Perturbaciones vs variaciones.ipynb

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   "source": [
    "# Métodos de resolución de la Ec. de Schrödinger estacionaria"
   ]
  },
  {
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   "source": [
    "Los estados estacionarios de la Ec. de Schrödinger son aquellos que satisfacen la Ec. \n",
    "\n",
    "$$\n",
    "E_n \\Psi_n = {\\bf H}\\Psi_n\n",
    "$$\n",
    "donde  ${\\bf H}$ es el Hamiltoniano del sistema y $E_n$ los posibles valores de la energía. En particular, muchas veces estamos interesados en el estado fundamental $\\Psi_0$ que satisface $E_0 \\leq E_n \\forall_n$. \n",
    "\n",
    "En general, el problema de encontrar el espectro y las correspondientes autofunciones no admite soluciones analíticas, aún si sólo nos restringimos al estado fundamental. Más aún, puede ocurrir que la solución directa de la Ec. de Schr\\\"odinger no sea siquiera numéricamente tratable, como veremos más adelante al analizar el caso de sistemas de muchas partículas.\n",
    "\n",
    "\n",
    "En estas notas nos enfocaremos en el caso del problema de encontrar el estado fundamental de una partícula no relativista en un potencial. En este caso, el Hamiltoniano es de la forma \n",
    "\n",
    "$$\n",
    "{\\bf H}= -\\frac{\\hbar^2}{2m}\\nabla^2 + V(r)\n",
    "$$\n",
    "\n",
    "Atacaremos a través de dos aproximaciones: el tratamiento perturbativo y el método variacional. Finalmente, discutiremos una técnica para encontrar numéricamente las soluciones de la Ec. de Schrödinger en el caso unidimensional, lo que nos permitirá comparar los resultados.\n",
    "\n",
    "\n",
    "Para los ejemplos, consideraremos el potencial unidimensional de la forma $V(x)=-U_0 e^{-(x/x_0)^2}$.  "
   ]
  },
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   "source": [
    "## Adimensionalización de la ecuación"
   ]
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   "source": [
    "Un paso previo a intentar resolver la Ec. de Schröedinger (o cualquier otra ecuación en física) reescribirla en términos de cantidades adimensionales. En el caso de la Ec. de Schrödinger no relativista, una forma de adimensionalizar es elegir una escala de longitudes $x_0$. \n",
    "\n",
    "En el caso de potenciales de corto alcance (como el del ejemplo) esta longitud puede elegirse igual al alcance del potencial. \n",
    "\n",
    "\n",
    "\n",
    "En el caso de potenciales de largo alcance ($\\approx C/r^{\\alpha}$), eligiendo $x_0=\\left(\\frac{ 2 m C}{\\hbar^2}\\right) ^{1/(\\alpha-2)} $ \n",
    "\n",
    "\n",
    "Para el caso del pseudopotencial \"de contacto\" unidimensional, $V(x)=\\lambda \\delta(x)$, podemos elegir $x_0=\\frac{\\hbar^2}{4 m \\lambda}$, de manera que \n",
    "$$\n",
    "-\\tilde{\\psi}''(u)-\\frac{1}{2}\\delta(u) \\tilde{\\psi}(u)-\\tilde{E}\\tilde{\\psi}(u)=0\n",
    "$$"
   ]
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   "source": [
    "En cualquier caso, las energías y autofunciones normalizadas del Hamiltoniano original toman la forma\n",
    "$$\n",
    "\\Psi_n(x)=\\tilde{\\Psi}_n(x/x_0)/x_0^{1/2}\n",
    "$$\n",
    "$$\n",
    "E=\\tilde{E} \\frac{\\hbar^2}{2 m x_0^2}\n",
    "$$\n",
    "donde $\\tilde{E}$ y $\\tilde{\\Psi}(u)$ son autovalores y autofunciones de la Ec. adimensional\n",
    "$$\n",
    "\\tilde{E}_n\\tilde{\\Psi}_n(u)=-\\tilde{\\Psi}''_n(u) + \\tilde{V}(u)\\tilde{\\Psi}_n(u)\n",
    "$$\n",
    "con \n",
    "$$\n",
    "\\tilde{V}(u)=\\frac{2 m x_0^2}{\\hbar^2}V(x_0 u)\n",
    "$$\n",
    "eligiendo $\\tilde{\\Psi}_n(u)$ normalizadas de manera que $\\int |\\tilde{\\Psi}_n(u)|^2 du=1$"
   ]
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   "source": [
    " "
   ]
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   "source": [
    "## Método de potenciales aproximados"
   ]
  },
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   },
   "source": [
    "### Aproximación por un pozo cuadrado"
   ]
  },
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   "source": [
    "Si el potencial está localizado, es posible aproximarlo por un pozo de potencial que lo contenga. La diferencia entre el potencial tipo pozo y el potencial real puede entonces tomarse como una perturbación."
