HPsi=Epsi_hydrogen.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "水素原子の2p1波動関数$\\Psi (r,\\theta,\\phi) = \\frac{r}{a} \\exp (-\\frac{r}{2a}) \\exp (i\\phi)\\sin(\\theta)$を使って、$H\\Psi = E \\Psi$ から $H\\Psi / \\Psi = E$ を確認する"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "極座標の定義"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "r = Symbol('r',real=True, positive=True)\n",
    "θ = Symbol('theta',real=True, positive=True,domain=Interval(0,pi))\n",
    "ϕ = Symbol('\\phi',real=True, positive=True,domain=Interval(0,2*pi))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = r * sin(θ) * cos(ϕ)\n",
    "y = r * sin(θ) * sin(ϕ)\n",
    "z = r * cos(θ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left [ r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )}, \\quad r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )}, \\quad r \\cos{\\left (\\theta \\right )}\\right ]$$"
      ],
      "text/plain": [
       "[r⋅sin(θ)⋅cos(\\phi), r⋅sin(\\phi)⋅sin(θ), r⋅cos(θ)]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xyz = [x, y, z]\n",
    "xyz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left [ r, \\quad \\theta, \\quad \\phi\\right ]$$"
      ],
      "text/plain": [
       "[r, θ, \\phi]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rtp = [r, θ, ϕ]\n",
    "rtp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = Symbol('a')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "水素原子の2p1波動関数$\\psi$を入れてやる"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{r}{a} e^{i \\phi} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}$$"
      ],
      "text/plain": [
       "           -r        \n",
       "           ───       \n",
       "   ⅈ⋅\\phi  2⋅a       \n",
       "r⋅ℯ      ⋅ℯ   ⋅sin(θ)\n",
       "─────────────────────\n",
       "          a          "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ψ =r/a * exp (-r/(2*a)) * exp (I*ϕ)* sin(θ)\n",
    "ψ"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "まず、クーロンポテンシャル部分 $U \\psi $ を先に計算する"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{e^{i \\phi}}{a^{2}} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}$$"
      ],
      "text/plain": [
       "         -r        \n",
       "         ───       \n",
       " ⅈ⋅\\phi  2⋅a       \n",
       "ℯ      ⋅ℯ   ⋅sin(θ)\n",
       "───────────────────\n",
       "          2        \n",
       "         a         "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Uψ = (1/r) * (1/a) * ψ\n",
    "Uψ"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "次に、ラプラシアンの計算を行う。あるスカラ関数$\\psi$に対するラプラシアンは$\\nabla^2\\psi$であり、直交座標系ではdiv grad $\\psi$ と等しい。これを球面座標系に変換する必要がある。球面座標系から直交座標系へのヤコビアン$J=\\frac{\\partial (r, \\theta, \\phi)}{\\partial (x, y, z)}$を使えば、"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "div grad $\\psi = (\\nabla \\cdot J) (J^{-1}  \\nabla \\psi)$ と書ける(ちょっとした手間で示せる)。まずヤコビアン$J$を求めよう"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{\\partial}{\\partial r}\\left(r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )}\\right) & \\frac{\\partial}{\\partial r}\\left(r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )}\\right) & \\frac{\\partial}{\\partial r}\\left(r \\cos{\\left (\\theta \\right )}\\right)\\\\\\frac{\\partial}{\\partial \\theta}\\left(r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )}\\right) & \\frac{\\partial}{\\partial \\theta}\\left(r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )}\\right) & \\frac{\\partial}{\\partial \\theta}\\left(r \\cos{\\left (\\theta \\right )}\\right)\\\\\\frac{\\partial}{\\partial \\phi}\\left(r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )}\\right) & \\frac{\\partial}{\\partial \\phi}\\left(r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )}\\right) & \\frac{\\partial}{\\partial \\phi}\\left(r \\cos{\\left (\\theta \\right )}\\right)\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ ∂                          ∂                          ∂             ⎤\n",
       "⎢ ──(r⋅sin(θ)⋅cos(\\phi))     ──(r⋅sin(\\phi)⋅sin(θ))     ──(r⋅cos(θ))  ⎥\n",
       "⎢ ∂r                         ∂r                         ∂r            ⎥\n",
       "⎢                                                                     ⎥\n",
       "⎢ ∂                          ∂                          ∂             ⎥\n",
       "⎢ ──(r⋅sin(θ)⋅cos(\\phi))     ──(r⋅sin(\\phi)⋅sin(θ))     ──(r⋅cos(θ))  ⎥\n",
       "⎢ ∂θ                         ∂θ                         ∂θ            ⎥\n",
       "⎢                                                                     ⎥\n",
