[4]_Module_4_Ordinary_Differential_Equations.ipynb

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      "source": [
        "# **AP155 Lab Assignment** \n",
        "## Module 5: Ordinary Differential Equations\n",
        "\n",
        "_Instructions_: Answer each problem as completely as you can. Discuss **all** your answers as clearly and concisely as possible. \n",
        "\n",
        "_Scoring Criteria_: 50% - *correctness of code*; 50% - *discussion of the code and results*. Maximum score is **100 points** (Problems 1 to 3 are worth 25, 25, and 50 points, respectively).\n",
        "\n",
        "_Due Date_: **5PM of December 7 (Tue)**\n"
      ]
    },
    {
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      "metadata": {
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      "source": [
        "### Student Information \n",
        "\n",
        "_Full Name (Last Name, First Name)_: Jonabel Eleanor B. Baldres\\\n",
        "_Student No._: 2020-07278\\\n",
        "_Section_: THU-HX-3"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Hc183co-J4eI"
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      "source": [
        "### Submission Information\n",
        "\n",
        "_Date and Time Submitted (most recent upload)_: 3 December 2022\n",
        "\n",
        "**HONOR PLEDGE** I affirm that I have upheld the highest principles of honesty and integrity in my academic work and that this lab assignment is my own work.\n",
        "\n",
        "**Sign here with your full name:** sgd. Jonabel Eleanor Baldres"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "aOuHbwPqGjZ4"
      },
      "source": [
        "### Grading Information (c/o Lab Instructor)\n",
        "\n",
        "TOTAL SCORE: **97**/100\n",
        "\n",
        "Score breakdown:\n",
        "* Problem 1 - 25/25\n",
        "* Problem 2 - 24/25\n",
        "* Problem 3 - 48/50\n",
        "\n",
        "_Date and Time Scored (MM/DD/YYYY HH:MM AM/PM):_ "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "hMqWOsbIMoUh"
      },
      "source": [
        "# importing important libraries\n",
        "import matplotlib.pyplot as plt\n",
        "import numpy as np\n",
        "from math import sin, pi, inf, sqrt"
      ],
      "execution_count": 670,
      "outputs": []
    },
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      "metadata": {
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      "source": [
        "### PROBLEM 1 (25 pts.; 50/50 for code & discussion) \n",
        "**The Lotka-Volterra equations**\n",
        "\n",
        "_Refer to Exercise 8.2 in the Newman text._ The Lotka-Volterra equations are a mathematical model of predator-prey interactions between biological species. Let two variables $x$ and $y$ be proportional to the size of the populations of two species, traditionally called \"rabbits\" (the prey) and \"foxes\" (the predators). You could think of $x$ and $y$ as being the population in thousands, say, so that $x = 2$ means there are 2000 rabbits. \n",
        "\n",
        "In the Lotka-Volterra model the rabbits reproduce at a rate proportional to their population, but are eaten by the foxes at a rate proportional to both their own population and the population of foxes:\n",
        "\n",
        "$$\\frac{{\\rm d}x}{{\\rm d}t} = \\alpha x - \\beta xy,$$\n",
        "\n",
        "where $\\alpha$ and $\\beta$ are constants. At the same time the foxes reproduce at a rate proportional to the rate at which they eat rabbits-- because they need food to grow and reproduce-- but also die of old age at a rate proportional to their own population:\n",
        "\n",
        "$$\\frac{{\\rm d}y}{{\\rm d}t}  = \\gamma xy - \\delta y,$$\n",
        "\n",
        "where $\\gamma$ and $\\delta$ are also constants. *(15 pts.)*\n",
        "\n",
        "1. Solve these equations using the fourth-order Runge-Kutta method for the case $\\alpha = 1, \\beta = \\gamma = 0.5$, and $\\delta = 2$, starting from the initial condition $x = y = 2$. \n",
        "\n",
