Radial_Hydrogen.ipynb

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    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
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  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
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      "source": [
        "<a href=\"https://colab.research.google.com/github/krislars/modphys/blob/master/Radial_Hydrogen.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "metadata": {
        "id": "jmLnVzcPctxL"
      },
      "cell_type": "markdown",
      "source": [
        "# Hydrogen Atom"
      ]
    },
    {
      "metadata": {
        "id": "9SDK50hfczur"
      },
      "cell_type": "markdown",
      "source": [
        "Kristen Larson, Western Washington University"
      ]
    },
    {
      "metadata": {
        "id": "gbcLGnLRZpLY"
      },
      "cell_type": "code",
      "source": [
        "from sympy import Symbol,integrate,conjugate,plot,diff,solve,sin,cos,oo,pi\n",
        "from sympy.plotting import plot_parametric,plot3d_parametric_surface\n",
        "from sympy.physics.hydrogen import R_nl,Psi_nlm,Ynm"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "metadata": {
        "id": "FDz2ePTlcez3"
      },
      "cell_type": "code",
      "source": [
        "r=Symbol(\"r\", real=True, positive=True)\n",
        "phi=Symbol(\"phi\", real=True)\n",
        "theta=Symbol(\"theta\", real=True)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "metadata": {
        "id": "hgebJvkUd_qi"
      },
      "cell_type": "markdown",
      "source": [
        "## Radial Part, $R_{nl}(r)$"
      ]
    },
    {
      "metadata": {
        "id": "233ib7M0c6lX"
      },
      "cell_type": "markdown",
      "source": [
        "Using Sympy (Symbolic Python, like Mathematica), we can access the hydrogen wavefunctions directly.  Let's start with the radial part, R_nl(r).  Here are some cool things that you can do.\n",
        "1. Print out the wavefunction"
      ]
    },
    {
      "metadata": {
        "id": "W9NNVbY4daJK",
        "outputId": "6de71ef6-d154-4782-eea5-2e265bb8b63f",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 55
        }
      },
      "cell_type": "code",
      "source": [
        "R_nl(2,1,r)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "sqrt(6)*r*exp(-r/2)/12"
            ],
            "text/latex": "$\\displaystyle \\frac{\\sqrt{6} r e^{- \\frac{r}{2}}}{12}$"
          },
          "metadata": {},
          "execution_count": 5
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "R_nl(3,2,r)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 55
        },
        "id": "ZtuxHpOVUWXt",
        "outputId": "1484a890-b5bb-4868-acb3-2c91f895debb"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "2*sqrt(30)*r**2*exp(-r/3)/1215"
            ],
            "text/latex": "$\\displaystyle \\frac{2 \\sqrt{30} r^{2} e^{- \\frac{r}{3}}}{1215}$"
          },
          "metadata": {},
          "execution_count": 8
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "R_nl(2,0,r)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 55
        },
        "id": "M0N4Za2cUoWM",
        "outputId": "3190109d-fe4a-4607-e2d6-99c58c394057"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "sqrt(2)*(2 - r)*exp(-r/2)/4"
            ],
            "text/latex": "$\\displaystyle \\frac{\\sqrt{2} \\left(2 - r\\right) e^{- \\frac{r}{2}}}{4}$"
          },
          "metadata": {},
          "execution_count": 11
        }
      ]
    },
    {
      "metadata": {
        "id": "93MTXe00epWc"
      },
      "cell_type": "markdown",
      "source": [
        "2. Integrate the wavefunction.  \n",
        "\n",
        "The function below is the probability density integrated over all radii.  How would you change it to be an average?  Try it!  Notice that if you are expecting a radius, the answer is reported as a multiple of the Bohr radius.  Try changing the limits of integration.  What is the change of finding the particle outside of its average?"
      ]
    },
    {
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          "height": 37
        }
      },
      "cell_type": "code",
      "source": [
        "integrate(R_nl(2, 0, r)**2 * r**2, (r, 0, oo))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "1"
            ],
            "text/latex": "$\\displaystyle 1$"
          },
          "metadata": {},
          "execution_count": 12
        }
      ]
    },
    {
      "metadata": {
        "id": "KRp3UsjDgPwH"
      },
      "cell_type": "markdown",
      "source": [
        "3.  Plot the wavefunction\n",
        "\n",
        "The plot below is the probability density.  How would you change it to plot the wavefunction?"