       "⎢  ∂                          ∂                          ∂            ⎥\n",
       "⎢─────(r⋅sin(θ)⋅cos(\\phi))  ─────(r⋅sin(\\phi)⋅sin(θ))  ─────(r⋅cos(θ))⎥\n",
       "⎣∂\\phi                      ∂\\phi                      ∂\\phi          ⎦"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "J = Matrix(3 , 3 , [Derivative(c1 , c2) for c2 in rtp for c1 in xyz])\n",
    "J"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} & \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} & \\cos{\\left (\\theta \\right )}\\\\r \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} & r \\sin{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} & - r \\sin{\\left (\\theta \\right )}\\\\- r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} & r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ sin(θ)⋅cos(\\phi)     sin(\\phi)⋅sin(θ)    cos(θ)  ⎤\n",
       "⎢                                                  ⎥\n",
       "⎢r⋅cos(\\phi)⋅cos(θ)   r⋅sin(\\phi)⋅cos(θ)  -r⋅sin(θ)⎥\n",
       "⎢                                                  ⎥\n",
       "⎣-r⋅sin(\\phi)⋅sin(θ)  r⋅sin(θ)⋅cos(\\phi)      0    ⎦"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "J = J.doit()\n",
    "J"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$J$の逆行列も求めておく。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} & \\frac{1}{r} \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} & - \\frac{\\sin{\\left (\\phi \\right )}}{r \\sin{\\left (\\theta \\right )}}\\\\\\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} & \\frac{1}{r} \\sin{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} & \\frac{\\cos{\\left (\\phi \\right )}}{r \\sin{\\left (\\theta \\right )}}\\\\\\cos{\\left (\\theta \\right )} & - \\frac{1}{r} \\sin{\\left (\\theta \\right )} & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                  cos(\\phi)⋅cos(θ)  -sin(\\phi) ⎤\n",
       "⎢sin(θ)⋅cos(\\phi)  ────────────────  ───────────⎥\n",
       "⎢                         r            r⋅sin(θ) ⎥\n",
       "⎢                                               ⎥\n",
       "⎢                  sin(\\phi)⋅cos(θ)   cos(\\phi) ⎥\n",
       "⎢sin(\\phi)⋅sin(θ)  ────────────────   ───────── ⎥\n",
       "⎢                         r            r⋅sin(θ) ⎥\n",
       "⎢                                               ⎥\n",
       "⎢                      -sin(θ)                  ⎥\n",
       "⎢     cos(θ)           ────────           0     ⎥\n",
       "⎣                         r                     ⎦"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "InvJ = simplify(J.inv())\n",
    "InvJ"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "さてラプラシアンを求めよう。div grad $\\psi = (\\nabla \\cdot J) (J^{-1}  \\nabla \\psi)$を順に計算してゆく。まず、$a= \\nabla \\psi$として"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{\\partial}{\\partial r}\\left(\\frac{r}{a} e^{i \\phi} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right)\\\\\\frac{\\partial}{\\partial \\theta}\\left(\\frac{r}{a} e^{i \\phi} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right)\\\\\\frac{\\partial}{\\partial \\phi}\\left(\\frac{r}{a} e^{i \\phi} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right)\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡   ⎛           -r        ⎞  ⎤\n",
       "⎢   ⎜           ───       ⎟  ⎥\n",
       "⎢   ⎜   ⅈ⋅\\phi  2⋅a       ⎟  ⎥\n",
       "⎢ ∂ ⎜r⋅ℯ      ⋅ℯ   ⋅sin(θ)⎟  ⎥\n",
       "⎢ ──⎜─────────────────────⎟  ⎥\n",
       "⎢ ∂r⎝          a          ⎠  ⎥\n",
       "⎢                            ⎥\n",
       "⎢   ⎛           -r        ⎞  ⎥\n",
       "⎢   ⎜           ───       ⎟  ⎥\n",
       "⎢   ⎜   ⅈ⋅\\phi  2⋅a       ⎟  ⎥\n",
       "⎢ ∂ ⎜r⋅ℯ      ⋅ℯ   ⋅sin(θ)⎟  ⎥\n",
       "⎢ ──⎜─────────────────────⎟  ⎥\n",
       "⎢ ∂θ⎝          a          ⎠  ⎥\n",
       "⎢                            ⎥\n",
       "⎢     ⎛           -r        ⎞⎥\n",
       "⎢     ⎜           ───       ⎟⎥\n",
       "⎢     ⎜   ⅈ⋅\\phi  2⋅a       ⎟⎥\n",
       "⎢  ∂  ⎜r⋅ℯ      ⋅ℯ   ⋅sin(θ)⎟⎥\n",
       "⎢─────⎜─────────────────────⎟⎥\n",
       "⎣∂\\phi⎝          a          ⎠⎦"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = Matrix(3 , 1 , [Derivative(ψ , c) for c in rtp])\n",