   ]
  },
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   "source": [
    "En nuestro ejemplo, $V(u)=-U_0 e^{-u^2}$ podría aproximarse por un pozo de potencial cuadrado de ancho $2$ y profundidad $U_0$, de manera que\n",
    "\n",
    "$$\n",
    "V(u)= - U_0  \\Theta(1-u^2) + \\delta V(u)\n",
    "$$\n",
    "con $\\delta V(u)=-U_0 (e^{-u^2}-\\Theta(1-u^2))$\n",
    "\n",
    "donde $\\Theta(u)=\\left\\{_{0 \\;\\;u>0}^{1 \\;\\; u<0}\\right.$"
   ]
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   "source": [
    "Los niveles para el pozo pueden calcularse como  $E_n=k_n^2 - U_0$\n",
    "\n",
    "\n",
    "\n",
    "donde $k_n$ son las soluciones de la ecuación trascendente\n",
    "\n",
    "$$\n",
    "\\sqrt{U_0/(k_n/2)^2-1} = \\tanh(2 k_n)\n",
    "$$\n",
    "que son de la forma $\\delta_n \\frac{\\pi (n+1) }{2}$ con $\\delta_n$ un número próximo a 1.\n",
    "\n",
    "Las correcciones a estas energías debido a la diferencia con el potencial real, a primer orden, serán de la forma $\\int_{-\\infty}^{\\infty} |\\psi_n(u)|^2 \\delta V(u) du$. Debido a la simetría de paridad del potencial, sólo tendran correcciones las soluciones pares, de manera que $\\delta E_n \\approx \\frac{\\int_0^{1} \\cos(k_n u)^2\\delta V(u) du + \\int_{1}^{\\infty}\\delta V(u)e^{-2\\alpha_n u} du}{\\int_0^{1} \\cos(k_n u)^2 du + \\int_{1}^{\\infty}e^{-2\\alpha_n u} du}$ \n",
    "\n",
    "\n",
    "Para $U_0\\gg 1$, sin embargo, las correcciones dejan de ser perturbativas y este método falla."
   ]
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   "source": [
    "### Aproximación armónica"
   ]
  },
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   "source": [
    "Si el potencial presenta un mínimo absoluto, y este es suficientemente \"profundo\", la función de onda del estado fundamental estará localizada en el entorno de ese mínimo. En tal caso, es posible aproximar el potencial en torno al mínimo por un potencial armónico. En el caso del potencial gaussiano obtenemos por ejemplo\n",
    "\n",
    "$$\n",
    "-\\tilde{U_0} e^{-u^2} = -\\tilde{U_0} + \\tilde{U_0} u^2 + \\delta V(u)\n",
    "$$\n",
    "donde $\\delta V(u)\\approx 0$ si $u<1$."
   ]
  },
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   "source": [
    "La aproximación será válida si la contribución de $\\delta V(u)$ sobre el soporte de las funciones de onda estimadas es suficientemente pequeño. En nuestro ejemplo, esta aproximación lleva a un espectro aproximado de la forma $E_n=\\frac{\\hbar^2}{2 m x_0^2}\\sqrt{\\tilde{U}_0}(2 n +1) -U_0$, con $E_n<0$.\n",
    "Las autofunciones correspondientes son de la forma\n",
    "$$\n",
    "\\Psi_{n}(u)=e^{-\\sqrt{U_0}u^2/2}H_n(\\sqrt[4]{U_0}u)\n",
    "$$\n",
    "donde $H_n(x)$ son los Polinomios de Hermite. El soporte efectivo de estas funciones de onda\n",
    "se extiende en la región $|u|<\\frac{\\sqrt{2n+1}}{\\sqrt[4]{U_0}}$. Por otro lado, $\\delta V(u)$ es despreciable en la zona $|u|<1$, de manera que las autofunciones serán razonablemente aproximadas si $2 n+1 <\\sqrt{U_0}\\Rightarrow  \\sqrt{U_0}(2 n+1)- U_0=E_n <0$. \n",
    "\n",
    "Vemos que en este caso, a diferencia de la aproximación por un pozo, resulta válida para valores grandes de $U_0$."
   ]
  },
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   "source": [
    "### Aproximación por un pseudopotencial"
   ]
  },
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   "metadata": {
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   "source": [
    "Otro potencial 1D para el que la Ec. de Schrödinger es exactamente soluble es el del pseudopotencial ${U}(x)= -\\frac{\\hbar ^2}{m x_0}\\delta(x)$. En este caso, el espectro ligado sólo tiene un único nivel con $E=-\\frac{\\hbar ^2}{2 m x_0^2}$ y autofuncion $\\Psi_0(u)=e^{-|x|/x_0}$. \n",
    "\n",
    "Podemos conectar este resultado con el resultado para el potencial gaussiano notando que\n",
    "\n",
    "$$\n",
    "\\delta(x)=\\lim_{\\xi \\rightarrow 0} \\frac{e^{-(x/(\\xi x_0))^2}}{\\sqrt{\\pi}\\xi x_0}\n",
    "$$"
   ]
  },
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   "source": [
    "Remplazando en la Ec. de Schrödinger,\n",
    "\n",
    "$$\\lim_{{\\xi\\rightarrow 0}}-\\psi^{''}(x)-\\frac{2}{\\sqrt{\\pi}\\xi x_0^2}e^{-(x/(\\xi x_0))^2}{\\psi}(x)+ \\frac{1}{x_0^2}{\\psi}(x)= 0$$\n",
    "\n",
    "ó, adimensionalizando  respecto a la escala $\\xi x_0$, tenemos\n",
    "\n",
    "\n",
    "$$-\\tilde{\\psi}^{''}(v)+\\frac{2\\xi}{\\sqrt{\\pi}}e^{-v^2}\\tilde{\\psi}(v)+\\xi^2 \\tilde{\\psi}(v)\\approx 0$$\n",
    "\n",
    "que es la ecuación de Schödinger adimensionalizada para el potencial gaussiano, remplazando \n",
    "\n",
    "$$\\tilde{U}_0=\\frac{2 m U_0 }{\\hbar^2 x_0^2} \\rightarrow \\xi/\\sqrt{\\pi/4}\\,.$$ Esto es válido si $\\xi \\ll 1$, por lo que la aproximación será válida si\n",
    "$\\tilde{U}_0\\ll 1$. En tal límite, el potencial gaussiano tendrá un único estado ligado con energía de ligadura\n",
    "\n",
    "$$\n",
    "E=-\\frac{\\hbar^2 x_0^2}{2 m }\\xi^2=  \\frac{ \\pi}{4} (\\frac{2 m U_0^2 }{\\hbar^2 x_0^2})\n",
    "$$"
   ]
  },
  {
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   "metadata": {
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   },
   "source": [
    "A grandes distancias, la función de onda se comportará también como $\\Psi(x)\\approx e^{-x_0 x/|\\sqrt{4\\pi} \\frac{2 m U_0}{\\hbar^2}|}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "Si bien por la forma en que lo derivamos, sugiere que este resultado es válido sólo para potenciales gaussianos, podríamos repetir el mismo razonamiento para otros potenciales acotados y localizados (por ejemplo, el pozo cuadrado) y llegar a la siguiente conclusión general: en el límite de acoplamiendo débil, estos potenciales poseen un espectro equivalente al del pseudo potencial tipo delta, con parámetro $x_0= \\frac{\\hbar^2 }{m}  |\\int V(x)dx|^{-1}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notemos que en este caso, como el sistema tiene un único estado ligado, la expansión perturbativa a segundo orden implica tener en cuenta las posibles transiciones en al continuo."
