Verifying Potassium Paper Equations.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## verifying equations in potassium paper"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### e states"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### first column"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True 0.3651483716701107 0.3651483716701107\n",
      "True -0.4472135954999579 -0.44721359549995787\n",
      "True 0.5163977794943222 0.5163977794943222\n",
      "True -0.6324555320336759 -0.6324555320336758\n",
      "True 0.0 0\n",
      "True 0.0 0\n"
     ]
    }
   ],
   "source": [
    "J=2\n",
    "ell = J-1\n",
    "s_a = s_b = j_a = j_b = 1/2\n",
    "j = L = 1\n",
    "S = Lambda = Sigma = 0\n",
    "Omega = Lambda+Sigma\n",
    "#p = (-1)**J\n",
    "p = 0 if (-1)**J>0 else 1\n",
    "\n",
    "N = 1/np.sqrt(6*(2*J+1))\n",
    "\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "predictVal = N*np.sqrt(2*J)\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J+1\n",
    "predictVal = -N*np.sqrt(2*(J+1))\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "j_b = 3/2\n",
    "ell = J-1\n",
    "predictVal = 2*N*np.sqrt(J)\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J+1\n",
    "predictVal = -2*N*np.sqrt(J+1)\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "j = 2\n",
    "ell = J-1\n",
    "predictVal = 0\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J+1\n",
    "predictVal = 0\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### second column"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True 0.5443310539518174 0.5443310539518174\n",
      "True -0.6085806194501846 -0.6085806194501846\n",
      "True -0.38490017945975047 -0.3849001794597505\n",
      "True 0.43033148291193524 0.43033148291193524\n",
      "True 0.0 0\n",
      "True 0.0 0\n"
     ]
    }
   ],
   "source": [
    "J=4\n",
    "ell = J-1\n",
    "s_a = s_b = j_a = j_b = 1/2\n",
    "j = L = 1\n",
    "Sigma = 0\n",
    "S = Lambda = 1\n",
    "\n",
    "Omega = 0\n",
    "Sigma = Omega - Lambda\n",
    "#Omega = Lambda+Sigma\n",
    "p = 0 if (-1)**J>0 else 1\n",
    "\n",
    "N = 1/np.sqrt(6*(2*J+1))\n",
    "\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "predictVal = 2*N*np.sqrt(J)\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J+1\n",
    "predictVal = -2*N*np.sqrt(J+1)\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "j_b = 3/2\n",
    "ell = J-1\n",
    "predictVal = -N*np.sqrt(2*J)\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J+1\n",
    "predictVal = N*np.sqrt(2*(J+1))\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "j = 2\n",
    "ell = J-1\n",
    "predictVal = 0\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J+1\n",
    "predictVal = 0\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### third Column"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True 0.4236592728681617 0.42365927286816163\n",
      "True 0.3922322702763681 0.392232270276368\n",
      "True 0.5991446895152781 0.599144689515278\n",
      "True 0.5547001962252291 0.554700196225229\n",
      "True 0.0 0\n",
      "True 0.0 0\n"
     ]
    }
   ],
   "source": [
    "J=6\n",
    "ell = J-1\n",
    "s_a = s_b = j_a = j_b = 1/2\n",
    "j = L = 1\n",
    "\n",
    "S = 0\n",
    "#Sigma = 0\n",
    "Lambda = 1\n",
    "\n",
    "Omega = 1\n",
    "Sigma = Omega - Lambda\n",
    "#Omega = Lambda+Sigma\n",
    "p = 0 if (-1)**J>0 else 1\n",
    "\n",
    "N = 1/np.sqrt(6*(2*J+1))\n",
    "\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "predictVal = N*np.sqrt(2*(J+1))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J+1\n",
    "predictVal = N*np.sqrt(2*J)\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "j_b = 3/2\n",
    "ell = J-1\n",
    "predictVal = N*2*np.sqrt(J+1)\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J+1\n",
