLS1yB0atu7gTStdZ9gReBl4sXKqWigOeBwcAA4DGlVIxSqgMwDugO/A7w+azVqVOni90smzZtomPHjsD/EvdVV13Fpk2bcDgc5ObmMn369Ct+XsmEf/bsWWJiYoiKimLHjh0cOXKE/Px8L3wTa5FnpFSQ1JOoAKNdN4OA992vVwDvlSjvBWzSWmcAKKXWAX2A9sAnWusCYIv7n0975JFHePbZZ/nkk0+w2+289NJLACQkJDBmzBgWLVpEp06duP3223E6ndx5551X/Lyi5Z566qmL76Oiohg3bhyJiYmMGzeOadOmkZiY6JH4w87ugR2LPfJZHnHhpNkR+KTwjH3W2o7ZnuheFJ5iK6vL4EqUUsuAp7TWW93vU4DWWus89/s7gR5a68fc76cDKUBXXF09bQA78HjRZxRJSkpyRkZGGvoyOTk5pV7qaDYrxmUrzKPt4iEEFeaYHUqp9l39CnnNB5gdxmUsty2dDtouHkpwvjWHtjzQ/QVyWlf+MmNvs9x2dKtKXFlZWSQmJpZ6iFdui14pNQWYUmJyrxLvyzt+tBX7PxgYjquFPwfoUXJmo/3s3h5K0CirxkWL7ezb/hOtW7UyO5JLhYSRdyLXknVmyW3ZfCv7tm2w3nYMDiVHtmOlVCWupKSkMsvKTfRa6zm4EvJFSql5QCyw1X1i1lbUmnc76i4v0hjY4P5/l9baCaxVSrWo2FcQXlGjPnnRraCB9XZ4TshgLRUWVde62/GkbEcrMHoydhkw1v36RmBlifKNQA+lVG2lVA1crfc1wLfAUAClVDtc3TlCCCG8yOjJ2IXA9UqptUAucDeAUmoqsEpr/aP79VJc18tNc5+Y3aCUGq6U+tH9OQ9WKXohhBDlMpTotdaFwKRSpr9S7PUiYFEp8/wJ+JOR9QohhKg8eXqlEEL4OUn0Qgjh5yTRCyGEn5NEL4QQfs7QnbHelJSUZK2AhBDCR5R1Z6zlEr0QQgjPkq4bIYTwc5LohRDCz/n0CFNKqRBgLtAa13d5Umu9tsQ8pQ6AUg2x9Qc+Be7RWn9dSnk+sK7YpEHuG9HMjqva66uCA9lUa30ppd4ErsZ1Z/fviw+Qo5QaDLzkjvUbrfWVBxmovrgO4nqsSFG9jNdaH6mmuDoC/wHe1Fq/XaLMzPq6UlwHMa++/h/QD1feellr/XmxMo/Xl08nemAikKm17use1ORfQM+iwmIDoPQE8oBNSqnFWus0bwallGoNPM6liamkDK31AG/GUVJ5cZlVX/xvIJvxSqkhuAayub3EPNVWX+4fw3itdW+lVAKu8RZ6F5tlBq5nNh0BVimlPtNa77RAXADDtdYXvB1LibiigJnAd2XMYlZ9lRcXmFNf1wEd3duxLvAz8HmxWTxeX77edTMfV+ICOAXULVF+cQAUrXU2rgTXpxriOgaMBjKqYV2VUV5cZtXXIKBo1IwV1bTOKxkEfAGgtU4G6iilagEopVoBaVrrFK21A/jGPb+pcZksFxiB66m1lzC5vsqMy2Sr+d9DIdOBKKVUMHivvny6Ra+1zsc19iy4uhs+LDFLLK4fgCIngbhqiCsLQCl1pdnClVIf4uqu+Exr/YYF4jKlvoqvV2vtUEo5lVKhJR59XZ31FQsUf7j3Kfe0c5ReR629GEtF4yryD/fjv9cCT7sfCe5V7hHjCsrYr0yrr3LiKmJGfRUCme63k3F1zxR1H3mlvnwm0ZcxAMqftNZLlVIPAt1wPTL5Sjw+wOaV4ipn0SdxHZE4gdVKqdVa680WiKu46qqvigxk49X6KseV6sHMQVtLrvt5YAmQhqvlfyulPFjQZFYa5NbU+lJKjcKV6IdcYTaP1JfPJPrSBkABUEpNxpXgb3a38IsrawAUr8dVgeX+UfRaKfUd0AnwWOIyGJcp9VWBgWy8Xl8llKyHRri6vUora0z1dQ1cKS601kXjOKOU+gZXHZmd6M2srysys76UUkOBZ4FhRWNru3mlvny6j97dn3UfMFprXdrgp2UNgGIq5fKhUsrmvnKoD7DD7Lgwr76uOJCNCfW1DBjjXnc34KjW+jyA1vogUEsp1cIdy0j3/NWhzLiUUtFKqaVKqVD3vP2B7dUUV5lMrq8ymVlfSqlo4K/AyJIXOnirvnz6zlil1EvAOOBwsclDcJ2gLRoAZQzwFK5D/pla6wXVENcN7nW2w9XfdkxrPaTEwCyvAgNxXcb4pdb6RYvEZUZ9BeNq5cfjHshGa51iZn0ppV4BrnWv70FcA9tnaK0XK6WuBV51z/qZ1vo1b8ZSibh+D9wFZOO6kuPh6uhzVkolAq8DLXCdMzsCfAkcMLO+KhCXWfX1W+AFYHexyd8D27xVXz6d6IUQQpTPp7tuhBBClE8SvRBC+DlJ9EII4eck0QshhJ+TRC+EEH5OEr0QQvg5SfRCCOHnJNELIYSf+/8oAdtZiIkIcQAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "g4pI4U_F4zsR",
        "outputId": "bf1151fd-7b18-4c59-e957-7b8b8c1debda"
      },
      "source": [
        "eval[1]"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "-3.5692471115960975"
            ]
          },
          "metadata": {},
          "execution_count": 17
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "v25qjrDTUk15"
      },
      "source": [
        "### Lennard Jones Potential\n",
        "\n",
        "\\begin{equation}\n",
        "V(x) = a\\left(\\frac{x^2 - b^2}{b^2}\\right)^2\n",
        "\\end{equation}\n",
        "\n",