   ]
  },
  {
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   },
   "source": [
    "## Método variacional"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "level": 7
   },
   "source": [
    "En la ecuación de Schrödinger unidimensional, a nivel numérico, la mayor dificultad se encuentra en determinar el espectro de energías del Hamiltoniano. Conocido el espectro de energías, resolver la Ec. se reduce a encontrar la solución de una ecuación diferencial ordinaria con condiciones sobre un único borde, lo que puede lograrse eficientemente por integración numérica.  Por otro lado, muchas veces nos interesa encontrar aproximaciones analíticas simples para el estado fundamental del sistema. El método de Rayleight-Ritz nos da una estrategia para atacar ambos problemas. \n",
    "\n",
    "El método se basa en que como las autofunciones del Hamiltoniano forman una base completa, dada cualquier función $\\chi(u)$, es posible expandirla como combinación lineal de las autofunciones de ${\\bf H}$:\n",
    "$$\n",
    "\\chi(u)=\\sum a_n \\phi_{n}(u)\n",
    "$$\n",
    "Luego,\n",
    "$$\n",
    "\\frac{\\int \\chi*(u){\\bf H}\\chi(u)du}{\\int \\chi^*(u)\\chi(u)du }=\n",
    "\\frac{\\sum_{n,n'}\\int \\phi_n^*(u){\\bf H}\\phi_{n'}(u)du}{\\sum_{n,n'}\\int \\phi_n^*(u)\\phi_{n'}(u)du }=\n",
    "\\frac{\\sum_n |a_n|^2 e_n}{|a_n|^2 } \\geq e_0\n",
    "$$\n",
    "Esto nos permite reformular el problema de encontrar la energía del estado fundamental como\n",
    "\n",
    "$$\n",
    "e_0 = \\min_{\\chi} \\frac{\\int \\chi*(u){\\bf H}\\chi(u)du}{\\int \\chi^*(u)\\chi(u)du }\n",
    "$$\n",
    "El problema de optimización sobre funciones generales resulta igual de difícil que el original. Sin embargo, podemos encontrar una cota superior al mínimo, (y una estimación del estado) optimizando sobre una sub-familia de funciones:\n",
    "\n",
    "$$\n",
    "e_0 < \\min_{\\chi \\in{\\cal F}} \\frac{\\int \\chi*(u){\\bf H}\\chi(u)du}{\\int \\chi^*(u)\\chi(u)du }\n",
    "=\\min_{\\chi \\in{\\cal F}} \\langle {\\bf H}_{\\chi} \\rangle\n",
    "$$\n",
    "\n",
    "\n",
    "Para el caso del problema unidimensional, podemos encontrar cotas razonables buscando sobre familias de funciones que no presenten nodos."