      ]
    },
    {
      "metadata": {
        "id": "4MCXimMvbuE3",
        "outputId": "c0d2012d-93f0-401d-cdcc-977158dbf535",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 315
        }
      },
      "cell_type": "code",
      "source": [
        "p1 = plot(R_nl(5, 0, r)**2 * r**2, (r, 0, 60),\n",
        "          xlabel='radius', ylabel='probability density')\n",
        "p1"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<sympy.plotting.plot.Plot at 0x7fb0621a6710>"
            ]
          },
          "metadata": {},
          "execution_count": 16
        }
      ]
    },
    {
      "metadata": {
        "id": "jBm-eTQAivtV"
      },
      "cell_type": "markdown",
      "source": [
        "4.  Differentiate the wavefunction.\n",
        "\n",
        "Here we differentiate the probability density and set that equal to zero to find the maxima.  Guess what the answer is from the plot above, and then check your understanding below."
      ]
    },
    {
      "metadata": {
        "id": "aH0eYDtvYeRr",
        "outputId": "9ca5be88-80d6-4393-be01-0614c2242294",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 56
        }
      },
      "cell_type": "code",
      "source": [
        "diff(R_nl(2, 0, r)**2 * r**2)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "-r**2*(2 - r)**2*exp(-r)/8 + r**2*(2*r - 4)*exp(-r)/8 + r*(2 - r)**2*exp(-r)/4"
            ],
            "text/latex": "$\\displaystyle - \\frac{r^{2} \\left(2 - r\\right)^{2} e^{- r}}{8} + \\frac{r^{2} \\left(2 r - 4\\right) e^{- r}}{8} + \\frac{r \\left(2 - r\\right)^{2} e^{- r}}{4}$"
          },
          "metadata": {},
          "execution_count": 17
        }
      ]
    },
    {
      "metadata": {
        "id": "lzPguUPRjgPS",
        "outputId": "b44d2e97-0dd9-4972-bcb9-e9dd76edb5e0",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "cell_type": "code",
      "source": [
        "zeros=solve(diff(R_nl(2, 0, r)**2 * r**2), r)\n",
        "zeros"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[2, 3 - sqrt(5), sqrt(5) + 3]"
            ]
          },
          "metadata": {},
          "execution_count": 18
        }
      ]
    },
    {
      "metadata": {
        "id": "KjP_1LyOaFYw",
        "outputId": "8ee8df83-c152-4d37-80ec-16a12daaeeed",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "cell_type": "code",
      "source": [
        "zeros_eval=[x.evalf(3) for x in zeros]\n",
        "zeros_eval"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[2.00, 0.764, 5.24]"
            ]
          },
          "metadata": {},
          "execution_count": 19
        }
      ]
    },
    {
      "metadata": {
        "id": "g1yFSgWsDDEH"
      },
      "cell_type": "markdown",
      "source": [
        "## Full Wavefunction"
      ]
    },
    {
      "metadata": {
        "id": "8A93k_f1hGBK"
      },
      "cell_type": "markdown",
      "source": [
        "Notice that you don't have to use just the radial part or the angular part, as sympy provides the whole wavefunction, $\\psi_{nlm}(r,\\theta,\\phi)$.\n"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "Psi_nlm(2,0,0,r,phi,theta,Z=1)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 60
        },
        "id": "XN-cytTQ7xu-",
        "outputId": "bd2a24a3-b7e4-4018-828f-ceb372c4a598"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "sqrt(2)*(2 - r)*exp(-r/2)/(8*sqrt(pi))"
            ],
            "text/latex": "$\\displaystyle \\frac{\\sqrt{2} \\left(2 - r\\right) e^{- \\frac{r}{2}}}{8 \\sqrt{\\pi}}$"
          },
          "metadata": {},
          "execution_count": 20
        }
      ]
    },
    {
      "metadata": {
        "id": "hy5Uil6rgsyt",
        "outputId": "7b23c0a1-d608-4e6d-b1cc-c6fc5ba3de50",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 316
        }
      },
      "cell_type": "code",
      "source": [
        "p2 = plot(Psi_nlm(2,0,0,r,phi=0,theta=0,Z=1)**2 * 4 * pi * r**2, (r, 0, 10),\n",
        "          xlabel='radius', ylabel='probability density')\n",
        "p2"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<sympy.plotting.plot.Plot at 0x7fb7fd718350>"
            ]
          },
          "metadata": {},
          "execution_count": 35
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [],
      "metadata": {
        "id": "lMOZqMS69JIb"
      },
      "execution_count": null,
      "outputs": []
    }
  ]
}