        "2. Make a graph showing both $x$ and $y$ as a function of time on the same axes from $t =0$ to $t = 30$. *(10 pts.)*"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "def f(r,t):\n",
        "  '''defining the equations given'''\n",
        "  x = r[0] #the value of x is stored as the first index of the array r\n",
        "  y = r[1] #the value of x is stored as the second index of the array r\n",
        "  fx = alpha * x - beta * x * y # dx/dt\n",
        "  fy = gamma * x * y - delta * y #dy/dt\n",
        "  return np.array([fx,fy],float) "
      ],
      "metadata": {
        "id": "fqZKdNQ5Ff3W"
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      "execution_count": 671,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "#listing all the constants provided\n",
        "alpha = 1\n",
        "beta = 0.5\n",
        "delta = 2\n",
        "gamma = 0.5\n",
        "a= 0.0 #starting time\n",
        "b = 30.0 #end time\n",
        "N= 1000 #number of steps\n",
        "h = (b-a) / N"
      ],
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        "id": "ARrfQnTOF4oP"
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      "execution_count": 672,
      "outputs": []
    },
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      "cell_type": "code",
      "source": [
        "tpoints = np.arange(a,b,h)\n",
        "xpoints = [] # where we store the values for x \n",
        "ypoints = [] # where we store the values for y"
      ],
      "metadata": {
        "id": "60dq-jD6GmX3"
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      "execution_count": 673,
      "outputs": []
    },
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      "cell_type": "code",
      "source": [
        "#code for the runge-kutta methood\n",
        "r = np.array([2.0, 2.0]) #here, the initial conditions of x and y starts at 2\n",
        "for t in tpoints:\n",
        "  xpoints.append(r[0])\n",
        "  ypoints.append(r[1])\n",
        "  k1 = h * f(r,t)\n",
        "  k2 = h * f(r+ 0.5 * k1, t+0.5*h)\n",
        "  k3 = h *f(r+0.5*k2, t+0.5*h)\n",
        "  k4 = h*f(r+k3,t+h)\n",
        "  r += (k1+2*k2+2*k3+k4) / 6\n",
        "\n",
        "fig, ax = plt.subplots()\n",
        "ax.plot(tpoints, xpoints, color = 'salmon', label = 'rabbits (prey)')\n",
        "ax.plot(tpoints,ypoints, color = 'goldenrod', label = 'foxes (predator)')\n",
        "ax.set_ylim(0,9)\n",
        "ax.set_xlabel('t')\n",
        "ax.set_ylabel('Population (in thousands)')\n",
        "ax.legend()"
      ],
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      "execution_count": 674,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7f87d5361c70>"
            ]
          },
          "metadata": {},
          "execution_count": 674
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "The Lotka-Volterra model predicts that the prey consumption rate of a predator has a direct relationship to the prey abundance. From the graph, we can support what the model predicts. The prey population has a positive effect on the fox population. On the other hand, the predator population has a negative effect on the prey population. When the population of the prey rises, there is abundant food source for the predator making their population grow as well."
      ],
      "metadata": {
        "id": "Y4fBkY8LpIeD"
      }
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "kBX9EyMLOnXj"
      },
      "source": [
        "### PROBLEM 2 (25 pts.; 50/50 for code & discussion)\n",
        "\n",
        "**Nonlinear pendulum**\n",
        "\n",
        "_Refer to Exercise 8.4 in the Newman text._ Building on the results from Example 8.6, calculate the motion of a nonlinear pendulum as follows.\n",
        "\n",
        "1. Solve the two first-order equations, Eqs. (8.45) and (8.46), using the fourth-order Runge-Kutta method for a pendulum with a 10 cm arm. Use your program to calculate the angle $\\theta$ of displacement for several periods of the pendulum when it is released from a standstill at $\\theta = 179^\\circ$ from the vertical. *(15 pts.)*\n",
        "\n",