   ]
  },
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   },
   "source": [
    "### Aproximación por gaussianas\n"
   ]
  },
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   "metadata": {
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   "source": [
    "\n",
    "Si el potencial está localizado, una familia conveniente de funciones de prueba normalizadas son las funciones gaussianas:\n",
    "$$\n",
    "    \\chi_{a,u_0}(u)=\\frac{e^{-(a/x_0)^2 (x-x_0)^2/2}}{\\sqrt[4]{\\pi}}\\sqrt{a/x_0}\n",
    "$$\n",
    "Estas resultan ser las funciones de onda exactas del estado fundamental si el potencial es un oscilador armónico centrado en $x_0$.\n",
    "\n",
    "Si además el potencial es sim\\'etrico ante reflecciones respecto al eje x, podemos asumir que $u_0$, con lo que tenemos una familia de funciones con un único parámetro libre.\n",
    "\n",
    "Para esta familia, el término cinético viene dado por\n",
    "$$\n",
    "\\langle T \\rangle =-\\int_{-\\infty}^{\\infty} \\chi(u)\\chi^{''}(u)du= \\frac{\\hbar^2}{2 m x_0^2}\\frac{a^2}{2}\n",
    "$$\n",
    "\n",
    "El termino asociado al potencial viene dado por\n",
    "$$\n",
    "\\langle V\\rangle = \\int_{-\\infty}^{\\infty} \\chi(u)U(u)\\chi(u)du\n",
    "$$\n",
    "que para el caso del potencial gaussiano,\n",
    "$$\n",
    "\\langle V\\rangle = \\frac{-{U}_0 a}{\\sqrt{\\pi}}\\int_{-\\infty}^{\\infty} e^{-(1+a^2)u^2}du=\n",
    "\\frac{-U_0 a}{\\sqrt{a^2+1}}\n",
    "$$\n",
    "\n",
    "\n",
    "de manera que la función a optimizar será\n",
    "$$\n",
    "\\langle E\\rangle_a = \\frac{\\hbar^2}{2 m x_0^2}\\left(\\frac{a^2}{2} - \\frac{\\tilde{U}_0 a}{\\sqrt{a^2+1}}\\right)\n",
    "$$\n"
   ]
  },
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   "source": [
    "Derivando respecto de $a$ obtenemos la condición estacionaria:\n",
    "\n",
    "$$\n",
    "a-\\frac{\\tilde{U}_0}{(a^2+1)^{3/2}}=0\n",
    "$$\n",
    "\n",
    "o, \n",
    "\n",
    "$$\n",
    "(a^2+1)^{3} a^2-{\\tilde{U}_0^2}=0\n",
    "$$\n",
    "que es una ecuación polinómica de grado 4 en $a^2$, por lo que admite una solución analítica:"
   ]
  },
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   "source": [
    "$a^2={\\frac{1}{4} \\sqrt{-\\frac{16 \\sqrt[3]{\\frac{2}{3}} \\tilde{U}_0\n",
    "   ^2}{\\sqrt[3]{\\sqrt{3} \\sqrt{256 \\tilde{U}_0^6 + 27 \\tilde{U}_0 ^4}-9 \\tilde{U}_0\n",
    "   ^2}}+2 \\left(\\frac{2}{3}\\right)^{2/3} \\sqrt[3]{\\sqrt{3} \\sqrt{256 \\tilde{U}_0\n",
    "   ^6+27 \\tilde{U}_0^4}-9 \\tilde{U}_0 ^2}+1}+\\frac{1}{2} \\sqrt{\\frac{4\n",
    "   \\sqrt[3]{\\frac{2}{3}} \\tilde{U}_0 ^2}{\\sqrt[3]{\\sqrt{3} \\sqrt{256 \\tilde{U}_0\n",
    "   ^6+27 \\tilde{U}_0^4}-9 \\tilde{U}_0 ^2}}-\\frac{\\sqrt[3]{\\sqrt{3} \\sqrt{256\n",
    "   \\tilde{U}_0^6+27 \\tilde{U}_0 ^4}-9 \\tilde{U}_0 ^2}}{\\sqrt[3]{2} 3^{2/3}}+\\frac{1}{2\n",
    "   \\sqrt{-\\frac{16 \\sqrt[3]{\\frac{2}{3}} \\tilde{U}_0 ^2}{\\sqrt[3]{\\sqrt{3}\n",
    "   \\sqrt{256 \\tilde{U}_0 ^6+27 \\tilde{U}_0 ^4}-9 \\tilde{U}_0 ^2}}+2\n",
    "   \\left(\\frac{2}{3}\\right)^{2/3} \\sqrt[3]{\\sqrt{3} \\sqrt{256 \\tilde{U}_0 ^6+27\n",
    "   \\tilde{U}_0 ^4}-9 \\tilde{U}_0 ^2}+1}}+\\frac{1}{2}}-\\frac{3}{4}}$"
   ]
  },
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   },
   "source": [
    "Lo que no nos dice mucho...  más útil es estimar su comportamiento asintótico. Para $\\lambda \\ll 1$, \n",
    "$a\\approx \\lambda$. Para ver esto podemos usar el \"método autoconsistente\": si reescribimos la ecuaci\\'on como\n",
    "\n",
    "$$\n",
    "a^2 = \\frac{\\tilde{U}_0^2}{(a^2+1)^3}\n",
    "$$\n",
    "\n",
    "lo que sugiere proponer la secuencia $a_0=0$, $a^2_{n+1}= \\frac{\\tilde{U}_0^2}{(a_{n}^2+1)^3}$ de manera que si la secuencia converge, esta lo hará a la solución de la ecuación. \n",
    "\n",
    "Inspeccionando los primeros términos, y asumiendo $\\tilde{U}_0\\ll 1$ encontramos\n",
    "\n",
    "$$\n",
    "a_{1}=\\tilde{U}_0\\;\\;\\;a_2= \\frac{\\tilde{U}_0}{(1+\\tilde{U}_0^2)^{3/2}}\\approx \\tilde{U}_0 + {\\cal O}^{5/2}(\\tilde{U}_0)\n",
    "$$\n",
    "\n",
    "Por otro lado,  si $\\tilde{U}_0\\rightarrow \\infty$, $a \\approx \\sqrt[4]{\\tilde{U}_0}$. Para verlo, proponemos que  $a_0=\\tilde{U}_0^{\\alpha}$, de manera que $\\alpha$ satisfaga la ecuación aproximada\n",
    "$$\n",
    "\\tilde{U}_0^{\\alpha} = \\frac{\\tilde{U}_0}{(1+\\tilde{U}_0^{2\\alpha})^{3/2}}\\approx \\tilde{U}_0^{1-3\\alpha}\\Rightarrow \\alpha=1/4\n",
    "$$\n",
    "\n",
    "Luego,\n",