    "predictVal = N*2*np.sqrt(J)\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "j = 2\n",
    "ell = J-1\n",
    "predictVal = 0\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J+1\n",
    "predictVal = 0\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### fourth column"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True -0.4236592728681617 -0.42365927286816163\n",
      "True -0.3922322702763681 -0.392232270276368\n",
      "True 0.29957234475763905 0.299572344757639\n",
      "True 0.2773500981126146 0.2773500981126145\n",
      "True -0.43852900965351466 -0.4385290096535146\n",
      "True 0.5547001962252291 0.554700196225229\n"
     ]
    }
   ],
   "source": [
    "J=6\n",
    "ell = J-1\n",
    "s_a = s_b = j_a = j_b = 1/2\n",
    "j = L = 1\n",
    "\n",
    "S = 1\n",
    "Lambda = 0\n",
    "\n",
    "Omega = 1\n",
    "Sigma = Omega - Lambda\n",
    "#Omega = Lambda+Sigma\n",
    "p = 0 if (-1)**J>0 else 1\n",
    "\n",
    "N = 1/np.sqrt(6*(2*J+1))\n",
    "\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "predictVal = -N*np.sqrt(2*(J+1))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J+1\n",
    "predictVal = -N*np.sqrt(2*J)\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "j_b = 3/2\n",
    "ell = J-1\n",
    "predictVal = N*np.sqrt(J+1)\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J+1\n",
    "predictVal = N*np.sqrt(J)\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "j = 2\n",
    "ell = J-1\n",
    "predictVal = -N*np.sqrt(3*(J-1))\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J+1\n",
    "predictVal = N*np.sqrt(3*(J+2))\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### fifth column"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True 0.4216370213557839 0.4216370213557839\n",
      "True 0.39440531887330776 0.3944053188733077\n",
      "True -0.29814239699997197 -0.29814239699997197\n",
      "True -0.27888667551135854 -0.27888667551135854\n",
      "True -0.447213595499958 -0.44721359549995787\n",
      "True 0.5477225575051661 0.5477225575051661\n"
     ]
    }
   ],
   "source": [
    "J=7\n",
    "ell = J-1\n",
    "s_a = s_b = j_a = j_b = 1/2\n",
    "j = L = 1\n",
    "\n",
    "S = 1\n",
    "Lambda = 1\n",
    "\n",
    "Omega = 1\n",
    "Sigma = Omega - Lambda\n",
    "#Omega = Lambda+Sigma\n",
    "p = 0 if (-1)**J>0 else 1\n",
    "\n",
    "\n",
    "N = 1/np.sqrt(6*(2*J+1))\n",
    "\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "predictVal = N*np.sqrt(2*(J+1))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J+1\n",
    "predictVal = N*np.sqrt(2*J)\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "j_b = 3/2\n",
    "ell = J-1\n",
    "predictVal = -N*np.sqrt(J+1)\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J+1\n",
    "predictVal = -N*np.sqrt(J)\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "j = 2\n",
    "ell = J-1\n",
    "predictVal = -N*np.sqrt(3*(J-1))\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J+1\n",
    "predictVal = N*np.sqrt(3*(J+2))\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### sixth column"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True 0.0 0\n",
      "True 0.0 0\n",
      "True 0.0 0\n",
      "True 0.0 0\n",
      "True -0.7745966692414834 -0.7745966692414834\n",
      "True -0.6324555320336759 -0.6324555320336759\n"
     ]
    }
   ],
   "source": [
    "J=7\n",
    "ell = J-1\n",
    "s_a = s_b = j_a = j_b = 1/2\n",
    "j = L = 1\n",
    "\n",
    "S = 1\n",
    "Lambda = 1\n",
    "Omega = 2\n",
    "\n",
    "Sigma = Omega - Lambda\n",
    "p = 0 if (-1)**J>0 else 1\n",
    "\n",
    "N = 1/np.sqrt(6*(2*J+1))\n",
    "\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "predictVal = 0\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J+1\n",
    "predictVal = 0\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "j_b = 3/2\n",
    "ell = J-1\n",