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "grad $\\psi = J^{-1} a$ だから"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\frac{\\partial}{\\partial r}\\left(\\frac{r}{a} e^{i \\phi} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right) - \\frac{\\sin{\\left (\\phi \\right )}}{r \\sin{\\left (\\theta \\right )}} \\frac{\\partial}{\\partial \\phi}\\left(\\frac{r}{a} e^{i \\phi} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right) + \\frac{1}{r} \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} \\frac{\\partial}{\\partial \\theta}\\left(\\frac{r}{a} e^{i \\phi} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right)\\\\\\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} \\frac{\\partial}{\\partial r}\\left(\\frac{r}{a} e^{i \\phi} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right) + \\frac{1}{r} \\sin{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} \\frac{\\partial}{\\partial \\theta}\\left(\\frac{r}{a} e^{i \\phi} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right) + \\frac{\\cos{\\left (\\phi \\right )}}{r \\sin{\\left (\\theta \\right )}} \\frac{\\partial}{\\partial \\phi}\\left(\\frac{r}{a} e^{i \\phi} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right)\\\\\\cos{\\left (\\theta \\right )} \\frac{\\partial}{\\partial r}\\left(\\frac{r}{a} e^{i \\phi} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right) - \\frac{1}{r} \\sin{\\left (\\theta \\right )} \\frac{\\partial}{\\partial \\theta}\\left(\\frac{r}{a} e^{i \\phi} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right)\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                                                            ⎛           -r   \n",
       "⎢                                                            ⎜           ───  \n",
       "⎢                   ⎛           -r        ⎞                  ⎜   ⅈ⋅\\phi  2⋅a  \n",
       "⎢                   ⎜           ───       ⎟               ∂  ⎜r⋅ℯ      ⋅ℯ   ⋅s\n",
       "⎢                   ⎜   ⅈ⋅\\phi  2⋅a       ⎟   sin(\\phi)⋅─────⎜────────────────\n",
       "⎢                 ∂ ⎜r⋅ℯ      ⋅ℯ   ⋅sin(θ)⎟             ∂\\phi⎝          a     \n",
       "⎢sin(θ)⋅cos(\\phi)⋅──⎜─────────────────────⎟ - ────────────────────────────────\n",
       "⎢                 ∂r⎝          a          ⎠                  r⋅sin(θ)         \n",
       "⎢                                                                             \n",
       "⎢                                                                ⎛           -\n",
       "⎢                                                                ⎜           ─\n",
       "⎢                   ⎛           -r        ⎞                      ⎜   ⅈ⋅\\phi  2\n",
       "⎢                   ⎜           ───       ⎟                    ∂ ⎜r⋅ℯ      ⋅ℯ \n",
       "⎢                   ⎜   ⅈ⋅\\phi  2⋅a       ⎟   sin(\\phi)⋅cos(θ)⋅──⎜────────────\n",
       "⎢                 ∂ ⎜r⋅ℯ      ⋅ℯ   ⋅sin(θ)⎟                    ∂θ⎝          a \n",
       "⎢sin(\\phi)⋅sin(θ)⋅──⎜─────────────────────⎟ + ────────────────────────────────\n",
       "⎢                 ∂r⎝          a          ⎠                       r           \n",
       "⎢                                                                             \n",
       "⎢                                                                          ⎛  \n",
       "⎢                                                                          ⎜  \n",
       "⎢                                       ⎛           -r        ⎞            ⎜  \n",
       "⎢                                       ⎜           ───       ⎟          ∂ ⎜r⋅\n",
       "⎢                                       ⎜   ⅈ⋅\\phi  2⋅a       ⎟   sin(θ)⋅──⎜──\n",
       "⎢                                     ∂ ⎜r⋅ℯ      ⋅ℯ   ⋅sin(θ)⎟          ∂θ⎝  \n",
       "⎢                              cos(θ)⋅──⎜─────────────────────⎟ - ────────────\n",
       "⎣                                     ∂r⎝          a          ⎠               \n",
       "\n",
       "     ⎞                      ⎛           -r        ⎞⎤\n",
       "     ⎟                      ⎜           ───       ⎟⎥\n",
       "     ⎟                      ⎜   ⅈ⋅\\phi  2⋅a       ⎟⎥\n",
       "in(θ)⎟                    ∂ ⎜r⋅ℯ      ⋅ℯ   ⋅sin(θ)⎟⎥\n",
       "─────⎟   cos(\\phi)⋅cos(θ)⋅──⎜─────────────────────⎟⎥\n",
       "     ⎠                    ∂θ⎝          a          ⎠⎥\n",
       "────── + ──────────────────────────────────────────⎥\n",
       "                             r                     ⎥\n",
       "                                                   ⎥\n",
       "r        ⎞                  ⎛           -r        ⎞⎥\n",
       "──       ⎟                  ⎜           ───       ⎟⎥\n",
       "⋅a       ⎟                  ⎜   ⅈ⋅\\phi  2⋅a       ⎟⎥\n",
       "  ⋅sin(θ)⎟               ∂  ⎜r⋅ℯ      ⋅ℯ   ⋅sin(θ)⎟⎥\n",
       "─────────⎟   cos(\\phi)⋅─────⎜─────────────────────⎟⎥\n",