    "\n",
    "$$\n",
    "a \\approx a_1 =\\frac{\\tilde{U}_0}{(1+\\tilde{U}_0^{1/2})^{3/2}}\\approx \\tilde{U}_0^{1/4}-\\frac{3}{2}\\tilde{U}_0^{-1/4}+{\\cal O}^{-3/4}(\\tilde{U}_0)\n",
    "$$\n",
    "\n",
    "\n"
   ]
  },
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   "metadata": {
    "level": 7
   },
   "source": [
    "Remplazando ambos casos en la energía obtenemos\n",
    "\n",
    "$$\n",
    "E_0 \\approx \\left\\{  \n",
    "\\begin{array}{l r}\n",
    "-\\frac{m x_0^2 {U}^2_0}{\\hbar^2} & \\tilde{U}_0 < 1\\\\\n",
    "-U_0 + \\sqrt{U_0} & \\tilde{U}_0 > 1\\\\\n",
    "\\end{array}\n",
    "\\right.\n",
    "$$"
   ]
  },
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   "metadata": {
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    "level": 3
   },
   "source": [
    "### Aproximación por funciones armónicas"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "level": 7
   },
   "source": [
    "Una tercera aproximación es aproximar la solución del problema por aquellas correspondientes a un pozo de potencial \"infinito\" de un cierto tamaño: \n",
    "$$\n",
    "\\Psi(a)=\\left\\{\\begin{array}{l r}\n",
    "\\frac{\\cos(a \\pi x/(2 x_0))}{\\sqrt{x_0/a}} &  |x|<x_0/a\\\\ \n",
    "0 & |x|>x_0/a\n",
    "\\end{array}\\right .\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "level": 7
   },
   "source": [
    "La energía cinética media para estas funciones viene dada por\n",
    "$$\n",
    "\\langle T\\rangle_a= (\\frac{\\pi}{2})^2 \\frac{\\hbar^2}{2 m x_0^2} a^2\n",
    "$$\n",
    "mientras que la energía potencial\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\langle U \\rangle_{a} &=& -U_0 \\int_{-x_0/a}^{x_0/a} \\cos^2(a \\pi x/(2 x_0)) e^{-(x/x_0)^2}  \\frac{dx}{x_0/a}\\\\\n",
    "&=&-\\frac{ U_0}{\\pi/2} \\int_{0}^{\\pi/2} 2\\cos^2(u) e^{-( 2/(a \\pi)   u )^2}   du\\\\\n",
    "&=&-\\frac{ U_0}{\\pi/2} \\int_{0}^{\\pi/2} (1+\\cos(2 u))e^{-( 2/(a \\pi)   u )^2}   du\\\\\n",
    "&=&-U_0  a \\left(\n",
    "\\int_{0}^{\\pi/2} e^{-   u ^2}   du+\n",
    "\\Re \\int_{0}^{\\pi/2} \\exp({\\bf i}a \\pi u)e^{-   u ^2}   du \\right)\\\\\n",
    "&=&-U_0  a  \\frac{\\sqrt{\\pi}}{2}\\left(\n",
    "{\\rm erf}(1/a)+ e^{-a^2 \\pi^2/4} \\Re \\left({\\rm erf}(1/a -{\\bf i} a \\pi/2  )\\right) \\right)\\\\\n",
    "\\end{eqnarray}\n",
    "\n",
    "\n",
    "de esta manera\n",
    "\n",
    "$$\n",
    "\\langle E \\rangle_a=\\frac{\\hbar^2}{2 m x_0^2}\\left(\\left(\\frac{ \\pi a}{2 }\\right)^2- \\tilde{U}_0 \n",
    "a \\frac{\\sqrt{\\pi}}{2} \\left(\n",
    "{\\rm erf}(1/a)+  e^{-a^2 \\pi^2/4} \\Re \\left({\\rm erf}(1/a -{\\bf i} a \\pi/2  )\\right) \\right)\n",
    "\\right)\n",
    "$$\n",
    "\n",
    "Notamos a partir de su forma integral (3ra linea), la energía potencial está acotada por $-U_0\\leq \\langle U \\rangle_a\\leq 0$, límite que se satura cuando $a \\gtrapprox 1$ de manera que para  $\\tilde{U}_0 \\leq 1$, $a\\leq 2/\\pi<1$. En esta región podemos aproximar $\\langle U \\rangle_a\\approx -\\frac{U_0}{\\pi/2}\\int_0^{\\infty} 2 e^{-(2/(a \\pi) u)^2}du= a U_0 \\sqrt{\\pi}$  de manera que el mínimo se alcanzará para $a=2\\tilde{U}_0/\\sqrt{\\pi^3}$, con una energía de \n",
    "$\\langle E\\rangle \\approx -\\frac{1}{\\pi} U_0^2 \\frac{2 m x_0^2 }{\\hbar^2}$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "heading_collapsed": false,
    "level": 3
   },
   "source": [
    "### Aproximación por estados \"cuasi-ligados\" "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "level": 7
   },
   "source": [
    "Los estados \"cuasi-ligados\" asociados al pseudopotencial deltiforme pertenecen a la familia\n",
    "$$\n",
    "\\Psi_a(x)=\\frac{e^{-a |x|/x_0}}{\\sqrt{x_0/a}}\n",
    "$$\n",
    "\n",
    "\n",
    "La energía cinética media viene dada por\n",
    "$$\n",
    "\\langle T\\rangle_a= -\\frac{\\hbar^2}{2 m}\\int_{-\\infty}^{\\infty} \\Psi_a(x) \\Psi_a^{''}(x)dx=\n",
    " a \\frac{\\hbar^2}{2 m x_0^2}\\int_{-\\infty}^{\\infty} -(e^{-a |u|}) (e^{-a |u|})^{''}(x)d u\n",
    "$$\n",
    "Notamos sin embargo al tratar de evaluar la integral que $(e^{-a |u|})^{''}$ no está bien definida. Para evaluar esta integral, integramos por partes con lo que obtenemos\n",
    "$$\n",
    "\\int_{-\\infty}^{\\infty} -(e^{-a |u|}) (e^{-a |u|})^{''}(x)d u=\n",
    "\\int_{-\\infty}^{\\infty} ((e^{-a |u|})^{'}(x))^2 d u = 2 a^2 \\int_{0}^{\\infty} e^{-2 u a} du= a\n",
    "$$\n",
    "y por lo tanto\n",
    "$$\n",
    "\\langle T\\rangle_a= \n",
    " \\frac{\\hbar^2 a^2 }{2 m x_0^2}\n",
