Notebooks/Propagação temporal de uma função de onda (método de Crank-Nicolson).ipynb

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   "source": [
    "# Propagação temporal de uma função de onda (método de Crank-Nicolson)\n",
    "\n",
    "* PET - Física UFRN\n",
    "* Petiano: Moisés da Rocha Alves\n",
    "* Data: 09 de Junho de 2023\n",
    "\n",
    "O objetivo deste `Notebook` é introduzirmos a propagação temporal de uma função de onda através da solução numérica da equação de Schrodinger dependente do tempo. Para isso, utilizaremos uma evolução unitária através do método de Crank-Nicolson [1].\n",
    "\n",
    "Vale lembrar que este `Notebook` é uma revisitação a um `Notebook` anterior sobre o mesmo tema [2], feito por outro petiano. A razão de o revisitarmos é introduzir um método diferente do que foi utilizado anteriormente, assim como animar as evoluções temporais. Cabe também uma menção a referência [3], que serviu de inspiração para a criação deste `Notebook`.\n",
    "\n",
    "Dito isto, vamos primeiro importar as bibliotecas que nos serão necessárias."
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Importando as bibliotecas necessárias"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy.sparse as spa\n",
    "from scipy.sparse.linalg import splu\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.animation as animation\n",
    "from scipy.integrate import simpson"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Além disso, definiremos também alguns parâmetros que melhoram a qualidade geral dos plots do pacote `matplotlib.pyplot`. Esses parâmetros podem ser chamados através da função `opt_plot`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def opt_plot():\n",
    "    plt.minorticks_on()\n",
    "    plt.tick_params(axis='both',which='minor', direction = \"in\",\n",
    "                    top = True,right = True, length=5,width=1,\n",
    "                    labelsize=15)\n",
    "    plt.tick_params(axis='both',which='major', direction = \"in\",\n",
    "                    top = True,right = True, length=8,width=1,\n",
    "                    labelsize=15)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Agora, vamos ao conteúdo do `Notebook` propriamente dito."
   ]
  },
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   "source": [
    "## 1. A equação de Schrödinger e a evolução temporal"
   ]
  },
  {
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   "metadata": {},
   "source": [
    "A equação de Schrödinger, que descreve uma evolução temporal de um estado $|\\psi(t)\\rangle$, pode ser enunciada da seguinte forma:"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "i\\hbar \\frac{d}{dt}|\\psi(t)\\rangle = \\mathcal{H}(t) |\\psi (t)\\rangle \\tag{1.1}.\n",
    "$$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Se o operador hamiltoniano $\\mathcal{H}(t)$ for independente do tempo, podemos escrever a evolução temporal como:"
   ]
  },
  {
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "|\\psi(t + \\Delta t)\\rangle = e^{-i\\mathcal{H}\\Delta t/\\hbar} |\\psi(t)\\rangle \\tag{1.2},\n",
    "$$"
   ]
  },
  {
   "attachments": {},
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   "metadata": {},
   "source": [
    "onde o operador $U(\\Delta t) = e^{-i\\mathcal{H}\\Delta t/\\hbar}$ é um operador unitário (isto é, preservará a norma do estado).\n",
    "\n",
    "Porém, estamos interessados no problema unidimensional. Para isso, é necessário que avaliemos nossa função no espaço da posição $x$. Assim, ao atuarmos $\\langle{x}|$ na equação (1.1), ficamos com a seguinte equação:"
   ]
  },
  {
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   "source": [
    "$$\n",
    "i \\hbar \\frac{\\partial}{\\partial t}\\psi(x,t) = \\mathcal{H} \\psi (x,t) \\tag{1.3},\n",
    "$$"
   ]
  },
  {
   "attachments": {},
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   "source": [
    "Note, no entanto, que podemos escrever a equação (1.3) expandindo o operador hamiltoniano na seguinte forma:"
   ]
  },
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   "source": [
    "$$\n",
    "\\mathcal{H} = \\mathcal{T}+\\hat{V},\n",
    "$$"
   ]
  },
  {
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   "source": [
    "que nos dá a equação de Schrodinger unidimensional dependente do tempo:"
   ]
  },
  {
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   "source": [
    "$$\n",
    "i\\hbar \\frac{\\partial}{\\partial t}\\psi(x,t) = -\\frac{\\hbar^2}{2m}\\frac{\\partial^2}{\\partial x^2}\\psi(x,t) + \\hat{V}\\psi(x,t)\\tag{1.4}.\n",
    "$$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Se considerarmos unidades onde $m = 1$ e $\\hbar = 1$ (para fins de simplificação numérica), teremos que a equação (1.4) e a equação (1.2) (no espaço da posição) ficarão na seguinte forma:"
   ]
  },
  {
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   "source": [
    "$$\n",
    "i\\frac{\\partial}{\\partial t}\\psi(x,t) = -\\frac{1}{2}\\frac{\\partial^2}{\\partial x^2}\\psi(x,t) + \\hat{V}\\psi(x,t)\\tag{1.5},\n",
    "$$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\psi(x, t + \\Delta t) =  e^{-i\\mathcal{H}\\Delta t}\\ \\psi(x,t)\\tag{1.6}.\n",
    "$$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Agora, estamos aptos a abordar as equações acima numericamente."