        "2. Make a graph of $\\theta$ as a function of time. *(10 pts.)*\n"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "To solve, we first turn the given second-order equation into two first-order equations since nonlinear equations are not easy to solve. Our initial equation of motion for the pendulum is \n",
        "\n",
        "\\begin{align}\n",
        "  \\frac{d^2 \\theta}{dt^2} = - \\frac{g}{l} \\sin{\\theta}\n",
        "  \\end{align}\n",
        "\n",
        "Defining a new variable $\\omega$ by\n",
        " \\begin{align}\n",
        "  \\frac{d \\theta}{dt} = \\omega\n",
        "  \\end{align}\n",
        "\n",
        "The initial equation becomes\n",
        "\n",
        "\\begin{align}\n",
        "  \\frac{d \\omega}{dt} = -\\frac{g}{l} \\sin{\\theta}\n",
        "  \\end{align}"
      ],
      "metadata": {
        "id": "HUZyFTw3ErLx"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#constants\n",
        "g = 9.81 # gravity\n",
        "l = 0.1 # length\n",
        "a = 0\n",
        "b = 10\n",
        "N = 1000\n",
        "h = (b-a)/N\n",
        "theta = np.deg2rad(179) #converting the degree to radians\n"
      ],
      "metadata": {
        "id": "1V6QBWG5F4C1"
      },
      "execution_count": 675,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "R0g43FSaZdBz"
      },
      "source": [
        "def f(r,t):\n",
        "  '''defining the equation'''\n",
        "  theta = r[0] # assigning theta values as the first index of array r\n",
        "  omega = r[1] # assigning theta values as the second index of array r\n",
        "  ftheta = omega # definition of the new variable\n",
        "  fomega = - (g/l) * sin(theta)  # modification of the initial equation\n",
        "  return np.array([ftheta, fomega], float)"
      ],
      "execution_count": 676,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "tpoints = np.arange(a,b,h) \n",
        "radians = [] # to store the values of radians"
      ],
      "metadata": {
        "id": "ggZpHvgNOyrC"
      },
      "execution_count": 677,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "r = np.array([theta, 0],float)\n",
        "#using the runge-kutta method\n",
        "for t in tpoints:\n",
        "  radians.append(r[0])\n",
        "  k1 = h * f(r,t)\n",
        "  k2 = h * f(r+ 0.5 * k1, t+0.5*h)\n",
        "  k3 = h *f(r+0.5*k2, t+0.5*h)\n",
        "  k4 = h*f(r+k3,t+h)\n",
        "  r += (k1+2*k2+2*k3+k4) / 6\n",
        "\n",
        "max, min = 0,0\n",
        "for x in range(len(radians)): #converting back to degrees\n",
        "  radians[x] = np.degrees(radians[x]) \n",
        "  if radians[x] > max:\n",
        "    max = radians[x]\n",
        "  if radians[x] < min:\n",
        "    min = radians[x]\n",
        "\n",
        "\n",
        "fig, ax = plt.subplots()\n",
        "ax.axhline(y=max, linestyle = '--', color = 'salmon', alpha = 0.7)\n",
        "ax.axhline(y=min, linestyle = '--', color = 'salmon', alpha = 0.7)\n",
        "ax.axhline(y =0, linestyle = '--', color = 'salmon', alpha = 0.7)\n",
        "ax.plot(tpoints, radians, color = 'darkcyan')\n",
        "ax.set_xlabel('t (seconds)')\n",
        "ax.set_ylabel('Angular displacement (degrees)')\n",
        "ax.set_xlim(0,10)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 296
        },
        "id": "DY5DQZEKNmQu",
        "outputId": "ec80dc3f-78ea-4f0f-e972-3d43af7a39c5"
      },
      "execution_count": 678,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(0.0, 10.0)"
            ]
          },
          "metadata": {},
          "execution_count": 678
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Hs6zy-yraBeY"
      },
      "source": [
        "### PROBLEM 3 (50 pts.; 50/50 for code & discussion)\n",
        "\n",
        "**Quantum Oscillators**\n",
        "\n",
        "_Refer to Exercise 8.14 of the Newman text_ Consider the one-dimensional, time-independent Schrodinger equation in a harmonic (i.e., quadratic) potential $V(x) = V_0 x^2 / a^2$, where $V_0$ and $a$ are constants. \n",
        "\n",
