SchrodingerSolver.ipynb

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      "name": "SchrodingerSolver.ipynb",
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      "source": [
        "#Import necessary library\n",
        "import numpy as np\n",
        "import pandas as pd"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "def potential(xmin, xmax):\n",
        "  '''\n",
        "  Determine the potential within the system in certain interval\n",
        "  Input   : x interval\n",
        "  Return  : potential\n",
        "  '''\n",
        "\n",
        "  potential_v = 5/2\n",
        "\n",
        "  return potential_v"
      ],
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      "execution_count": null,
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      "cell_type": "code",
      "source": [
        "def normalize(n, h, psi):\n",
        "  '''\n",
        "  Normalize wave function\n",
        "  Input   : initial wave function, number of points, step\n",
        "  Return  : normalized wave function\n",
        "  '''\n",
        "  norm = psi[0]**2+psi[n]**2\n",
        "  for i in range(1,n-3,2):\n",
        "    norm = norm+4*psi[i]**2+2*psi[i+1]**2\n",
        "\n",
        "  norm = norm + 4*psi[n-1]**2\n",
        "  norm = 1/np.sqrt((norm*h/3))\n",
        "\n",
        "  for i in range(0, len(psi)):\n",
        "   psi[i] = psi[i]*norm\n",
        "\n",
        "  return psi"
      ],
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      "execution_count": null,
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      "source": [
        "def numerov(nx, x, e_init, psi):\n",
        "  '''\n",
        "  Integrates the wave function using Numerov's method\n",
        "  Input   : number of points, x inteval, initial energy, wave function\n",
        "  Return  : integrated wave function\n",
        "  '''\n",
        "\n",
        "  #Calculates the steps\n",
        "  h = xmax/nx\n",
        "  h2 = h**2\n",
        "  h12 = h2/12\n",
        "\n",
        "  #Initialize wave function\n",
        "  psi[0] = 0\n",
        "  psi[1] = 0.0001\n",
        "\n",
        "  #Calculates using Numerov's nethod\n",
        "  fn = 2*(potential(-x, x)-e_init)\n",
        "  q0 = psi[0]*(1-h12*fn)\n",
        "  fn = 2*(potential(-x+h, x)-e_init)\n",
        "  q1 = psi[1]*(1-h12*fn)\n",
        "  \n",
        "  for n in range(2, 2*nx+1):\n",
        "    q2 = h2*fn*psi[n-1]+2*q1-q0\n",
        "    fn = 2*(potential(n*h-x, x)-e_init)\n",
        "    psi[n] = q2/(1-h12*fn)\n",
        "    q0 = q1\n",
        "    q1 = q2\n",
        "\n",
        "  psi = normalize(2*nx, h, psi)\n",
        "\n",
        "  return psi"
      ],
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      "source": [
        "def calculates(nx, x, e_init, de, eps, psi, cs_energy, cs_dev, bis_energy, bis_dev):\n",
        "  '''\n",
        "  Starting course search and approximate energy level using bisection method\n",
        "  Input   : number of points, x inteval, initial energy,\n",
        "            eps, lists to store energy and deviation\n",
        "  Return  : Energy approximation\n",
        "  '''\n",
        "  #Starting course search\n",
        "  psi = numerov(nx, x, e_init, psi)\n",
        "  dev1 = psi[2*nx]\n",
        "\n",
        "  #Store initial energy and initial boundary deviation\n",
        "  cs_energy.append(e_init)\n",
        "  cs_dev.append(dev1)\n",
        "\n",
        "  # for i in range(1, len(psi)):\n",
        "  while True:\n",
        "    #Update energy with de step\n",
        "    e2 = e_init + de\n",
        " \n",
        "    psi = numerov(nx, x, e2, psi)\n",
        "    dev2 = psi[2*nx]\n",
        "\n",
        "    #Store energy and boundary deviation each iteration\n",
        "    cs_energy.append(e2)\n",
        "    cs_dev.append(dev2)\n",
        "\n",
        "    #Check if deviation is zero\n",
        "    if dev1*dev2 < 0:\n",
        "      break\n",
        "\n",
        "    #Update energy and deviation value for next iteration  \n",
        "    e_init = e2\n",
        "    dev1 = dev2\n",
        "\n",
        "  #Starting bisection\n",
        "  # e3 = (e_init+2)/2\n",
        "  # print(e_init-e2)\n",
        "  while abs(e_init-e2) > eps:\n",
        "    e3 = (e_init+e2)/2\n",
        "    psi = numerov(nx,x,e3,psi)\n",
        "    dev3 = psi[2*nx]\n",
        "\n",
        "    #Store energy and boundary deviation each iteration\n",
        "    bis_energy.append(e3)\n",
        "    bis_dev.append(dev3)\n",
        "\n",
        "    if dev3*dev1 <= 0.000000 :\n",
        "      e2 = e3\n",
        "      dev2 = dev3\n",
        "    else:\n",
        "      e_init = e3\n",
        "      dev2 = dev3\n",
        "\n",
        "  return e3, psi"
      ],
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      "source": [
        "#Initialize Parameters\n",
        "nx = 100 #number of poins\n",
        "xmax = 2 #x maximum\n",
        "e0 = 0 #initial/guessed energy\n",
        "de = 0.001 #energy step\n",
        "eps = 10e-6 #error limit\n",
        "dx = xmax/nx #x step\n",
        "\n",
        "#Initialize wave function\n",
        "psi = np.zeros(2*nx+1)\n",
        "\n",
        "#Make lists to store energy and deviation\n",
        "cs_energy = []\n",
        "cs_dev = []\n",
        "bis_energy = []\n",
        "bis_dev = []\n",
        "\n",
        "#Calculates energy and deviation\n",
        "energy, psi = calculates(nx, xmax, e0, de, eps, psi, cs_energy, cs_dev, bis_energy, bis_dev)\n",
        "\n",
        "print('Guessed energy: ', e0)\n",
        "print('Approximated energy: ', energy)\n",
        "\n"
      ],
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          "name": "stdout",
          "text": [
            "Guessed energy:  0\n",
            "Approximated energy:  2.8084296874998014\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#Make dataframe to store energy and deviation\n",
        "\n",
        "output_cs = pd.DataFrame(columns=['Energy', 'Boundary Deviation'])\n",
        "output_bisection = pd.DataFrame(columns=['Energy', 'Boundary Deviation'])\n",
        "\n",
        "for i in range(len(cs_energy)):\n",
        "  output_cs.loc[i] = [cs_energy[i], cs_dev[i]]\n",
        "\n",
        "for i in range(len(bis_energy)):\n",
        "  output_bisection.loc[i] = [bis_energy[i], bis_dev[i]]"
      ],