       "         ⎠             ∂\\phi⎝          a          ⎠⎥\n",
       "────────── + ──────────────────────────────────────⎥\n",
       "                            r⋅sin(θ)               ⎥\n",
       "                                                   ⎥\n",
       "         -r        ⎞                               ⎥\n",
       "         ───       ⎟                               ⎥\n",
       " ⅈ⋅\\phi  2⋅a       ⎟                               ⎥\n",
       "ℯ      ⋅ℯ   ⋅sin(θ)⎟                               ⎥\n",
       "───────────────────⎟                               ⎥\n",
       "        a          ⎠                               ⎥\n",
       "────────────────────                               ⎥\n",
       "   r                                               ⎦"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gradψ = InvJ*a\n",
    "gradψ"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "さらにgrad $\\psi$のdivを求めると、ラプラシアンが求まる"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$$\\left(\\left(- \\frac{e^{i \\phi}}{a^{2}} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )} + \\frac{r e^{i \\phi}}{4 a^{3}} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right) \\cos{\\left (\\theta \\right )} + \\frac{\\cos{\\left (\\theta \\right )}}{2 a^{2}} e^{i \\phi} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right) \\cos{\\left (\\theta \\right )} + \\left(\\left(- \\frac{e^{i \\phi}}{a^{2}} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )} + \\frac{r e^{i \\phi}}{4 a^{3}} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right) \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} - \\frac{\\cos^{2}{\\left (\\theta \\right )}}{2 a^{2}} e^{i \\phi} e^{- \\frac{r}{2 a}} \\sin{\\left (\\phi \\right )} - \\frac{i e^{i \\phi}}{2 a^{2}} e^{- \\frac{r}{2 a}} \\cos{\\left (\\phi \\right )}\\right) \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} + \\left(\\left(- \\frac{e^{i \\phi}}{a^{2}} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )} + \\frac{r e^{i \\phi}}{4 a^{3}} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right) \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + \\frac{i e^{i \\phi}}{2 a^{2}} e^{- \\frac{r}{2 a}} \\sin{\\left (\\phi \\right )} - \\frac{\\cos^{2}{\\left (\\theta \\right )}}{2 a^{2}} e^{i \\phi} e^{- \\frac{r}{2 a}} \\cos{\\left (\\phi \\right )}\\right) \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + \\frac{1}{r} \\left(\\left(\\frac{e^{i \\phi}}{a} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )} - \\frac{r e^{i \\phi}}{2 a^{2}} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right) \\sin{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} + \\left(\\frac{e^{i \\phi}}{a} e^{- \\frac{r}{2 a}} \\cos{\\left (\\theta \\right )} - \\frac{r e^{i \\phi}}{2 a^{2}} e^{- \\frac{r}{2 a}} \\cos{\\left (\\theta \\right )}\\right) \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} - \\frac{2}{a} e^{i \\phi} e^{- \\frac{r}{2 a}} \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )}\\right) \\sin{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} + \\frac{1}{r} \\left(\\left(\\frac{e^{i \\phi}}{a} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )} - \\frac{r e^{i \\phi}}{2 a^{2}} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right) \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} + \\left(\\frac{e^{i \\phi}}{a} e^{- \\frac{r}{2 a}} \\cos{\\left (\\theta \\right )} - \\frac{r e^{i \\phi}}{2 a^{2}} e^{- \\frac{r}{2 a}} \\cos{\\left (\\theta \\right )}\\right) \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} - \\frac{2}{a} e^{i \\phi} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )}\\right) \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} - \\frac{1}{r} \\left(- \\left(\\frac{e^{i \\phi}}{a} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )} - \\frac{r e^{i \\phi}}{2 a^{2}} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right) \\sin{\\left (\\theta \\right )} + \\left(\\frac{e^{i \\phi}}{a} e^{- \\frac{r}{2 a}} \\cos{\\left (\\theta \\right )} - \\frac{r e^{i \\phi}}{2 a^{2}} e^{- \\frac{r}{2 a}} \\cos{\\left (\\theta \\right )}\\right) \\cos{\\left (\\theta \\right )} + \\frac{e^{i \\phi}}{a} e^{- \\frac{r}{2 a}} \\sin^{2}{\\left (\\theta \\right )} - \\frac{e^{i \\phi}}{a} e^{- \\frac{r}{2 a}} \\cos^{2}{\\left (\\theta \\right )}\\right) \\sin{\\left (\\theta \\right )} - \\frac{\\sin{\\left (\\phi \\right )}}{r \\sin{\\left (\\theta \\right )}} \\left(- \\left(\\frac{e^{i \\phi}}{a} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )} - \\frac{r e^{i \\phi}}{2 