        "1. Modify the code from Example 8.9 to find the energies of the ground state and the first two excited states for this potential when $m$ is the electron mass, $V_0 = 50$ eV, and $a = 10^{-11}$ m. Note that in theory the wavefunction goes all the way out to $x = \\pm \\infty$, but you can get good answers by using a large but finite interval. Try using $x = - 10a$ to $+10a$, with the wavefunction $\\psi = 0$ at both boundaries. (In effect, you are putting the harmonic oscillator in a box with impenetrable walls.) The wavefunction is real everywhere, so you don't need to use complex variables, and you can use evenly spaced points for the solution-- there is no need to use an adaptive method for this problem.\n",
        "\n",
        "   The quantum harmonic oscillator is known to have energy states that are equally spaced. Check that this is true, to the precision of your calculation, for your answers. (Hint: The ground state has energy in the range 100 to 200eV.) *(20 pts.)*\n",
        "\n",
        "2. Modify the code from (1) to calculate the same three energies for the anharmonic oscillator with $V(x) = V_0 x^4 /a^4$, with the same parameter values. *(10 pts.)*\n",
        "\n",
        "3. Calculate the properly normalized wavefunctions of the anharmonic oscillator for the three states. To normalize the wavefunctions you will have to calculate the value of the integral $\\int_{-\\infty}^{+\\infty} |\\psi(x)|^2 {\\rm d}x$ and then rescale $\\psi$ appropriately to ensure that the area under the square of each of the wavefunctions is 1. Either the trapezoidal rule or Simpson's rule will give you a reasonable value for the integral. *(15 pts.)*\n",
        "\n",
        "   Note that you may find a few very large values at the end of the array holding the wavefunction. Where do these large values come from? Are they real, or spurious? One simple way to deal with the large values is to make use of the fact that the system is symmetric about its midpoint and calculate the integral of the wavefunction over only the left-hand half of the system, then double the result. This neatly misses out the large values. \n",
        "\n",
        "4. Make a plot of the normalized wavefunctions from (3), all on the same axes, as a function of $x$ over a modest range near the origin-- say $x = -5a$ to $x =+5a$. *(5 pts.)*\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "The time-independent Schrodinger equation is given by the formula\n",
        "\n",
        "\\begin{align}\n",
        "  - \\frac{\\hbar}{2m} \\frac{d^2 \\psi}{dx^2} + V(x) \\psi (x) = E \\psi (x)\n",
        "  \\end{align}\n",
        "where $\\psi(x)$ is the wavefunction, $V(x)$ is the potential energy at position $x$, and $E$ is the total energy of the particle, plus kinetic. \n",
        "\n",
        "Since the equation is in second-order, we turn them into first-order equations. We let\n",
        "\n",
        "\\begin{align}\n",
        "  \\frac{d \\psi }{dx} = \\phi \\hspace{1cm} \\frac{d \\phi}{dx} = \\frac{2m}{\\hbar^2} [V(x) - E] \\psi \n",
        "  \\end{align}\n",
        "\n",
        "Note that we know that $psi = 0$ at $x = 0$ however, we lack the knowledge on the value of $\\frac{d\\psi}{dx}$ resorting to varying the energy $E$ and calculating it from the range provided. "
      ],
      "metadata": {
        "id": "RizBwAzFI3WN"
      }
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "nMx3AX6iaF8Q"
      },
      "source": [
        "#constants\n",
        "m = 9.1094e-31\n",
        "e = 1.6022e-19\n",
        "V_0 = 50. * e #eV\n",
        "a = 1.e-11 #m\n",
        "x_1 = -10. * a\n",
        "x_2 = 10. *a\n",
        "hbar = 1.0546e-34 #Planck's constant\n",
        "N = 1000\n",
        "h = (x_2 - x_1) / N\n"
      ],
      "execution_count": 679,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# Function\n",
        "def V(x): \n",
        "  return V_0 * (x ** 2) / (a**2)\n",
        "\n",
        "def f(r,x,E): \n",
        "  psi = r[0]\n",
        "  phi = r[1]\n",
        "  fpsi = phi\n",
        "  fphi = (2*m/ hbar **2) * (V(x)-E) * psi\n",
        "  return np.array([fpsi,fphi], float)\n",
        "\n",
        "# Calculate the wavefunction for a particular energy\n",