   ]
  },
  {
   "attachments": {},
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   "source": [
    "## 2. O método de Crank-Nicolson"
   ]
  },
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   "source": [
    "A partir da equação (1.6), uma abordagem natural seria considerarmos a definição da exponenciação de uma matriz e expandirmos em seus termos de primeira ordem para buscarmos uma evolução temporal numérica. Assim, ao exponenciarmos uma matriz $A$, teremos, por definição, que:"
   ]
  },
  {
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   "metadata": {},
   "source": [
    "$$\n",
    "e^{A} = \\sum_{k=0}^{\\infty}\\frac{1}{k!}A^k \\tag{2.1}.\n",
    "$$"
   ]
  },
  {
   "attachments": {},
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   "metadata": {},
   "source": [
    "Portanto, ao denotarmos um passo temporal $n+1$ para um certo valor discretizado de $\\psi(x,t+\\Delta t)$ e um passo $n$ para um valor de $\\psi(x,t)$, ficamos com a seguinte expansão em primeira ordem para uma evolução temporal (que é análoga a utilizar o método de Euler \"para frente\" ou explícito):"
   ]
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   "source": [
    "$$\n",
    "\\psi^{n+1} = e^{-i\\mathcal{H}\\Delta t}\\ \\psi^{n} \\approx (\\mathbb{1} - i\\mathcal{H}\\Delta t)\\ \\psi^{n} \\tag{2.2}.\n",
    "$$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Entretanto, repare que após fazermos esta aproximação teremos uma evolução que não é mais unitária, já que $(1 - i\\mathcal{H}\\Delta t)$ não é um resultado unitário.\n",
    "\n",
    "Se evoluirmos pelo método de Euler \"para trás\" (ou implícitamente) teremos a seguinte forma:"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\psi^{n} = e^{i\\mathcal{H}\\Delta t}\\ \\psi^{n+1} \\approx (\\mathbb{1} + i\\mathcal{H}\\Delta t)\\ \\psi^{n} \\tag{2.3}.\n",
    "$$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note que esta forma também nos trás o mesmo problema da não-unitariedade. Podemos, entretanto, combinar os métodos explícitos e implícitos através do método de Crank-Nicolson. Faremos a evolução da seguinte forma:"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\psi^{n+1} = e^{-i\\mathcal{H}\\Delta t}\\ \\psi^{n} = e^{-i\\mathcal{H}\\Delta t/2}e^{-i\\mathcal{H}\\Delta t/2}\\ \\psi^{n} = e^{-i\\mathcal{H}\\Delta t/2}(e^{i\\mathcal{H}\\Delta t/2})^{-1}\\ \\psi^{n}.\n",
    "$$"
   ]
  },
  {
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   "source": [
    "Agora, utilizamos as aproximações feitas nas formas explícita e implícita (2.2 e 2.3) para obtermos o seguinte resultado:"
   ]
  },
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   "source": [
    "$$\n",
    "\\psi^{n+1} \\approx (\\mathbb{1} - i\\mathcal{H}\\Delta t/2)(\\mathbb{1} + i\\mathcal{H}\\Delta t/2)^{-1}\\ \\psi^{n}. \\tag{2.4}\n",
    "$$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note agora que o operador de evolução temporal que obtemos é unitário, já que, se $U = (\\mathbb{1} - i\\mathcal{H}\\Delta t/2)(\\mathbb{1} + i\\mathcal{H}\\Delta t/2)^{-1}$, então:"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
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   "source": [
    "$$\n",
    "U^{\\dagger} = (\\mathbb{1} + i\\mathcal{H}\\Delta t/2)(\\mathbb{1} - i\\mathcal{H}\\Delta t/2)^{-1},\n",
    "$$"
   ]
  },
  {
   "attachments": {},
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   "source": [
    "e, portanto:"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "UU^{\\dagger} = \\mathbb{1}.\n",
    "$$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assim, teremos uma evolução do tipo:"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\psi^{n+1} \\approx U(\\Delta t)\\ \\psi^{n} \\tag{2.5},\n",
    "$$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "que preserva a norma ao longo de toda a evolução temporal que faremos.\n",
    "\n",
    "Por fim, é importante notarmos que podemos escrever a evolução na forma:"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "(\\mathbb{1} + i\\mathcal{H}\\Delta t/2)\\ \\psi^{n+1} \\approx (\\mathbb{1} - i\\mathcal{H}\\Delta t/2)\\ \\psi^{n} \\tag{2.6},\n",
    "$$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "onde ficamos com uma equação matricial do tipo $Ax = b$ que pode ser determinada através de métodos como a decomposição LU. Para este fim, utilizaremos o pacote `scipy.sparse` e o pacote `scipy.sparse.linalg`.\n",
    "\n",
    "Agora, nos resta um problema: como descrever $\\mathcal{H}$?"