a^{2}} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right) \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} + \\left(\\frac{i}{a} e^{i \\phi} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )} - \\frac{i r e^{i \\phi}}{2 a^{2}} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right) \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} - \\frac{e^{i \\phi}}{a} e^{- \\frac{r}{2 a}} \\sin{\\left (\\phi \\right )} \\cos^{2}{\\left (\\theta \\right )} + \\frac{e^{i \\phi}}{a} e^{- \\frac{r}{2 a}} \\sin{\\left (\\phi \\right )} + \\frac{i}{a} e^{i \\phi} e^{- \\frac{r}{2 a}} \\cos{\\left (\\phi \\right )} \\cos^{2}{\\left (\\theta \\right )} - \\frac{i}{a} e^{i \\phi} e^{- \\frac{r}{2 a}} \\cos{\\left (\\phi \\right )}\\right) + \\frac{\\cos{\\left (\\phi \\right )}}{r \\sin{\\left (\\theta \\right )}} \\left(\\left(\\frac{e^{i \\phi}}{a} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )} - \\frac{r e^{i \\phi}}{2 a^{2}} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right) \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + \\left(\\frac{i}{a} e^{i \\phi} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )} - \\frac{i r e^{i \\phi}}{2 a^{2}} e^{- \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}\\right) \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} + \\frac{i}{a} e^{i \\phi} e^{- \\frac{r}{2 a}} \\sin{\\left (\\phi \\right )} \\cos^{2}{\\left (\\theta \\right )} - \\frac{i}{a} e^{i \\phi} e^{- \\frac{r}{2 a}} \\sin{\\left (\\phi \\right )} + \\frac{e^{i \\phi}}{a} e^{- \\frac{r}{2 a}} \\cos{\\left (\\phi \\right )} \\cos^{2}{\\left (\\theta \\right )} - \\frac{e^{i \\phi}}{a} e^{- \\frac{r}{2 a}} \\cos{\\left (\\phi \\right )}\\right)$$"
      ],
      "text/plain": [
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "⎛⎛           -r                      -r        ⎞                   -r         \n",
       "⎜⎜           ───                     ───       ⎟                   ───        \n",
       "⎜⎜   ⅈ⋅\\phi  2⋅a             ⅈ⋅\\phi  2⋅a       ⎟           ⅈ⋅\\phi  2⋅a        \n",
       "⎜⎜  ℯ      ⋅ℯ   ⋅sin(θ)   r⋅ℯ      ⋅ℯ   ⋅sin(θ)⎟          ℯ      ⋅ℯ   ⋅sin(θ)⋅\n",
       "⎜⎜- ─────────────────── + ─────────────────────⎟⋅cos(θ) + ────────────────────\n",
       "⎜⎜            2                       3        ⎟                        2     \n",
       "⎝⎝           a                     4⋅a         ⎠                     2⋅a      \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "      ⎞          ⎛⎛           -r                      -r        ⎞             \n",
       "      ⎟          ⎜⎜           ───                     ───       ⎟             \n",
       "      ⎟          ⎜⎜   ⅈ⋅\\phi  2⋅a             ⅈ⋅\\phi  2⋅a       ⎟             \n",
       "cos(θ)⎟          ⎜⎜  ℯ      ⋅ℯ   ⋅sin(θ)   r⋅ℯ      ⋅ℯ   ⋅sin(θ)⎟             \n",
       "──────⎟⋅cos(θ) + ⎜⎜- ─────────────────── + ─────────────────────⎟⋅sin(\\phi)⋅si\n",
       "      ⎟          ⎜⎜            2                       3        ⎟             \n",
       "      ⎠          ⎝⎝           a                     4⋅a         ⎠             \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                -r                                 -r           ⎞             \n",
       "                ───                                ───          ⎟             \n",
       "        ⅈ⋅\\phi  2⋅a              2         ⅈ⋅\\phi  2⋅a          ⎟             \n",
       "       ℯ      ⋅ℯ   ⋅sin(\\phi)⋅cos (θ)   ⅈ⋅ℯ      ⋅ℯ   ⋅cos(\\phi)⎟             \n",
       "n(θ) - ────────────────────────────── - ────────────────────────⎟⋅sin(\\phi)⋅si\n",
       "                       2                             2          ⎟             \n",
       "                    2⋅a                           2⋅a           ⎠             \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "       ⎛⎛           -r                      -r        ⎞                       \n",
       "       ⎜⎜           ───                     ───       ⎟                       \n",
       "       ⎜⎜   ⅈ⋅\\phi  2⋅a             ⅈ⋅\\phi  2⋅a       ⎟                       \n",
       "       ⎜⎜  ℯ      ⋅ℯ   ⋅sin(θ)   r⋅ℯ      ⋅ℯ   ⋅sin(θ)⎟                    ⅈ⋅ℯ\n",
       "n(θ) + ⎜⎜- ─────────────────── + ─────────────────────⎟⋅sin(θ)⋅cos(\\phi) + ───\n",
       "       ⎜⎜            2                       3        ⎟                       \n",
       "       ⎝⎝           a                     4⋅a         ⎠                       \n",
       "\n",
       "                                                                           ⎛⎛ \n",