        "I am taking $a$ as natuarl energy salce and $l_0 = \\sqrt{\\dfrac{\\hbar^2}{2ma^2}}$ as natural length scale\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "7Jjl8J18KNLJ"
      },
      "source": [
        "import numpy as np\n",
        "import math\n",
        "import matplotlib.pyplot as plt\n",
        "import numpy.linalg as lin"
      ],
      "execution_count": 46,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "hXKasQnV7pin"
      },
      "source": [
        "h = 6.67e-34 #plank constant\n",
        "hbar = h/math.pi\n",
        "qe = 1.6e-19 #chrage of electron\n",
        "m =9.1e-31 #mass of electron"
      ],
      "execution_count": 47,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "hBZvpCcGyaIu"
      },
      "source": [
        "a = 10*qe"
      ],
      "execution_count": 50,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "zyJSFm2Dyfma"
      },
      "source": [
        "b = 2e-10"
      ],
      "execution_count": 51,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "v47Hybeey5l1"
      },
      "source": [
        "l0 =math.sqrt( hbar**2/(2*m*a))"
      ],
      "execution_count": 52,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ZV5gBpYdzAuP"
      },
      "source": [
        "def V(x):\n",
        "    return ((l0*x/b)**2-1)**2"
      ],
      "execution_count": 53,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "65Wwk49VzMlF"
      },
      "source": [
        "n = 100 #number of states we want to know\n",
        "y = np.linspace(-3,3,2*n)\n",
        "Delta = y[1]-y[0]"
      ],
      "execution_count": 54,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "4a4hKjjfzT2J"
      },
      "source": [
        "def k(i,j):\n",
        "    return 1 if i==j else 0\n",
        "\n",
        "def Hemiltonian(i,j):\n",
        "    return (-1/Delta**2)*(k(i+1,j)+k(i-1,j)-2*k(i,j)) + V(Delta*i)*k(i,j)\n",
        "    \n",
        "H = np.array([[Hemiltonian(i,j) for i in range(-n,n)] for j in range(-n,n)])"
      ],
      "execution_count": 55,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "1JQN6p4hzWQS"
      },
      "source": [
        "eval, evac = lin.eig(H)"
      ],
      "execution_count": 56,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "MmXXjqK8za79"
      },
      "source": [
        "idx = eval.argsort()\n",
        "eval = eval[idx]\n",
        "evac = evac[:,idx]\n",
        "psi1 = evac[:,1]\n",
        "psi2 = evac[:,2]\n",
        "psi0 = evac[:,0]\n",
        "\n"
      ],
      "execution_count": 57,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 308
        },
        "id": "z-AwZdWOzey6",
        "outputId": "f669ac8b-fd3a-49b8-a69e-c72391cb013d"
      },
      "source": [
        "plt.plot(y,psi0, label=\"n=0\")\n",
        "plt.ylabel(r'$\\psi_1$')\n",
        "plt.xlabel(r'$x$')\n",
        "plt.title('First Excited State')"
      ],
      "execution_count": 59,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Text(0.5, 1.0, 'First Excited State')"
            ]
          },
          "metadata": {},
          "execution_count": 59
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "g5_T1jKczic1",
        "outputId": "712edf8f-03aa-41db-8bf5-d7190b25ecf1"