        "def solve(E):\n",
        "  psi = 0.0\n",
        "  phi = 1.0\n",
        "  r = np.array([psi,phi], float)\n",
        "\n",
        "  for x in np.arange(x_1,x_2,h):\n",
        "    k1 = h * f(r,x,E)\n",
        "    k2 = h * f(r+ 0.5 * k1, x+0.5*h, E)\n",
        "    k3 = h * f(r+ 0.5 * k2, x+0.5*h, E)\n",
        "    k4 = h*f(r+k3,x+h,E)\n",
        "    r += (k1+2*k2+2*k3+k4) / 6\n",
        "\n",
        "  return r[0]"
      ],
      "metadata": {
        "id": "w8YR6XCQWjZ-"
      },
      "execution_count": 680,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "E1 = 0.0\n",
        "E2 = e\n",
        "psi2 = solve(E1)\n",
        "\n",
        "target = e/1000\n",
        "while abs(E1-E2) > target:\n",
        "  psi1,psi2 = psi2,solve(E2)\n",
        "  E1,E2 = E2, E2 - psi2*(E2 - E1) / (psi2 -psi1)\n",
        "\n",
        "GS = E2/e\n",
        "print(\"The ground state energy is \", E2/e, \"eV\")\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "HM3IEqflXUYU",
        "outputId": "808226fc-4b4d-47dc-f72d-c33217954bb0"
      },
      "execution_count": 681,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The ground state energy is  138.02397130603683 eV\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "E1 = 140 * e\n",
        "E2 = 300 * e\n",
        "psi2 = solve(E1)\n",
        "\n",
        "target = e/1000\n",
        "while abs(E1-E2) > target:\n",
        "  psi1,psi2 = psi2,solve(E2)\n",
        "  E1,E2 = E2, E2 - psi2*(E2 - E1) / (psi2 -psi1)\n",
        "\n",
        "FE = E2/e\n",
        "print(\"The first excited state energy is \", FE, \"eV\")"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "ZkOixceuXkkt",
        "outputId": "d6fb28e7-9e70-4a25-a631-a6df327d6813"
      },
      "execution_count": 682,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The first excited state energy is  414.07191654124284 eV\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "E1 = 420 * e\n",
        "E2 = 700 * e\n",
        "psi2 = solve(E1)\n",
        "\n",
        "target = e/1000\n",
        "while abs(E1-E2) > target:\n",
        "  psi1,psi2 = psi2,solve(E2)\n",
        "  E1,E2 = E2, E2 - psi2*(E2 - E1) / (psi2 -psi1)\n",
        "\n",
        "SE = E2/e\n",
        "print(\"The second excited state energy is \", SE, \"eV\")"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "jjHuSvOuZB5M",
        "outputId": "5dbfa129-5e10-413f-f39f-dfda8a60e697"
      },
      "execution_count": 683,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The second excited state energy is  690.1198621200641 eV\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "To check that the quantum harmonic oscillator's energy states are equally spaced, we divide the first and second excited state to the ground state. Shown below is the result that shows that the energies are quantized."
      ],
      "metadata": {
        "id": "MYM4c8b_QYMY"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "print(round(FE / GS)) # dividing the first excited state by the ground state energy\n",
        "print(round(SE/GS)) # dividing the second excited state by ground state energy"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "1a49_W4vkAZy",
        "outputId": "0daa8fd5-e5d7-443a-c835-2d9ce36937cf"
      },
      "execution_count": 684,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "3\n",
            "5\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "We do the same thing for number 2 however, with different value for $V(x)$"
      ],
      "metadata": {
        "id": "8JCahG--iCOb"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Function\n",
        "def V(x): \n",
        "  return V_0 * (x ** 4) / (a**4)\n",
        "\n",
        "def f(r,x,E): \n",
        "  psi = r[0]\n",
        "  phi = r[1]\n",
        "  fpsi = phi\n",
        "  fphi = (2*m/ hbar **2) * (V(x)-E) * psi\n",
        "  return np.array([fpsi,fphi], float)\n",
        "\n",
        "def solve(E):\n",
        "  psi = 0.0\n",
        "  phi = 1.0\n",
        "  r = np.array([psi,phi], float)\n",
        "\n",