       "                                                                           ⎜⎜ \n",
       "                                                                           ⎜⎜ \n",
       "        -r                       -r                   ⎞                    ⎜⎜ℯ\n",
       "        ───                      ───                  ⎟                    ⎜⎜─\n",
       "ⅈ⋅\\phi  2⋅a              ⅈ⋅\\phi  2⋅a              2   ⎟                    ⎜⎜ \n",
       "      ⋅ℯ   ⋅sin(\\phi)   ℯ      ⋅ℯ   ⋅cos(\\phi)⋅cos (θ)⎟                    ⎝⎝ \n",
       "───────────────────── - ──────────────────────────────⎟⋅sin(θ)⋅cos(\\phi) + ───\n",
       "          2                             2             ⎟                       \n",
       "       2⋅a                           2⋅a              ⎠                       \n",
       "\n",
       "        -r                      -r        ⎞                    ⎛         -r   \n",
       "        ───                     ───       ⎟                    ⎜         ───  \n",
       "ⅈ⋅\\phi  2⋅a             ⅈ⋅\\phi  2⋅a       ⎟                    ⎜ ⅈ⋅\\phi  2⋅a  \n",
       "      ⋅ℯ   ⋅sin(θ)   r⋅ℯ      ⋅ℯ   ⋅sin(θ)⎟                    ⎜ℯ      ⋅ℯ   ⋅c\n",
       "────────────────── - ─────────────────────⎟⋅sin(\\phi)⋅cos(θ) + ⎜──────────────\n",
       "        a                        2        ⎟                    ⎜         a    \n",
       "                              2⋅a         ⎠                    ⎝              \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                   -r        ⎞                               -r               \n",
       "                   ───       ⎟                               ───              \n",
       "           ⅈ⋅\\phi  2⋅a       ⎟                       ⅈ⋅\\phi  2⋅a              \n",
       "os(θ)   r⋅ℯ      ⋅ℯ   ⋅cos(θ)⎟                    2⋅ℯ      ⋅ℯ   ⋅sin(\\phi)⋅sin\n",
       "───── - ─────────────────────⎟⋅sin(\\phi)⋅sin(θ) - ────────────────────────────\n",
       "                    2        ⎟                                      a         \n",
       "                 2⋅a         ⎠                                                \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "            r                                                                 \n",
       "                                                                              \n",
       "\n",
       "          ⎞                    ⎛⎛         -r                      -r        ⎞ \n",
       "          ⎟                    ⎜⎜         ───                     ───       ⎟ \n",
       "          ⎟                    ⎜⎜ ⅈ⋅\\phi  2⋅a             ⅈ⋅\\phi  2⋅a       ⎟ \n",
       "(θ)⋅cos(θ)⎟                    ⎜⎜ℯ      ⋅ℯ   ⋅sin(θ)   r⋅ℯ      ⋅ℯ   ⋅sin(θ)⎟ \n",
       "──────────⎟⋅sin(\\phi)⋅cos(θ)   ⎜⎜─────────────────── - ─────────────────────⎟⋅\n",
       "          ⎟                    ⎜⎜         a                        2        ⎟ \n",
       "          ⎠                    ⎝⎝                               2⋅a         ⎠ \n",
       "──────────────────────────── + ───────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                   ⎛         -r                      -r        ⎞              \n",
       "                   ⎜         ───                     ───       ⎟              \n",
       "                   ⎜ ⅈ⋅\\phi  2⋅a             ⅈ⋅\\phi  2⋅a       ⎟              \n",
       "                   ⎜ℯ      ⋅ℯ   ⋅cos(θ)   r⋅ℯ      ⋅ℯ   ⋅cos(θ)⎟              \n",
       "cos(\\phi)⋅cos(θ) + ⎜─────────────────── - ─────────────────────⎟⋅sin(θ)⋅cos(\\p\n",
       "                   ⎜         a                        2        ⎟              \n",
       "                   ⎝                               2⋅a         ⎠              \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                              r                               \n",
       "                                                                              \n",
       "\n",
       "                 -r                         ⎞                    ⎛  ⎛         \n",
       "                 ───                        ⎟                    ⎜  ⎜         \n",
       "         ⅈ⋅\\phi  2⋅a                        ⎟                    ⎜  ⎜ ⅈ⋅\\phi  \n",
       "      2⋅ℯ      ⋅ℯ   ⋅sin(θ)⋅cos(\\phi)⋅cos(θ)⎟                    ⎜  ⎜ℯ      ⋅ℯ\n",
       "hi) - ──────────────────────────────────────⎟⋅cos(\\phi)⋅cos(θ)   ⎜- ⎜─────────\n",
       "                        a                   ⎟                    ⎜  ⎜         \n",