   ]
  },
  {
   "attachments": {},
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   "source": [
    "## 3. Discretizando a equação de Schrödinger"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Durante a descrição feita ao longo de toda a seção 2 utilizamos somente a evolução unitária para tratarmos nosso problema. Agora, utilizaremos a equação de Schrödinger descrita na equação (1.5) de maneira discretizada para descrevermos nosso operador hamiltoniano numa forma matricial. Para discretizar a parte temporal, basta que façamos:"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\frac{\\partial}{\\partial t} \\psi(x_j,t_n) = \\frac{\\psi^{n+1}_{j} -\\psi^{n}_{j}}{\\Delta t} \\tag{3.1},\n",
    "$$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "onde o índice $j$ representa uma discretização espacial e o índice $n$ representa a discretização temporal. O termo $\\Delta t$ determinará o tamanho do passo temporal.\n",
    "\n",
    "Para a parte espacial, podemos usar diferenças finitas centradas em sua aproximação de segunda ordem:"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\frac{\\partial^2}{\\partial x^2} \\psi(x_j,t_n) = \\frac{\\psi^{n}_{j+1} + \\psi^{n}_{j-1} - 2\\psi^{n}_{j}}{(\\Delta x)^2} \\tag{3.2},\n",
    "$$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "onde o termo $\\Delta x$ determinará o tamanho do passo espacial.\n",
    "\n",
    "Mas como isso ajuda em descrever $\\mathcal{H}$? \n",
    "\n",
    "Para entender essa questão, consideremos as funções de onda $\\Psi$ em um tempo $t_{n+1}$ e $t_{n}$ como vetores coluna, onde cada componente da coluna é uma discretização no espaço definido dentro do intervalo $[0, J]$, onde $J + 1$ representa o tamanho da discretização no espaço (tamanho do grid). Assim, ficamos tal que:"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\Psi^{n} = \n",
    "\\begin{bmatrix}\n",
    "\\psi^{n}_0 & \\psi^{n}_1 & \\psi^{n}_2 & \\ldots & \\psi^{n}_{J}\\\\\n",
    "\\end{bmatrix}^T \\tag{3.3},\n",
    "$$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\Psi^{n+1} = \n",
    "\\begin{bmatrix}\n",
    "\\psi^{n+1}_0 & \\psi^{n+1}_1 & \\psi^{n+1}_2 & \\ldots & \\psi^{n+1}_{J}\\\\\n",
    "\\end{bmatrix}^T \\tag{3.4}.\n",
    "$$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ao compactarmos toda a informação espacial das funções de onda desta maneira, podemos notar, através da equação (3.2), que o operador $\\mathcal{T}$ pode ser escrito como uma matriz tridiagonal tal que:"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\mathcal{T} = -\\frac{1}{2}\\frac{\\partial^2}{\\partial x^2} = -\\frac{1}{2\\Delta x^2}\n",
    "\\begin{bmatrix}\n",
    "-2 & 1 & 0 & 0 & \\ldots & 0 & 0 & 0\\\\\n",
    "1 & -2 & 1 & 0 & \\ldots & 0 & 0 & 0\\\\\n",
    "0 & 1 & -2 & 1 & \\ldots & 0 & 0 & 0\\\\\n",
    "\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots\\\\\n",
    "0 & 0 & 0 & 0 & \\ldots & 1 & -2 & 1\\\\\n",
    "0 & 0 & 0 & 0 & \\ldots & 0 & 1 & -2\\\\\n",
    "\\end{bmatrix} \\tag{3.5}.\n",
    "$$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Já o operador $\\hat{V}$ poderá ser escrito como uma matriz diagonal tal que:"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\hat{V} = diag\n",