        "  for x in np.arange(x_1,x_2,h):\n",
        "    k1 = h * f(r,x,E)\n",
        "    k2 = h * f(r+ 0.5 * k1, x+0.5*h, E)\n",
        "    k3 = h * f(r+ 0.5 * k2, x+0.5*h, E)\n",
        "    k4 = h*f(r+k3,x+h,E)\n",
        "    r += (k1+2*k2+2*k3+k4) / 6\n",
        "\n",
        "  return r[0]\n"
      ],
      "metadata": {
        "id": "tC9SMyNthDpW"
      },
      "execution_count": 685,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "E1 = 0.0\n",
        "E2 = e\n",
        "psi2 = solve(E1)\n",
        "\n",
        "target = e/1000\n",
        "while abs(E1-E2) > target:\n",
        "  psi1,psi2 = psi2,solve(E2)\n",
        "  E1,E2 = E2, E2 - psi2*(E2 - E1) / (psi2 -psi1)\n",
        "\n",
        "GS = E2/e\n",
        "print(\"The ground state energy is\", GS, \"eV\")"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "9FGSEDfDjI6-",
        "outputId": "54079fbb-2451-4837-b60a-850c4bd3ab35"
      },
      "execution_count": 686,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The ground state energy is 205.30690346934932 eV\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "E1 = 0 * e\n",
        "E2 = 400 * e\n",
        "psi2 = solve(E1)\n",
        "\n",
        "target = e/1000\n",
        "while abs(E1-E2) > target:\n",
        "  psi1,psi2 = psi2,solve(E2)\n",
        "  E1,E2 = E2, E2 - psi2*(E2 - E1) / (psi2 -psi1)\n",
        "\n",
        "FS = E2/e\n",
        "print(\"The first excited state energy is = \", FS, \"eV\")"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "3JndhXp1jQnJ",
        "outputId": "6aeec985-d178-4154-d86d-256e0c6e8e38"
      },
      "execution_count": 687,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The first excited state energy is =  735.691247040332 eV\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "E1 = 0 * e\n",
        "E2 = 1500 * e\n",
        "psi2 = solve(E1)\n",
        "\n",
        "target = e/1000\n",
        "while abs(E1-E2) > target:\n",
        "  psi1,psi2 = psi2,solve(E2)\n",
        "  E1,E2 = E2, E2 - psi2*(E2 - E1) / (psi2 -psi1)\n",
        "\n",
        "SE = E2/e\n",
        "print(\"The second excited state energy is \", SE , \"eV\")"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "AX56bV5fjT0Z",
        "outputId": "c50d0a73-e4af-4945-daca-9bb8425d4160"
      },
      "execution_count": 688,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The second excited state energy is  1443.5694213261647 eV\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#converting the ground state, first and second state energy into an array and setting the precision up to 14 decimal places\n",
        "anharmonic = np.array([GS, FS, SE])\n",
        "np.set_printoptions(precision=14)\n",
        "anharmonic"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "cDTCGnieGTwG",
        "outputId": "9e64c62c-87bf-4aa3-8f8c-59e69510e7d5"
      },
      "execution_count": 689,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([ 205.30690346934932,  735.691247040332  , 1443.5694213261647 ])"
            ]
          },
          "metadata": {},
          "execution_count": 689
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Number 3 \\\n",
        "In this problem, we normalize the wavefunction. To do that, we use the equation\n",
        "\n",
        "\\begin{align}\n",
        "  \\int_{-∞}^{∞} |\\psi (x)|^2  = 1\n",
        "  \\end{align}\n",
        "\n",
        "Here, we use the Simpsons rule and taking into account that the boundaries are equal to zero, meaning $f(a) = f(b) = 0$, and that the values of $\\psi$ are already evenly spaced,  our Simpsons rule becomes\n",
        "\n",
        "\\begin{align}\n",
        "\\frac{1}{3} \\left[ 4 \\sum_{k \\ odd} f(a+kh) + 2 \\sum_{k \\ even} f(a+kh) \\right]\n",
        "  \\end{align}"
      ],
      "metadata": {
        "id": "B3Gp-xkZSIxm"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "a = 1.e-11 #m\n",
        "x1 = -5 * a\n",
        "x2 = 5 * a\n",
        "N = 1000\n",
        "h = (x2 - x1) / N\n"
      ],
      "metadata": {
        "id": "eeQp6ybiKmi3"
      },