       "                                            ⎠                    ⎝  ⎝         \n",
       "────────────────────────────────────────────────────────────── - ─────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "-r                      -r        ⎞          ⎛         -r                     \n",
       "───                     ───       ⎟          ⎜         ───                    \n",
       "2⋅a             ⅈ⋅\\phi  2⋅a       ⎟          ⎜ ⅈ⋅\\phi  2⋅a             ⅈ⋅\\phi \n",
       "   ⋅sin(θ)   r⋅ℯ      ⋅ℯ   ⋅sin(θ)⎟          ⎜ℯ      ⋅ℯ   ⋅cos(θ)   r⋅ℯ      ⋅\n",
       "────────── - ─────────────────────⎟⋅sin(θ) + ⎜─────────────────── - ──────────\n",
       "a                        2        ⎟          ⎜         a                      \n",
       "                      2⋅a         ⎠          ⎝                               2\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                    r         \n",
       "                                                                              \n",
       "\n",
       " -r        ⎞                   -r                     -r         ⎞          ⎛ \n",
       " ───       ⎟                   ───                    ───        ⎟          ⎜ \n",
       " 2⋅a       ⎟           ⅈ⋅\\phi  2⋅a    2       ⅈ⋅\\phi  2⋅a    2   ⎟          ⎜ \n",
       "ℯ   ⋅cos(θ)⎟          ℯ      ⋅ℯ   ⋅sin (θ)   ℯ      ⋅ℯ   ⋅cos (θ)⎟          ⎜ \n",
       "───────────⎟⋅cos(θ) + ──────────────────── - ────────────────────⎟⋅sin(θ)   ⎜-\n",
       "  2        ⎟                   a                      a          ⎟          ⎜ \n",
       "⋅a         ⎠                                                     ⎠          ⎝ \n",
       "───────────────────────────────────────────────────────────────────────── - ──\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       " ⎛         -r                      -r        ⎞                    ⎛           \n",
       " ⎜         ───                     ───       ⎟                    ⎜           \n",
       " ⎜ ⅈ⋅\\phi  2⋅a             ⅈ⋅\\phi  2⋅a       ⎟                    ⎜   ⅈ⋅\\phi  \n",
       " ⎜ℯ      ⋅ℯ   ⋅sin(θ)   r⋅ℯ      ⋅ℯ   ⋅sin(θ)⎟                    ⎜ⅈ⋅ℯ      ⋅ℯ\n",
       " ⎜─────────────────── - ─────────────────────⎟⋅sin(\\phi)⋅sin(θ) + ⎜───────────\n",
       " ⎜         a                        2        ⎟                    ⎜          a\n",
       " ⎝                               2⋅a         ⎠                    ⎝           \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "-r                        -r        ⎞                             -r          \n",
       "───                       ───       ⎟                             ───         \n",
       "2⋅a               ⅈ⋅\\phi  2⋅a       ⎟                     ⅈ⋅\\phi  2⋅a         \n",
       "   ⋅sin(θ)   ⅈ⋅r⋅ℯ      ⋅ℯ   ⋅sin(θ)⎟                    ℯ      ⋅ℯ   ⋅sin(\\phi\n",
       "────────── - ───────────────────────⎟⋅sin(θ)⋅cos(\\phi) - ─────────────────────\n",
       "                          2         ⎟                                  a      \n",
       "                       2⋅a          ⎠                                         \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                 r⋅sin(θ)                     \n",
       "                                                                              \n",
       "\n",
       "                     -r                         -r                            \n",
       "                     ───                        ───                           \n",
       "     2       ⅈ⋅\\phi  2⋅a                ⅈ⋅\\phi  2⋅a              2         ⅈ⋅\\\n",
       ")⋅cos (θ)   ℯ      ⋅ℯ   ⋅sin(\\phi)   ⅈ⋅ℯ      ⋅ℯ   ⋅cos(\\phi)⋅cos (θ)   ⅈ⋅ℯ   \n",
       "───────── + ────────────────────── + ──────────────────────────────── - ──────\n",
       "                      a                             a                         \n",
       "                                                                              \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "     -r           ⎞             ⎛⎛         -r                      -r        ⎞\n",
       "     ───          ⎟             ⎜⎜         ───                     ───       ⎟\n",
       "phi  2⋅a          ⎟             ⎜⎜ ⅈ⋅\\phi  2⋅a             ⅈ⋅\\phi  2⋅a       ⎟\n",
       "   ⋅ℯ   ⋅cos(\\phi)⎟             ⎜⎜ℯ      ⋅ℯ   ⋅sin(θ)   r⋅ℯ      ⋅ℯ   ⋅sin(θ)⎟\n",
       "──────────────────⎟⋅sin(\\phi)   ⎜⎜─────────────────── - ─────────────────────⎟\n",
       "     a            ⎟             ⎜⎜         a                        2        ⎟\n",