    "\\begin{bmatrix}\n",
    "v_0 & v_1 & v_2 & \\ldots & v_{J}\\\\\n",
    "\\end{bmatrix} \\tag{3.6}.\n",
    "$$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note que assim que escrevemos os operadores desta maneira, colocamos condições de contorno implícitas na equação tais que:"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\psi(x = L_1,t) = \\psi(x = L_2, t) = 0.\n",
    "$$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ou seja, estamos assumindo que o potencial nas bordas é infinito, resultando em um poço de potencial infinito definido no intervalo $[L_1, L_2]$.\n",
    "\n",
    "Dito isto, ao obtermos uma forma de representar $\\mathcal{H}$, conforme o método descrito na seção 2, podemos reescrever a equação (2.6) da seguinte forma:"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "U_2\\ \\Psi^{n+1} = U_1\\ \\Psi^{n} \\tag{3.7},\n",
    "$$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "onde:"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "U_2 = \\mathbb{1} + \\frac{i\\Delta t}{2} \\ (\\mathcal{T}+\\hat{V}),\n",
    "$$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "U_1 = \\mathbb{1} - \\frac{i\\Delta t}{2} \\ (\\mathcal{T}+\\hat{V}).\n",
    "$$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Agora, é possível resolvermos nosso problema de maneira númerica ao especificarmos uma certa função de onda inicial $\\psi(x,t = 0)$ e um potencial $\\hat{V}$."
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4. Resolvendo o problema numericamente"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Como dito anteriormente, devemos especificar uma certa função de onda inicial. Para o nosso problema, considere um pacote gaussiano centrado em $x_0$ e com um momento médio $k_0$ que é dado por:"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "$$\n",
    "\\psi(x,t = 0) =\n",
    "  \\dfrac{1}{\\sqrt[4]{2\\pi\\sigma_x^2}}\n",
    "  \\cdot\n",
    "  e^{i\\cdot k_0(x - x_0)}\n",
    "  \\cdot\n",
    "  e^{-\\frac{(x - x_0)^2}{4\\sigma_x^2}}, \\tag{4.1}\n",
    "$$\n",
    "\n"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assim, como devemos transcrever nosso resultado para a matriz coluna que foi proposta, em código é cômodo definirmos uma função que retorne com os valores desejados do pacote gaussiano conforme os parâmetros inseridos nesta função."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gaussian_wavepacket(x, x0, k, sigma = 0.1):\n",
    "    x = np.asarray(x)\n",
    "    g = np.sqrt(1/(np.sqrt(2*np.pi)*sigma)) * np.exp(-(x - x0)**2/(4*sigma**2))\n",
    "    return np.exp(1j * k*(x-x0)) * g"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Agora, precisamos definir nossos grids espaciais e temporais. Em código, utilizando o pacote do `numpy`, pode ser feito da maneira a seguir."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "L = 20\n",
    "Nx = 200\n",
    "x_points = np.linspace(-L,L,Nx)\n",
    "dx = x_points[1] - x_points[0]\n",
    "\n",
    "Lt = 500\n",
    "Nt = 300\n",
    "t_points = np.linspace(0, Lt, Nt)\n",
    "dt = t_points[1] - t_points[0]"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note que definimos um poço simétrico de potencial infinito com parâmetro $L = 10$."