    "predictVal = 0\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J+1\n",
    "predictVal = 0\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "j = 2\n",
    "ell = J-1\n",
    "predictVal = -N*np.sqrt(6*(J+2))\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J+1\n",
    "predictVal = -N*np.sqrt(6*(J-1))\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### f states"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### total"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 739,
   "metadata": {},
   "outputs": [],
   "source": [
    "fS_ae = np.zeros((6,6))\n",
    "J=21\n",
    "ell = J\n",
    "s_a = s_b = j_a = j_b = 1/2\n",
    "L = 1\n",
    "j = 0\n",
    "\n",
    "S = 1\n",
    "Lambda = 0\n",
    "Omega = 0\n",
    "\n",
    "Sigma = Omega - Lambda\n",
    "C = (2*J+3)*(2*J-1)\n",
    "D = (2*J+3)*(2*J+1)\n",
    "p = 0 if (-1)**(J+1)>0 else 1\n",
    "#p = -1\n",
    "fS_ae[0,0] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "j = 1\n",
    "fS_ae[1,0] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "j_b = 3/2\n",
    "fS_ae[2,0] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "ell = J-2\n",
    "j = 2\n",
    "fS_ae[3,0] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "ell = J\n",
    "fS_ae[4,0] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "ell = J+2\n",
    "fS_ae[5,0] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "\n",
    "ell = J\n",
    "s_a = s_b = j_a = j_b = 1/2\n",
    "L = 1\n",
    "j = 0\n",
    "\n",
    "S = 1\n",
    "Lambda = 1\n",
    "Omega = 0\n",
    "\n",
    "Sigma = Omega - Lambda\n",
    "p = 0 if (-1)**(J+1)>0 else 1\n",
    "#p = -1\n",
    "fS_ae[0,1] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "j = 1\n",
    "fS_ae[1,1] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "j_b = 3/2\n",
    "fS_ae[2,1] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "ell = J-2\n",
    "j = 2\n",
    "fS_ae[3,1] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "ell = J\n",
    "fS_ae[4,1] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "ell = J+2\n",
    "fS_ae[5,1] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "\n",
    "ell = J\n",
    "s_a = s_b = j_a = j_b = 1/2\n",
    "L = 1\n",
    "j = 0\n",
    "\n",
    "S = 0\n",
    "Lambda = 1\n",
    "Omega = 1\n",
    "\n",
    "Sigma = Omega - Lambda\n",
    "p = 0 if (-1)**(J+1)>0 else 1\n",
    "#p = -1\n",
    "fS_ae[0,2] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "j = 1\n",
    "fS_ae[1,2] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "j_b = 3/2\n",
    "fS_ae[2,2] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "ell = J-2\n",
    "j = 2\n",
    "fS_ae[3,2] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "ell = J\n",
    "fS_ae[4,2] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "ell = J+2\n",
    "fS_ae[5,2] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "\n",
    "ell = J\n",
    "s_a = s_b = j_a = j_b = 1/2\n",
    "L = 1\n",
    "j = 0\n",
    "\n",
    "S = 1\n",
    "Lambda = 0\n",
    "Omega = 1\n",
    "\n",
    "Sigma = Omega - Lambda\n",
    "p = 0 if (-1)**(J+1)>0 else 1\n",
    "#p = -1\n",
    "fS_ae[0,3] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "j = 1\n",
    "fS_ae[1,3] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "j_b = 3/2\n",
    "fS_ae[2,3] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "ell = J-2\n",
    "j = 2\n",
    "fS_ae[3,3] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "ell = J\n",
    "fS_ae[4,3] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "ell = J+2\n",
    "fS_ae[5,3] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "\n",
    "ell = J\n",
    "s_a = s_b = j_a = j_b = 1/2\n",
    "L = 1\n",
    "j = 0\n",
    "\n",
    "S = 1\n",
    "Lambda = 1\n",
    "Omega = 1\n",
    "Sigma = Omega - Lambda\n",
    "p = 0 if (-1)**(J+1)>0 else 1\n",
    "#p = -1\n",
    "fS_ae[0,4] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "j = 1\n",