      "execution_count": 690,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "t_point = np.arange(x1, x2, h)\n",
        "def get_psi(E):\n",
        "  '''to get the values of psi given the energies'''\n",
        "\n",
        "  x_points = []\n",
        "  psi = 0.0\n",
        "  phi = 1.0\n",
        "  r = np.array([psi,phi], float)\n",
        "\n",
        "  for t in t_point:\n",
        "    x_points.append(r[0])\n",
        "    k1 = h * f(r,t,E)\n",
        "    k2 = h * f(r+ 0.5 * k1, t+0.5*h, E)\n",
        "    k3 = h * f(r+ 0.5 * k2, t+0.5*h, E)\n",
        "    k4 = h*f(r+k3,t+h,E)\n",
        "    r += (k1+2*k2+2*k3+k4) / 6\n",
        "\n",
        "  x = np.array(x_points)\n",
        "  return x\n",
        "\n"
      ],
      "metadata": {
        "id": "p1Xj33wUKzi5"
      },
      "execution_count": 691,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "To get the normalization constant $N$, we use the equation \n",
        "\\begin{align}\n",
        "  N = \\sqrt{\\frac{1}{\\int_{-∞}^{∞} |\\psi (x)|^2}}\n",
        "  \\end{align}\n",
        "\n",
        "In this problem, we restrain our limits of integration from $-5a$ to $5a$ "
      ],
      "metadata": {
        "id": "kUKZkdLmf7bg"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# to get the values of psi\n",
        "ground_psi = np.array(get_psi(anharmonic[0] * e))\n",
        "first_psi = np.array(get_psi(anharmonic[1] * e))\n",
        "second_psi = np.array(get_psi(anharmonic[2] * e))\n",
        "np.set_printoptions(precision=14)"
      ],
      "metadata": {
        "id": "7b0rjjVzaQFs"
      },
      "execution_count": 692,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "def simpsons(psi):\n",
        "  '''defining simpsons rule to solve the given integral'''\n",
        "  I_1, I_2 = 0,0\n",
        "  I = 0\n",
        "  for k in range(1,N,2):\n",
        "    I_1 += psi[k]\n",
        "  for k in range(2,N,2):\n",
        "    I_2 += psi[k]\n",
        "\n",
        "  I = (1/3) * (4*I_1 + 2*I_2)\n",
        "  return I"
      ],
      "metadata": {
        "id": "X3habGKGeLwL"
      },
      "execution_count": 693,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# Getting the normalization constants\n",
        "N0 = np.sqrt(1/simpsons(ground_psi ** 2))\n",
        "N1 = np.sqrt(1/simpsons(first_psi ** 2))\n",
        "N2 = np.sqrt(1/simpsons(second_psi ** 2))\n",
        "\n",
        "#Values for normalized wavefunctions\n",
        "psi_normal = N0 * ground_psi\n",
        "psi1_normal = N1 * first_psi\n",
        "psi2_normal  = N2 * second_psi\n",
        "\n",
        "print(\"The normalization constants are \" + str(N0) + \", \"+ str(N1) + \" and \"+ str(N2))"
      ],
      "metadata": {
        "id": "ZLq9UwsiXqYx",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "aa8727f5-ac46-42cb-8818-87e0ce77ff75"
      },
      "execution_count": 694,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The normalization constants are 11111.845815185508, 39521.62179691883 and 120632.37150139798\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#plotting the wavefunctions\n",
        "tvalue = t_point / a\n",
        "\n",
        "fig,ax = plt.subplots(figsize=(8,6))\n",
        "ax.axvline(0, color = 'lightskyblue', linestyle = '--', alpha = 0.75)\n",
        "ax.axhline(0, color = 'lightskyblue', linestyle = '--', alpha = 0.75)\n",
        "ax.plot(tvalue, psi_normal, label = 'psi 0', color = 'midnightblue', linewidth = 2)\n",
        "ax.plot(tvalue, psi1_normal,label = 'psi 1', color = 'gold', linestyle = '--',linewidth = 2.5)\n",
        "ax.plot(tvalue, psi2_normal,label = 'psi 2', color = 'orange', linestyle = 'dotted',linewidth = 2.5)\n",
        "ax.legend()\n",
        "ax.set_xlabel('x/a')\n",
        "ax.set_ylabel('psi(x)')\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 388
        },
        "id": "vrT9gtR5jVFj",
        "outputId": "43d29a0e-2425-4bc9-fcfd-47759b1317a3"
      },
      "execution_count": 695,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 576x432 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    }
  ]
}