      },
      "source": [
        "eval[0]"
      ],
      "execution_count": 60,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.8251261941677884"
            ]
          },
          "metadata": {},
          "execution_count": 60
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "xk6TP6JUzpiQ",
        "outputId": "83b5308b-4c38-4eb0-a052-370da0a435f1"
      },
      "source": [
        "eval[1]\n",
        "eval[1]-eval[0]"
      ],
      "execution_count": 61,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.6435520524651223"
            ]
          },
          "metadata": {},
          "execution_count": 61
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 308
        },
        "id": "o10hGWsNzrCv",
        "outputId": "da9e1aeb-a36a-479b-dbc9-e05b9f901d13"
      },
      "source": [
        "plt.plot(y, V(y))\n",
        "plt.ylabel(r'$V(x)$')\n",
        "plt.xlabel(r'$x$')\n",
        "plt.title('Potential')"
      ],
      "execution_count": 62,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Text(0.5, 1.0, 'Potential')"
            ]
          },
          "metadata": {},
          "execution_count": 62
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "l12PQrAA0W8E"
      },
      "source": [
        "def Va(a):\n",
        "    h = 6.64e-34\n",
        "    qe = 1.6e-19\n",
        "    m = 9.1e-31\n",
        "    b = 2*1e-10\n",
        "    l0 =math.sqrt(h**2/(2*m*a*qe))\n",
        "    def f(x):\n",
        "        return ((l0*x/b)**2-1)**2\n",
        "    return f"
      ],
      "execution_count": 64,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "YkPuSWBa4xGI"
      },
      "source": [
        "A_list =np.linspace(10,50,50)\n",
        "E_requard = []\n",
        "for a in A_list:\n",
        "    n = 100 \n",
        "    y = np.linspace(-3,3,2*n)\n",
        "    Delta = y[1]-y[0]\n",
        "    V = Va(a)\n",
        "    def Hemiltonian(i,j):\n",
        "        return (-1/Delta**2)*(k(i+1,j)+k(i-1,j)-2*k(i,j)) + V(Delta*i)*k(i,j)\n",
        "    H = np.array([[Hemiltonian(i,j) for i in range(-n,n)] for j in range(-n,n)])\n",
        "    eval, evac = lin.eig(H)\n",
        "    idx = eval.argsort()\n",
        "    eval = eval[idx]\n",
        "    E_requard.append(eval[1]-eval[0])"
      ],
      "execution_count": 65,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 294
        },
        "id": "TuilEWie5f6B",
        "outputId": "04246184-046e-4ac3-fbd0-6dd2ed6eadf1"
      },
      "source": [
        "plt.plot(A_list,E_requard)\n",
        "plt.xlabel('a')\n",
        "plt.ylabel('E')"
      ],
      "execution_count": 76,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Text(0, 0.5, 'E')"
            ]
          },
          "metadata": {},
          "execution_count": 76
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "wRCdQcHsWppI"
      },
      "source": [
        "## Hydrogen Atom"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ocT08yBrX77x"
      },
      "source": [
        "import numpy as np\n",
        "import math\n",
        "import matplotlib.pyplot as plt\n",