    "fS_ae[1,4] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "j_b = 3/2\n",
    "fS_ae[2,4] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "ell = J-2\n",
    "j = 2\n",
    "fS_ae[3,4] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "ell = J\n",
    "fS_ae[4,4] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "ell = J+2\n",
    "fS_ae[5,4] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "\n",
    "ell = J\n",
    "s_a = s_b = j_a = j_b = 1/2\n",
    "L = 1\n",
    "j = 0\n",
    "\n",
    "S = 1\n",
    "Lambda = 1\n",
    "Omega = 2\n",
    "\n",
    "Sigma = Omega - Lambda\n",
    "p = 1 if (-1)**J>0 else 0\n",
    "#p = -1\n",
    "fS_ae[0,5] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "j = 1\n",
    "fS_ae[1,5] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "j_b = 3/2\n",
    "fS_ae[2,5] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "ell = J-2\n",
    "j = 2\n",
    "fS_ae[3,5] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "ell = J\n",
    "fS_ae[4,5] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "ell = J+2\n",
    "fS_ae[5,5] = float(coef_fs(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 740,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.577, -0.816,  0.   ,  0.   ,  0.   ,  0.   ],\n",
       "       [ 0.   ,  0.   , -0.577,  0.577, -0.577,  0.   ],\n",
       "       [ 0.   ,  0.   , -0.816, -0.408,  0.408,  0.   ],\n",
       "       [ 0.488,  0.345,  0.   ,  0.5  ,  0.5  ,  0.379],\n",
       "       [-0.409, -0.289,  0.   , -0.029, -0.029,  0.865],\n",
       "       [ 0.511,  0.362,  0.   , -0.5  , -0.5  ,  0.329]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "array([[ 1., -0.,  0., -0., -0.,  0.],\n",
       "       [-0.,  1.,  0., -0., -0., -0.],\n",
       "       [ 0.,  0.,  1.,  0., -0.,  0.],\n",
       "       [-0., -0.,  0.,  1., -0., -0.],\n",
       "       [-0., -0., -0., -0.,  1., -0.],\n",
       "       [ 0., -0.,  0., -0., -0.,  1.]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(fS_ae)\n",
    "display(fS_ae.T@fS_ae)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### First Column"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True 0.5773502691896257 0.5773502691896257\n",
      "True 0.0 0\n",
      "True 0.0 0\n",
      "True 0.4880884031398789 0.48808840313987895\n",
      "True -0.408580064830253 -0.40858006483025305\n",
      "True 0.511369443758948 0.511369443758948\n"
     ]
    }
   ],
   "source": [
    "J=21\n",
    "ell = J\n",
    "s_a = s_b = j_a = j_b = 1/2\n",
    "L = 1\n",
    "j = 0\n",
    "\n",
    "S = 1\n",
    "Lambda = 0\n",
    "Omega = 0\n",
    "\n",
    "Sigma = Omega - Lambda\n",
    "C = (2*J+3)*(2*J-1)\n",
    "D = (2*J+3)*(2*J+1)\n",
    "p = 0 if (-1)**(J+1)>0 else 1\n",
    "#p = -1\n",
    "\n",
    "N = 1/np.sqrt(6*(2*J+1))\n",
    "\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "predictVal = np.sqrt(1/3)\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "j = 1\n",
    "predictVal = 0\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "j_b = 3/2\n",
    "predictVal = 0\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J-2\n",
    "j = 2\n",
    "predictVal = np.sqrt(J*(J-1)/(4*J**2-1))\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J\n",
    "predictVal = -np.sqrt((2*J*(J+1))/(3*C))\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J+2\n",
    "predictVal = np.sqrt(((J+2)*(J+1))/D)\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### second column"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True -0.816496580927726 -0.816496580927726\n",
      "True 0.0 0\n",
      "True 0.0 0\n",
      "True 0.34723790501531626 0.3472379050153163\n",
      "True -0.28880854373754017 -0.2888085437375402\n",
      "True 0.3596509359338702 0.35965093593387026\n"
     ]
    }
   ],
   "source": [
    "J=28\n",
    "ell = J\n",