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Agora, precisamos descrever as matrizes de $\\mathcal{T}$ e $\\hat{V}$ utilizando os pacotes de álgebra linear importados. Foi feito da seguinte maneira:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "O = np.ones((Nx))\n",
    "Z = np.zeros((Nx))\n",
    "tridiagonal = spa.spdiags([O, -2*O, O], [-1, 0, 1], Nx, Nx)\n",
    "T = -1/(2*dx)*tridiagonal\n",
    "V = spa.spdiags([Z], [0], Nx, Nx)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "É necessário, agora, descrever as matrizes $U_1$ e $U_2$ através dos valores de $\\mathcal{T}$ e $\\hat{V}$. Faremos, depois disso, a decomposição LU de $U_2$. A usaremos para computar os valores da solução de $\\Psi^{n+1}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "U2 = spa.eye(Nx) + (1j * 0.5 * dt) * (T + V)\n",
    "U1 = spa.eye(Nx) - (1j * 0.5 * dt) * (T + V)\n",
    "U2 = U2.tocsc()\n",
    "LU = splu(U2)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Por último, e não menos importante, definimos a função de onda inicial através da função que definimos para um pacote gaussiano lá em cima. Utilizaremos $x_0 = 0$, $k_0 = 0$ e $\\sigma_x = 1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "psi = np.zeros((Nx, Nt), dtype=complex)\n",
    "psi[:, 0] = gaussian_wavepacket(x_points, 0, 0, 1)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Agora, resta só evoluirmos nossa função de onda no tempo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "for tt in range(Nt-1):\n",
    "    b = U1.dot(psi[:,tt])\n",
    "    psi[:,tt+1] = LU.solve(b)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note que o array `psi` contém todos os valores de $\\psi(x,t)$ ao longo da posição e do tempo. Os utilizaremos para fazer uma animação. Para este fim, utilizaremos o pacote `matplotlib.animation`.\n",
    "\n",
    "Ainda, por meio do método de Simpson para calcular integrais numéricas, que pode ser chamado através da função `simpson` do pacote `scipy.integrate`, calculamos o desvio padrão e o valor médio da posição do pacote para cada passo da evolução. Também checaremos o fator de normalização da função de onda, que deve ser sempre igual a 1 se a evolução temporal é de fato unitária."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "MovieWriter ffmpeg unavailable; using Pillow instead.\n"
     ]
    },
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 792x504 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Inicialização do gráfico em animação\n",
    "plt.style.use('dark_background')\n",
    "fig, ax = plt.subplots(figsize=(11,7))\n",
    "plt_data, = ax.plot([], [], label=' ')\n",
    "opt_plot()\n",
    "ax.set_ylim(0,1)\n",
    "ax.set_xlim(-20, 20)\n",
    "ax.set_xlabel(\"x\", fontsize = 15)\n",
    "ax.set_ylabel(\"$|\\psi(x,t)|$\", fontsize = 15)\n",
    "T = plt.legend(loc=1)\n",
    "\n",
    "# Função que atualiza a animação a cada frame t\n",
    "def wave_plot(t):\n",
    "    # Atualização dos parâmetros no gráfico\n",
    "    x_mean_value = simpson(x_points*np.abs(psi[:,t-1])**2, x_points)\n",
    "    x_squared_mean_value = simpson(x_points**2*np.abs(psi[:,t-1])**2, x_points)\n",
    "    deviation = np.sqrt(x_squared_mean_value - x_mean_value**2)\n",
    "    norm_factor = simpson(np.abs(psi[:,t-1])**2, x_points)\n",
    "    lab = ('frame = ' + str(t) \n",
    "            + '\\n $\\langle x \\\\rangle$ = ' + str(round(x_mean_value, 4)) \n",
    "            + \"\\n $\\sigma _x = $\" + str(round(deviation, 4)) \n",
    "            + \"\\n norm = \" + str(round(norm_factor, 4)))\n",
    "    T.get_texts()[0].set_text(lab)\n",
    "\n",
    "    # Plots do gráfico da função de onda\n",
    "    y_points = np.abs(psi[:,t-1])\n",
    "    plt_data.set_data(x_points, y_points)\n",
    "    return plt_data,\n",
    "\n",
    "ani = animation.FuncAnimation(fig, wave_plot, range(0, Nt), interval=1, blit=False, repeat=False)\n",
    "ani.save('wave.gif')"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Após a função ser chamada, o Jupyter plota apenas o estado final da animação. Já que o Jupyter não é compatível com esse tipo de animação, salvamos a animação em um .gif para que possamos visualizar o resultado final. Assim:"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![.gif da evolução no tempo da função de onda](../Imagens/Propaga%C3%A7%C3%A3o%20temporal%20de%20uma%20fun%C3%A7%C3%A3o%20de%20onda%20(m%C3%A9todo%20de%20Crank-Nicolson)/wave.gif)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Conforme previsto pela mecânica quântica, o pacote deve evoluir de maneira que o $\\sigma _x$ aumente com o tempo, onde é possível verificar o mesmo comportamento ocorrendo neste gráfico."