       "                  ⎠             ⎝⎝                               2⋅a         ⎠\n",
       "───────────────────────────── + ──────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                    ⎛           -r                        -r        ⎞         \n",
       "                    ⎜           ───                       ───       ⎟         \n",
       "                    ⎜   ⅈ⋅\\phi  2⋅a               ⅈ⋅\\phi  2⋅a       ⎟         \n",
       "                    ⎜ⅈ⋅ℯ      ⋅ℯ   ⋅sin(θ)   ⅈ⋅r⋅ℯ      ⋅ℯ   ⋅sin(θ)⎟         \n",
       "⋅sin(θ)⋅cos(\\phi) + ⎜───────────────────── - ───────────────────────⎟⋅sin(\\phi\n",
       "                    ⎜          a                          2         ⎟         \n",
       "                    ⎝                                  2⋅a          ⎠         \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                      -r                                 -r                   \n",
       "                      ───                                ───                  \n",
       "              ⅈ⋅\\phi  2⋅a              2         ⅈ⋅\\phi  2⋅a              ⅈ⋅\\p\n",
       "           ⅈ⋅ℯ      ⋅ℯ   ⋅sin(\\phi)⋅cos (θ)   ⅈ⋅ℯ      ⋅ℯ   ⋅sin(\\phi)   ℯ    \n",
       ")⋅sin(θ) + ──────────────────────────────── - ──────────────────────── + ─────\n",
       "                          a                              a                    \n",
       "                                                                              \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "    r⋅sin(θ)                                                                  \n",
       "                                                                              \n",
       "\n",
       "    -r                               -r           ⎞          \n",
       "    ───                              ───          ⎟          \n",
       "hi  2⋅a              2       ⅈ⋅\\phi  2⋅a          ⎟          \n",
       "  ⋅ℯ   ⋅cos(\\phi)⋅cos (θ)   ℯ      ⋅ℯ   ⋅cos(\\phi)⎟          \n",
       "───────────────────────── - ──────────────────────⎟⋅cos(\\phi)\n",
       "         a                            a           ⎟          \n",
       "                                                  ⎠          \n",
       "─────────────────────────────────────────────────────────────\n",
       "                                                             \n",
       "                                                             "
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L2ψ=sum([ (InvJ * Matrix(3,1,[ Derivative(gradψ[i],e) for e in rtp ]).doit())[i] for i in range(3) ])\n",
    "L2ψ"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "かなり複雑な式になったので、簡単に(simplify)してみると..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{\\sin{\\left (\\theta \\right )}}{4 a^{3}} \\left(- 8 a + r\\right) e^{i \\phi - \\frac{r}{2 a}}$$"
      ],
      "text/plain": [
       "                      r        \n",
       "            ⅈ⋅\\phi - ───       \n",
       "                     2⋅a       \n",
       "(-8⋅a + r)⋅ℯ            ⋅sin(θ)\n",
       "───────────────────────────────\n",
       "                 3             \n",
       "              4⋅a              "
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(L2ψ)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "ずいぶん簡単になった。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "やっと$H\\Psi / \\Psi = E$ を確認するところまできた。$H\\psi$を求める"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "Hψ = -0.5 * L2ψ - Uψ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{0.125 r}{a^{3}} e^{i \\phi - \\frac{r}{2 a}} \\sin{\\left (\\theta \\right )}$$"
      ],
      "text/plain": [
       "                    r         \n",
       "          ⅈ⋅\\phi - ───        \n",
       "                   2⋅a        \n",
       "-0.125⋅r⋅ℯ            ⋅sin(θ) \n",
       "──────────────────────────────\n",
       "               3              \n",
       "              a               "
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(Hψ)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "これを$\\psi$で割ると実際に定数になっているだろうか? "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{0.125}{a^{2}}$$"
      ],
      "text/plain": [
       "-0.125 \n",
       "───────\n",
       "    2  \n",
       "   a   "
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(Hψ / ψ)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "確かに定数になった"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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