    "$$\n",
    "\n",
    "Por otro lado, la energía potencial media vendrá dada por\n",
    "\\begin{eqnarray}\n",
    "\\langle U\\rangle_a &=& U_0 \\int_{-\\infty}^{\\infty} e^{-(x/x_0)^2}e^{-2 a |x|/x_0}\\frac{ dx}{x_0/a}\\\\\n",
    "&=& 2 a U_0 \\int_{0}^{\\infty} e^{-u^2}e^{-2 a u} du\\\\\n",
    "&=& 2 a U_0 e^{-a^2} \\int_{0}^{\\infty} e^{-(u+a )^2} du\\\\\n",
    "&=& 2 a U_0 e^{-a^2} \\int_{a}^{\\infty} e^{-u^2} du =  a U_0 e^{-a^2} \\sqrt{\\pi} (1-{\\rm erf}(a) )\n",
    "\\end{eqnarray}\n",
    "De manera que \n",
    "$$\n",
    "\\langle E\\rangle_a= \\frac{\\hbar^2}{2 m x_0^2}\\left(a^2 -\\sqrt{\\pi} \\tilde{U}_0 a e^{-a^2}(1-{\\rm erf}(a) )  \\right)\n",
    "$$\n",
    "\n",
    "Si $a\\ll 1$, podemos desarrollar la energía media a segundo orden en torno a $a=0$ de manera que\n",
    "$$\n",
    "\\langle E\\rangle_a\\approx \\frac{\\hbar^2}{2 m x_0^2}\\left(a^2 -\\sqrt{\\pi} \\tilde{U}_0 a (1- \\frac{2 a}{\\sqrt{\\pi}}  )  \\right)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "level": 7
   },
   "source": [
    "con lo que el mínimo de energía se alcanza para $a_0=\\sqrt{\\pi}\\frac{\\tilde{U}_0}{2(1+2 \\tilde{U}_0)}$ donde \n",
    "$\\langle E\\rangle_{a_0}=  -\\frac{\\hbar^2}{2 m x_0^2}\\frac{\\pi}{4+8 \\tilde{U}_0}\\tilde{U}_0^2 \\approx -\n",
    "\\frac{\\pi}{4}\\frac{2 m x_0^2 {U}_0^2}{\\hbar^2}$ que es el resultado que obtendríamos para el pseudopotencial. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-04-16T18:19:52.192862",
     "start_time": "2017-04-16T18:19:52.189559"
    },
    "heading_collapsed": false,
    "level": 2
   },
   "source": [
    "## Solución del problema de Sturm-Liouvile 1D. Método de Shooting"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "level": 7
   },
   "source": [
    "Un método alternativo para resolver numéricamente el espectro y las auto funciones de un Hamiltoniano en una dimensión consiste en definir la función\n",
    "$$\n",
    "b(E)=\\psi(x_2,E)\n",
    "$$\n",
    "donde $\\psi(x,E)$ es una solución de la ecuación diferencial $-\\frac{\\hbar^2}{2m} \\psi''(x)+(V(x)-E) \\psi(x)=0$\n",
    "con la condición $\\psi(x_1)=0$  $\\psi'(x_1)=1$ (o viceversa).\n",
    "\n",
    "Para cada $E$, es posible evaluar en forma relativamente eficiente $b(E)$ simplemente integrando la ecuación diferencial con métodos estandar como Runge-Kutta.\n",
    "\n",
    "Para encontrar el espectro ligado, buscamos luego los valores para los que $b(E)=0$. Para que el método resulte eficiente, es conveniente saber de antemano los valores aproximados de las energías que buscamos."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Tratamiento numérico y comparación de los resultados"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
   "metadata": {
    "collapsed": true,
    "level": 7
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from numpy import sqrt,exp,pi,linspace,array\n",
    "from scipy.special import erf\n",
    "from scipy.optimize import minimize_scalar\n",
    "from scipy.integrate import ode\n",
    "\n",
    "# Calculo variacional con las diferentes familias\n",
    "\n",
    "def energia_gauss(a,u):\n",
    "    return a**2/2- u*a/sqrt(1+a**2)\n",
    "\n",
    "\n",
    "def energia_armonica(a,u):\n",
    "    return (pi/2*a)**2  - \\\n",
    "            sqrt(pi)/2.* u*a*(erf(1./(a+1e-12)) +  \\\n",
    "            exp(-(pi*a/2)**2)*(erf(1./(a+1e-12) +1j*a*pi/2.)).real  ) \n",
    "\n",
    "def energia_exp(a,u):\n",
    "    return (a**2- sqrt(pi)* u * a * exp(-a**2)*(1-erf(a)))\n",
    "\n",
    "\n",
    "def variational_energy(u,en):\n",
    "    return minimize_scalar(en,args=(u),bounds=(.0001,10)).fun\n",
    "\n",
    "def variational_energy_and_coeff(u,en):\n",
    "    res = minimize_scalar(en,args=(u),bounds=(.0001,10))\n",
    "    return (res.fun,res.x)\n",
    "\n",
    "\n",
    "# Solución por método de Shooting\n",
    "\n",
    "\n",
    "def schroedinger_equation2(x,psi2,en,utilde):\n",
    "    \"\"\"\n",
    "    A partir del valor de x, psi2=(psi,psi'),\n",
    "    la energía, y la constante de acoplamiento reducida,\n",
    "    calcula la derivada segunda de psi(x) de acuerdo a la\n",
    "    ecuación de Schrödinger, y devuelve el array\n",
    "    np.array([psi',psi''])\n",
    "    \n",
    "    \"\"\"\n",
    "    ddpsi = -(en + utilde * np.exp(-(x)**2))*psi2[0]\n",
    "    #ddpsi = -(en + utilde * (1-(x)**2))*psi2[0]\n",
    "    return np.array([psi2[1],ddpsi])\n",
    "\n",