    "s_a = s_b = j_a = j_b = 1/2\n",
    "L = 1\n",
    "j = 0\n",
    "\n",
    "S = 1\n",
    "Lambda = 1\n",
    "Omega = 0\n",
    "\n",
    "Sigma = Omega - Lambda\n",
    "C = (2*J+3)*(2*J-1)\n",
    "D = (2*J+3)*(2*J+1)\n",
    "p = 0 if (-1)**(J+1)>0 else 1\n",
    "#p = -1\n",
    "\n",
    "N = 1/np.sqrt(6*(2*J+1))\n",
    "\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "predictVal = -np.sqrt(2/3)\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "j = 1\n",
    "predictVal = 0\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "j_b = 3/2\n",
    "predictVal = 0\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J-2\n",
    "j = 2\n",
    "predictVal = np.sqrt(J*(J-1)/(2*(4*J**2-1)))\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J\n",
    "predictVal = -np.sqrt((J*(J+1))/(3*C))\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J+2\n",
    "predictVal = np.sqrt(((J+2)*(J+1))/(2*D))\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 3rd column"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True 0.0 0\n",
      "False -0.5773502691896257 0.5773502691896258\n",
      "False -0.816496580927726 0.816496580927726\n",
      "True 0.0 0\n",
      "True 0.0 0\n",
      "True 0.0 0\n"
     ]
    }
   ],
   "source": [
    "J=4\n",
    "ell = J\n",
    "s_a = s_b = j_a = j_b = 1/2\n",
    "L = 1\n",
    "j = 0\n",
    "\n",
    "S = 0\n",
    "Lambda = 1\n",
    "Omega = 1\n",
    "\n",
    "Sigma = Omega - Lambda\n",
    "C = (2*J+3)*(2*J-1)\n",
    "D = (2*J+3)*(2*J+1)\n",
    "p = 0 if (-1)**(J+1)>0 else 1\n",
    "#p = -1\n",
    "\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "predictVal = 0\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "j = 1\n",
    "predictVal = (-1)**J/np.sqrt(3)\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "j_b = 3/2\n",
    "predictVal = (-1)**J*np.sqrt(2/3)\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J-2\n",
    "j = 2\n",
    "predictVal = 0\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J\n",
    "predictVal = 0\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)\n",
    "\n",
    "ell = J+2\n",
    "predictVal = 0\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal), coefVal,predictVal)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 4th"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True 0.0 0\n",
      "False -0.5773502691896257 0.5773502691896258\n",
      "True -0.408248290463863 -0.408248290463863\n",
      "False 0.47809144373375745 0.4140393356054125\n",
      "True -0.18257418583505536 -0.18257418583505536\n",
      "True -0.4879500364742666 -0.4879500364742666\n"
     ]
    }
   ],
   "source": [
    "J=3\n",
    "ell = J\n",
    "s_a = s_b = j_a = j_b = 1/2\n",
    "L = 1\n",
    "j = 0\n",
    "\n",
    "S = 1\n",
    "Lambda = 0\n",
    "Omega = 1\n",
    "\n",
    "Sigma = Omega - Lambda\n",
    "C = (2*J+3)*(2*J-1)\n",
    "D = (2*J+3)*(2*J+1)\n",
    "p = 0 if (-1)**(J+1)>0 else 1\n",
    "#p = -1\n",
    "\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "predictVal = 0\n",
    "print(math.isclose(coefVal,predictVal, rel_tol=1e-7), coefVal,predictVal)\n",
    "\n",
    "j = 1\n",
    "predictVal = -(-1)**J/np.sqrt(3)\n",
    "coefVal = -float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal, rel_tol=1e-7), coefVal,predictVal)\n",
    "\n",
    "j_b = 3/2\n",
    "predictVal = (-1)**J*np.sqrt(1/6)\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal, rel_tol=1e-7), coefVal,predictVal)\n",
    "\n",
    "ell = J-2\n",
    "j = 2\n",
    "predictVal = (J*(J-1)/(4*J**2-1))**(1/2)\n",
    "#predictVal = np.sqrt(J*(J-1)/((2*J-1)*(2*J+1)))\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal, rel_tol=1e-7), coefVal,predictVal)\n",
    "\n",
    "ell = J\n",
    "predictVal = -np.sqrt(3/(2*C))\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal, rel_tol=1e-7), coefVal,predictVal)\n",
