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    "<center><h2> Quarks </h2></center>\n",
    "\n",
    "\"\"\" From \"COMPUTATIONHL PHYSICS\" & \"COMPUTER PROBLEMS in PHYSICS\" by RH Landau, MJ Paez, and CC Bordeianu (deceased). Copyright R Landau, Oregon State Unv, MJ Paez, Univ Antioquia, C Bordeianu, Univ Bucharest, 2020. Please respect copyright & acknowledge our work.\"\"\"\n",
    "\n",
    "Verification of the raising and lowering operations on the 8 SU(3) generators representing quarks using liniear algebra routines.\n",
    "$I_+$ raises  d to u, $V_+$ raises s to u, $U_+$ raises s to d.\n",
    "\n",
    "$$\n",
    "\\lambda_1=\n",
    "\\begin{pmatrix}\n",
    "0 & 1 &0 \\\\[2pt]\n",
    " 1& 0 & 0\\\\[2pt]\n",
    " 0& 0 & 0 \\end{pmatrix}, \\ \\  \n",
    " \\lambda_2=\\begin{pmatrix}\n",
    " 0& -i &0 \\\\[2pt]\n",
    " i&  0& 0\\\\[2pt]\n",
    " 0&  0& 0 \\end{pmatrix},\n",
    "\\ \\ \\lambda_3=\\begin{pmatrix}\n",
    "1 &  0& 0 \\\\[2pt]\n",
    " 0&  -1& 0\\\\[2pt]\n",
    " 0&  0&  0\\end{pmatrix},\t\\ \\  \n",
    " \\lambda_4=\\begin{pmatrix}\n",
    "0 &0  &1 \\\\[2pt]\n",
    " 0& 0 &0 \\\\[2pt]\n",
    " 1&  0&0  \\end{pmatrix},\n",
    "\\ \\ \\lambda_5=\\begin{pmatrix}\n",
    "0 &0  & -i\\\\[2pt]\n",
    "0 & 0 & 0\\\\[2pt]\n",
    "i & 0 & 0 \\end{pmatrix},\n",
    "\\ \\ \\lambda_6=\\begin{pmatrix}\n",
    " 0&0  &0 \\\\[2pt]\n",
    " 0&0  &1 \\\\[2pt]\n",
    " 0&1  &0  \\end{pmatrix},\n",
    "\\lambda_7=\\begin{pmatrix}\n",
    " 0&0  &0\\\\[2pt]\n",
    " 0&0  &-i \\\\[2pt]\n",
    " 0&1  &0  \\end{pmatrix},\n",
    "\\ \\ \\lambda_8=\\frac{1}{\\sqrt{3}}\\begin{pmatrix}\n",
    " 1&0  &0 \\\\[2pt]\n",
    " 0&1  &0 \\\\[2pt]\n",
    " 0&0  &-2  \\end{pmatrix}. \n",
    "$$ "
   ]
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     "name": "stdout",
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     "text": [
      "Ipxd [1.+0.j 0.+0.j 0.+0.j]\n",
      "Vpxs [1.+0.j 0.+0.j 0.+0.j]\n",
      "Upxs  [0.+0.j 1.+0.j 0.+0.j]\n"
     ]
    }
   ],
   "source": [
    "from numpy import *\n",
    "from numpy.linalg import *\n",
    "\n",
    "L1 = array([[0,1,0],[1,0,0],[0,0,0]])           # eight generators\n",
    "L2 = array([[0,-1j,0],[1j,0,0],[0,0,0]])\n",
    "L3 = array([[1,0,0],[0,-1,0],[0,0,0]])\n",
    "L4 = array([[0,0,1],[0,0,0],[1,0,0]])\n",
    "L5 = array([[0,0,-1j],[0,0,0],[1j,0,0]])\n",
    "L6 = array([[0,0,0],[0,0,1],[0,1,0]])\n",
    "L7 = array([[0,0,0],[0,0,-1j],[0,1j,0]])\n",
    "L8 = array([[1,0,0],[0,1,0],[0,0,-2]])*1/sqrt(3)\n",
    "u = array([1,0,0])   # up quark\n",
    "d = array([0,1,0])   # down quark\n",
    "s = array([0,0,1])   # strange quark\n",
    "Ip = 0.5*(L1+1j*L2)  # raising operators   \n",
    "Up = 0.5*(L6+1j*L7)\n",
    "Vp = 0.5*(L4+1j*L5)\n",
    "Im = 0.5*(L1-1j*L2)  # lowering operators   \n",
    "Um = 0.5*(L6-1j*L7)\n",
    "Vm = 0.5*(L4-1j*L5)\n",
    "Ipxd = dot(Ip,d)    # raices d to u\n",
    "print(\"Ipxd\",Ipxd)\n",
    "Vpxs = dot(Vp,s)    # raises s to u\n",
    "print(\"Vpxs\",Vpxs)\n",
    "Upxs = dot(Up,s)    #  raises s to d\n",
    "print(\"Upxs \",Upxs) "
   ]
  }
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