    "def ln_boundary_value(E,utilde,bc1=\"dirichlet\",bc2=\"dirichlet\",xmax=10):\n",
    "    \"\"\"\n",
    "    Dada la energía, el valor de la constante de acoplamiento,\n",
    "    y las condiciones de borde en x=0, calcula el valor de la función \n",
    "    (o su derivada) en x=xmax\n",
    "    \"\"\"\n",
    "    bc_init_vals={'dirichlet':np.array([0,1]),\n",
    "                  'spherical':np.array([0,1]),\n",
    "                  'neumann':np.array([1,0]),\n",
    "                 }\n",
    "    target_bc={'dirichlet':0,\n",
    "                'spherical':0,\n",
    "                  'neumann':1,\n",
    "             }\n",
    "    \n",
    "    solver=ode(schroedinger_equation2)\n",
    "    solver.set_integrator(\"dop853\",nsteps=1000)\n",
    "    solver.set_f_params(E,utilde)\n",
    "    solver.set_initial_value(bc_init_vals[bc1])\n",
    "    return (np.log(np.abs(solver.integrate(xmax)[target_bc[bc2]])+1.e-12))\n",
    "\n",
    "\n",
    "def eigenfunction(values,E,utilde,bc1=\"dirichlet\",bc2=\"dirichlet\",xmax=10):\n",
    "    \"\"\"\n",
    "    Dada la energía, el valor de la constante de acoplamiento,\n",
    "    y las condiciones de borde en x=0, calcula el valor de la función \n",
    "    (o su derivada) en x=xmax\n",
    "    \"\"\"\n",
    "    bc_init_vals={'dirichlet':np.array([0,1]),\n",
    "                  'spherical':np.array([0,1]),\n",
    "                  'neumann':np.array([1,0]),\n",
    "                 }\n",
    "    target_bc={'dirichlet':0,\n",
    "                'spherical':0,\n",
    "                  'neumann':1,\n",
    "             }\n",
    "    \n",
    "    solver=ode(schroedinger_equation2)\n",
    "    solver.set_integrator(\"dop853\",nsteps=1000)\n",
    "    solver.set_f_params(E,utilde)\n",
    "    solver.set_initial_value(bc_init_vals[bc1])\n",
    "    res = []\n",
    "    for x in values:\n",
    "        res.append(solver.integrate(x)[0])\n",
    "    return res\n",
    "\n",
    "\n",
    "def find_spectrum(utilde,bounds=None,bc1=\"dirichlet\",bc2=\"dirichlet\",xmax=10):\n",
    "    if bounds is None:\n",
    "        if utilde<.3:  #aprovechamos lo que aprendimos con el método variacional\n",
    "            bounds=(-utilde,0.)\n",
    "        else:\n",
    "            bounds=(-utilde,min(-utilde+np.sqrt(utilde),0))\n",
    "    \n",
    "    energies = np.linspace(bounds[0],bounds[1],400)     \n",
    "    values = np.array([ln_boundary_value(en, utilde,bc1,bc2,xmax) for en in energies])\n",
    "    spectrum = []         \n",
    "    for i in range(len(energies)-2):\n",
    "        if values[i-1]> values[i] and values[i+1]> values[i]:\n",
    "            spectrum.append(energies[i])\n",
    "    return spectrum\n",
    "        \n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 152,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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uXsjk9ZOZvH4yaw+tJUf2HNxU/ibuqnwXd1S6g5J5S150zP7ISG75/nvWPdKN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\nRERE0k4NiYiIiKSdGhIRERFJu/8HBqZZv/26ZlUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fd96d0015c0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "urange = linspace(1e-3,.3,20)\n",
    "exactgse=[]\n",
    "for u in urange:\n",
    "    newspectrum = find_spectrum(u,bc1=\"neumann\",xmax=30)\n",
    "    if len(newspectrum)>0:\n",
    "        exactgse.append(newspectrum[0])\n",
    "    else:\n",
    "        exactgse.append(0.)\n",
    "\n",
    "\n",
    "ax1=plt.subplot(111)\n",
    "urange = linspace(1e-3,.3,20)\n",
    "opt_arm = array([ variational_energy(u,energia_armonica)   for u in urange])\n",
    "opt_exp = array([ variational_energy(u,energia_exp)   for u in urange])\n",
    "opt_gauss = array([ variational_energy(u,energia_gauss)   for u in urange])\n",
    "\n",
    "\n",
    "\n",
    "ax1.plot(urange,opt_gauss,label=\"var Gaussiana\")\n",
    "ax1.plot(urange,opt_exp,label=\"var exp\")\n",
    "ax1.plot(urange,opt_arm,label=\"var armonica\")\n",
    "ax1.plot(urange,array([-pi*u**2/5. for u in urange]),\".\",label=\"pseudo pot\")\n",
    "ax1.plot(urange,exactgse,\"*\",label=\"exact\")\n",
    "ax1.legend(loc=0)\n",
    "plt.title(\"U<<1 variacional, pseudo potencial y exacto\")\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 150,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-0.0245614035088\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/opt/anaconda/lib/python3.5/site-packages/scipy/integrate/_ode.py:1018: UserWarning: dop853: step size becomes too small\n",