    "\n",
    "ell = J+2\n",
    "predictVal = -np.sqrt(J*(J+2)/D)\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "\n",
    "print(math.isclose(coefVal,predictVal, rel_tol=1e-7), coefVal,predictVal)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True 0.0 0\n",
      "True 0.5773502691896257 0.5773502691896258\n",
      "False 0.408248290463863 -0.408248290463863\n",
      "False 0.447213595499958 0.3651483716701107\n",
      "True -0.2672612419124244 -0.2672612419124244\n",
      "True -0.47809144373375745 -0.47809144373375745\n"
     ]
    }
   ],
   "source": [
    "J=2\n",
    "ell = J\n",
    "s_a = s_b = j_a = j_b = 1/2\n",
    "L = 1\n",
    "j = 0\n",
    "\n",
    "S = 1\n",
    "Lambda = 1\n",
    "Omega = 1\n",
    "\n",
    "Sigma = Omega - Lambda\n",
    "C = (2*J+3)*(2*J-1)\n",
    "D = (2*J+3)*(2*J+1)\n",
    "p = 0 if (-1)**(J+1)>0 else 1\n",
    "#p = -1\n",
    "\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "predictVal = 0\n",
    "print(math.isclose(coefVal,predictVal, rel_tol=1e-7), coefVal,predictVal)\n",
    "\n",
    "j = 1\n",
    "predictVal = (-1)**J/np.sqrt(3)\n",
    "coefVal = -float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal, rel_tol=1e-7), coefVal,predictVal)\n",
    "\n",
    "j_b = 3/2\n",
    "predictVal = -(-1)**J*np.sqrt(1/6)\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal, rel_tol=1e-7), coefVal,predictVal)\n",
    "\n",
    "ell = J-2\n",
    "j = 2\n",
    "predictVal = np.sqrt(J*(J-1)/(4*J**2-1))\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal, rel_tol=1e-7), coefVal,predictVal)\n",
    "\n",
    "ell = J\n",
    "predictVal = -np.sqrt(3/(2*C))\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal, rel_tol=1e-7), coefVal,predictVal)\n",
    "\n",
    "ell = J+2\n",
    "predictVal = -np.sqrt(J*(J+2)/D)\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "\n",
    "print(math.isclose(coefVal,predictVal, rel_tol=1e-7), coefVal,predictVal)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True 0.0 0\n",
      "True -0.0 0\n",
      "True 0.0 0\n",
      "True 0.36915463698711004 0.36915463698711004\n",
      "True 0.8655701520031057 0.8655701520031059\n",
      "True 0.33839794023048536 0.33839794023048536\n"
     ]
    }
   ],
   "source": [
    "J=34\n",
    "ell = J\n",
    "s_a = s_b = j_a = j_b = 1/2\n",
    "L = 1\n",
    "j = 0\n",
    "\n",
    "S = 1\n",
    "Lambda = 1\n",
    "Omega = 2\n",
    "\n",
    "Sigma = Omega - Lambda\n",
    "C = (2*J+3)*(2*J-1)\n",
    "D = (2*J+3)*(2*J+1)\n",
    "p = 1 if (-1)**J>0 else 0\n",
    "#p = -1\n",
    "\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "predictVal = 0\n",
    "print(math.isclose(coefVal,predictVal, rel_tol=1e-7), coefVal,predictVal)\n",
    "\n",
    "j = 1\n",
    "predictVal = 0\n",
    "coefVal = -float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal, rel_tol=1e-7), coefVal,predictVal)\n",
    "\n",
    "j_b = 3/2\n",
    "predictVal = 0\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal, rel_tol=1e-7), coefVal,predictVal)\n",
    "\n",
    "ell = J-2\n",
    "j = 2\n",
    "predictVal = np.sqrt((J+1)*(J+2)/(2*(4*J**2-1)))\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal, rel_tol=1e-7), coefVal,predictVal)\n",
    "\n",
    "ell = J\n",
    "predictVal = np.sqrt((3*(J-1)*(J+2)/C))\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "print(math.isclose(coefVal,predictVal, rel_tol=1e-7), coefVal,predictVal)\n",
    "\n",
    "ell = J+2\n",
    "predictVal = np.sqrt(J*(J-1)/(2*D))\n",
    "coefVal = float(coef(j_a,j_b,j,ell,L,S,s_a,s_b,Lambda, Sigma, Omega,p,J))\n",
    "\n",
    "print(math.isclose(coefVal,predictVal, rel_tol=1e-7), coefVal,predictVal)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Hyperfine Structure"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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