        "import numpy.linalg as lin"
      ],
      "execution_count": 76,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "dWu0CxxqSi1h",
        "outputId": "6bbff9c4-6106-4146-d04a-d2e0d6a59361"
      },
      "source": [
        "h = 6.62e-34 #plank constant\n",
        "hbar = h/2*math.pi\n",
        "m = 9.1e-31 #electron mass\n",
        "K = 9e9 #1/4pie \n",
        "e = 1.6e-19 #electron charge\n",
        "E0 = 13.6*e #ground state energy in joule\n",
        "\n",
        "l0 = math.sqrt((hbar**2)/(2*m*E0)) #natrual length scale\n",
        "b = (K*e**2)/(l0*E0)\n",
        "b"
      ],
      "execution_count": 94,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.20263331844755425"
            ]
          },
          "metadata": {},
          "execution_count": 94
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "JRFm5p3YQFLW"
      },
      "source": [
        "n = 400 #number of states we want to know\n",
        "h = 0.01"
      ],
      "execution_count": 90,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "WNMwl1TKW7Qe"
      },
      "source": [
        "def k(i,j): #kronekar delta function\n",
        "    return 1 if i==j else 0\n",
        "\n",
        "def V(x):\n",
        "    return - b/x  # for hygrogen atom "
      ],
      "execution_count": 100,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "XicEiQmrXCpw"
      },
      "source": [
        "def Hemiltonian(i,j):\n",
        "    return (-1/Delta**2)*(k(i+1,j)+k(i-1,j)-2*k(i,j)) + V(Delta*i)*k(i,j)\n",
        "\n",
        "H = np.array([[Hemiltonian(i,j) for i in range(1,2*n+1)] for j in range(1,2*n+1)])\n",
        "ys = np.array([i*h for i in range(1,2*n+1)])\n",
        "eval, evac = lin.eig(H)\n",
        "sort_eval = np.sort(eval)\n",
        "idx = eval.argsort()\n",
        "eval = eval[idx]\n",
        "evac = evac[:,idx]\n",
        "psi1 = evac[:,1]\n",
        "psi2 = evac[:,2]\n",
        "psi0 = evac[:,0]\n"
      ],
      "execution_count": 101,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        },
        "id": "r9dDjLyuXdFz",
        "outputId": "afd27146-f8fa-4772-d88e-5e2f069a5679"
      },
      "source": [
        "potential = V(y)\n",
        "# plt.plot(y,psi1, label=\"n=1\")\n",
        "# plt.plot(y,psi2, label=\"n=2\")\n",
        "plt.plot(y,psi0, label=\"n=0\")\n",
        "# plt.plot(y,psi0/y, label='R')\n",
        "# plt.plot(y, potential, label=\"Potential\")\n",
        "plt.ylabel(r'$\\Psi$')\n",
        "plt.xlabel('x')\n",
        "plt.legend();"
      ],
      "execution_count": 106,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "LHP1-joeYZht",
        "outputId": "ae2723dd-0f90-49d6-8b7b-d0fc47e6f86d"
      },
      "source": [
        "eval[2]"
      ],
      "execution_count": 103,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.23293420348368932"
            ]
          },
          "metadata": {},
          "execution_count": 103
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "5DSm-5Dra6AE"
      },
      "source": [
        ""
      ],
      "execution_count": null,
      "outputs": []
    }
  ]
}