      "  self.messages.get(idid, 'Unexpected idid=%s' % idid))\n"
     ]
    },
    {
     "data": {
      "image/png": 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QJRERifeUJMSy9OlhwQI4fx4a1EvGiBoTOfL3EXp82SPYoYmIiFyXkoQ4kD07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\n6sAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fd96d2e0fd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "alpha=variational_energy_and_coeff(.2,energia_exp)[1]\n",
    "xvals=np.linspace(0,2/alpha,40)\n",
    "yvals=np.array([exp(-alpha*x) for x in xvals])\n",
    "\n",
    "e0shoot=find_spectrum(.2,bc1=\"neumann\",xmax=50)\n",
    "if len(e0shoot)>0:\n",
    "    e0shoot=e0shoot[0]\n",
    "    print(e0shoot)\n",
    "    psishoot=eigenfunction(xvals,e0shoot,.2,bc1=\"neumann\")\n",
    "    plt.plot(xvals,yvals)\n",
    "    plt.plot(xvals,psishoot)\n",
    "    plt.title(\"Función de onda del estado fundamental\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 148,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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SvoSX577M3qy9ANSvVb9IkJQUm0Tbhm2pFVTLm7urlFJK+TwNiI5TcHAwg+4c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\n+PXXX1m4cCGHDh3ip59+Mu+X0m+mS5cuVKlShWnTpt23FefYsWN8880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hQ4dSrlw5bGxs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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fd96d4ad978>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "urange = linspace(1e-3,25.,20)\n",
    "exactgse=[]\n",
    "for u in urange:\n",
    "    newspectrum = find_spectrum(u,bc1=\"neumann\")\n",
    "    if len(newspectrum)>0:\n",
    "        exactgse.append(newspectrum[0])\n",
    "    else:\n",
    "        exactgse.append(0.)\n",
    "\n",
    "\n",
    "ax2=plt.subplot(111)\n",
    "urange = linspace(1e-3,25.,20)\n",
    "opt_arm = array([ variational_energy(u,energia_armonica)   for u in urange])\n",
    "opt_exp = array([ variational_energy(u,energia_exp)   for u in urange])\n",
    "opt_gauss = array([ variational_energy(u,energia_gauss)   for u in urange])\n",
    "\n",
    "\n",
    "\n",
    "ax2.plot(urange,opt_gauss,label=\"var Gaussiana\")\n",
    "ax2.plot(urange,opt_exp,label=\"var exp\")\n",
    "ax2.plot(urange,opt_arm,label=\"var armonica\")\n",
    "ax2.plot(urange,array([-u + sqrt(u) for u in urange]),\".\",label=\"oscilador armónico\")\n",
    "ax2.plot(urange,exactgse,\"*\",label=\"exacta\")\n",
    "ax2.legend(loc=0)\n",
    "plt.title(\"U>>1 variacional, aproximación como oscilador armónico y exacto\")\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Discusión"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El método variacional nos permite encontrar, sin necesidad de resolver la ecuación de Schröedinger, una cota superior para la energía del estado fundamental, que sólo requiere optimizar una función con un número finito de parámetros (en este caso, un único parámetro). La aproximación perturbativa, si bien no prové de una cota a la energía, puede resultar muy precisa si el potencial se parece lo suficiente a un potencial soluble, al menos en la región donde la función de onda del estado fundamental tiene su soporte. Por otro lado, para Hamiltonianos en una dimensión vimos que el metodo de \"shooting\" permite, si tenemos una idea aproximada del valor de la energía en el nivel fundamental, resolver la ecuación de Schrödinger con mucha precisión."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para el caso del potencial gaussiano, vimos que cuando el parámetro adimensional $\\tilde{U}_{0}$ resulta pequeño, la aproximación variacional basada en exponenciales resulta ser mejor que la aproximación con funciones gaussianas. Sin embargo, vemos que la solución vía el método de Shooting sugiere que la energía del verdadero estado fundamental sería más baja. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En el límite de $\\tilde{U}_{0}\\gg 1$, vemos que la familia de funciones gaussianas da una mejor estimación a la energía del estado fundamental, que resulta ser prácticamente igual al resultado vía el método de shooting, mientras que la aproximación \"perturbativa\" vía el potencial de oscilador armónico provee un valor ligeramente más alto.  "
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "hide_input": false,
  "kernelspec": {
   "display_name": "Python",
   "language": "python",
   "name": "python3"
  },
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