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Agora, faremos a mesma coisa que anteriormente, porém definiremos um momento diferente de zero para o pacote (definiremos $k_0 = 1$). Assim, descrevendo o pacote e o evoluindo no tempo, teremos:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "psi2 = np.zeros((Nx, Nt), dtype=complex)\n",
    "psi2[:, 0] = gaussian_wavepacket(x_points, 0, 1, 1)\n",
    "\n",
    "for tt in range(Nt-1):\n",
    "    b = U1.dot(psi2[:,tt])\n",
    "    psi2[:,tt+1] = LU.solve(b)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "E, por fim, um outro plot em formato de gif:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "MovieWriter ffmpeg unavailable; using Pillow instead.\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 792x504 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Inicialização do gráfico em animação\n",
    "plt.style.use('dark_background')\n",
    "fig, ax = plt.subplots(figsize=(11,7))\n",
    "plt_data, = ax.plot([], [], label=' ')\n",
    "opt_plot()\n",
    "ax.set_ylim(0,1)\n",
    "ax.set_xlim(-20, 20)\n",
    "ax.set_xlabel(\"x\", fontsize = 15)\n",
    "ax.set_ylabel(\"$|\\psi(x,t)|$\", fontsize = 15)\n",
    "T = plt.legend(loc=1)\n",
    "\n",
    "# Função que atualiza a animação a cada frame t\n",
    "def wave_plot(t):\n",
    "    # Atualização dos parâmetros no gráfico\n",
    "    x_mean_value = simpson(x_points*np.abs(psi2[:,t-1])**2, x_points)\n",
    "    x_squared_mean_value = simpson(x_points**2*np.abs(psi2[:,t-1])**2, x_points)\n",
    "    deviation = np.sqrt(x_squared_mean_value - x_mean_value**2)\n",
    "    norm_factor = simpson(np.abs(psi2[:,t-1])**2, x_points)\n",
    "    lab = ('frame = ' + str(t) \n",
    "            + '\\n $\\langle x \\\\rangle$ = ' + str(round(x_mean_value, 4)) \n",
    "            + \"\\n $\\sigma _x = $\" + str(round(deviation, 4)) \n",
    "            + \"\\n norm = \" + str(round(norm_factor, 4)))\n",
    "    T.get_texts()[0].set_text(lab)\n",
    "    \n",
    "    # Plots do gráfico da função de onda\n",
    "    y_points = np.abs(psi2[:,t-1])\n",
    "    plt_data.set_data(x_points, y_points)\n",
    "    return plt_data,\n",
    "\n",
    "ani = animation.FuncAnimation(fig, wave_plot, range(0, Nt), interval=1, blit=False, repeat=False)\n",
    "ani.save('wave2.gif')"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Novamente, colocamos o .gif resultante da animação abaixo:"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![text](../Imagens/Propaga%C3%A7%C3%A3o%20temporal%20de%20uma%20fun%C3%A7%C3%A3o%20de%20onda%20(m%C3%A9todo%20de%20Crank-Nicolson)/wave2.gif)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "É interessante observar como o valor esperado da posição do pacote evolui com o tempo. Inicialmente este valor aumenta, dado o momento inicial do pacote. A partir do momento que o pacote encontra a barreira de potencial infinita, este valor começa a diminuir. Ao final, como a função de onda está confinada, o valor esperado da posição do pacote se aproxima cada vez mais de zero, a medida que a evolução temporal ocorre, e fica oscilando em valores próximos a zero."
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5. Conclusão"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Os dois pacotes gaussianos simulados, com e sem momento, se alargam, conforme como é previsto que o desvio padrão $\\sigma _x$ aumente a medida que o pacote é evoluído temporalmente. Além disso, foi interessante observar o comportamento dos pacotes a medida que encontram a barreira de potencial.\n",
    "\n",
    "Dito isto, este `Notebook` termina por aqui."
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Referências"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**[1]** *O método de Crank-Nicolson: <https://en.wikipedia.org/wiki/Crank–Nicolson_method>*\n",
    "\n",
    "**[2]** *Evolução e propagação temporal da função de onda, postado no projeto PET.py em 04/11/2020, por Gabriel Wendell Celestino Rocha (ex-petiano):\n",
    "<https://github.com/GabrielWendell/PET.py/blob/main/Notebooks/Evolução%20e%20propagação%20temporal%20da%20função%20de%20onda.ipynb>.*\n",
    "\n",
    "**[3]** *Inspiração para a criação do Notebook: <http://staff.ustc.edu.cn/~zqj/posts/Numerical_TDSE/>*"
   ]
  }
 ],
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