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              "      Energy  Boundary Deviation\n",
              "0      0.000            2.281782\n",
              "1      0.001            2.281525\n",
              "2      0.002            2.281268\n",
              "3      0.003            2.281011\n",
              "4      0.004            2.280754\n",
              "...      ...                 ...\n",
              "2805   2.805            0.012335\n",
              "2806   2.806            0.008734\n",
              "2807   2.807            0.005132\n",
              "2808   2.808            0.001531\n",
              "2809   2.809           -0.002070\n",
              "\n",
              "[2810 rows x 2 columns]"
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              "      <th>Energy</th>\n",
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              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
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              "      <th>2</th>\n",
              "      <td>0.002</td>\n",
              "      <td>2.281268</td>\n",
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              "      <th>3</th>\n",
              "      <td>0.003</td>\n",
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              "      <td>2.808</td>\n",
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              "    <tr>\n",
              "      <th>2809</th>\n",
              "      <td>2.809</td>\n",
              "      <td>-0.002070</td>\n",
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              "<p>2810 rows × 2 columns</p>\n",
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            "text/plain": [
              "     Energy  Boundary Deviation\n",
              "0  2.808500           -0.000270\n",
              "1  2.808250            0.000631\n",
              "2  2.808375            0.000181\n",
              "3  2.808437           -0.000045\n",
              "4  2.808406            0.000068\n",
              "5  2.808422            0.000012\n",
              "6  2.808430           -0.000016"
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              "      <th>1</th>\n",
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              "      <th>2</th>\n",
              "      <td>2.808375</td>\n",
              "      <td>0.000181</td>\n",
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              "      <th>3</th>\n",
              "      <td>2.808437</td>\n",
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              "      <th>4</th>\n",
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              "      <td>2.808422</td>\n",
              "      <td>0.000012</td>\n",
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              "      <th>6</th>\n",
              "      <td>2.808430</td>\n",
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              "      background-color: #E2EBFA;\n",
              "      box-shadow: 0px 1px 2px rgba(60, 64, 67, 0.3), 0px 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
              "      fill: #174EA6;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert {\n",
              "      background-color: #3B4455;\n",
              "      fill: #D2E3FC;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert:hover {\n",
              "      background-color: #434B5C;\n",
              "      box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);\n",
              "      filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));\n",
              "      fill: #FFFFFF;\n",
              "    }\n",
              "  </style>\n",
              "\n",
              "      <script>\n",
              "        const buttonEl =\n",
              "          document.querySelector('#df-0dd6cc65-650e-4e99-89ed-e6bb1bbb75dc button.colab-df-convert');\n",
              "        buttonEl.style.display =\n",
              "          google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "\n",
              "        async function convertToInteractive(key) {\n",
              "          const element = document.querySelector('#df-0dd6cc65-650e-4e99-89ed-e6bb1bbb75dc');\n",
              "          const dataTable =\n",
              "            await google.colab.kernel.invokeFunction('convertToInteractive',\n",
              "                                                     [key], {});\n",
              "          if (!dataTable) return;\n",
              "\n",
              "          const docLinkHtml = 'Like what you see? Visit the ' +\n",
              "            '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
              "            + ' to learn more about interactive tables.';\n",
              "          element.innerHTML = '';\n",
              "          dataTable['output_type'] = 'display_data';\n",
              "          await google.colab.output.renderOutput(dataTable, element);\n",
              "          const docLink = document.createElement('div');\n",
              "          docLink.innerHTML = docLinkHtml;\n",
              "          element.appendChild(docLink);\n",
              "        }\n",
              "      </script>\n",
              "    </div>\n",
              "  </div>\n",
              "  "
            ]
          },
          "metadata": {},
          "execution_count": 67
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#Save as dat file\n",
        "\n",
        "filename = 'psi-out2.dat'\n",
        "x = np.zeros(2*nx+1)\n",
        "for i in range(1, 2*nx+2):\n",
        "  x[i-1] = (i-1)*dx-xmax\n",
        "  \n",
        "with open(filename, 'w') as out_file:\n",
        "  for item in range(len(psi)):\n",
        "    # out_file.write(str(x[item]) + '      ' + (psi[item]) + '\\n')\n",
        "    out_file.write(\"{}      {}      \\n\".format(x[item], psi[item]))\n"
      ],
      "metadata": {
        "id": "rkPmLfNFyfkn"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "import matplotlib.pyplot as plt \n",
        "\n",
        "plt.plot(x, psi)\n",
        "plt.show()"
      ],
      "metadata": {
        "id": "rzWZ65t-Dx6B",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "outputId": "6a430bf2-5064-4beb-dbdd-aa6307352b1c"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
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          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    }
  ]
}