Schirmer Extension

SchirmerExtension.ipynb

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   "source": [
    "import sympy\n",
    "from sympy import Matrix, symbols, I"
   ]
  },
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   "source": [
    "H, μ1, μ2, dt, Ψ = symbols('H, μ_1, μ_2, dt, Psi')"
   ]
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     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- i H dt & 0\\\\- i dt μ_{1} & - i H dt\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[  -I*H*dt,       0],\n",
       "[-I*dt*μ_1, -I*H*dt]])"
      ]
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     "execution_count": 3,
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   "source": [
    "schirmer = -I * Matrix([[H, 0], [μ1, H]]) * dt;\n",
    "schirmer"
   ]
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   "source": [
    "state = Matrix([[Ψ], [0]])"
   ]
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     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}e^{- i H dt} & 0\\\\- i dt μ_{1} e^{- i H dt} & e^{- i H dt}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[          exp(-I*H*dt),            0],\n",
       "[-I*dt*μ_1*exp(-I*H*dt), exp(-I*H*dt)]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
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   ],
   "source": [
    "sympy.exp(schirmer)"
   ]
  },
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    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\Psi e^{- i H dt}\\\\- i \\Psi dt μ_{1} e^{- i H dt}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[          Psi*exp(-I*H*dt)],\n",
       "[-I*Psi*dt*μ_1*exp(-I*H*dt)]])"
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   ],
   "source": [
    "sympy.exp(schirmer) * state"
   ]
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- i H dt & 0 & 0\\\\- i dt μ_{1} & - i H dt & 0\\\\- i dt μ_{2} & 0 & - i H dt\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[  -I*H*dt,       0,       0],\n",
       "[-I*dt*μ_1, -I*H*dt,       0],\n",
       "[-I*dt*μ_2,       0, -I*H*dt]])"
      ]
     },
     "execution_count": 7,
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   ],
   "source": [
    "schirmer2 =  -I * Matrix([[H, 0, 0], [μ1, H, 0], [μ2, 0, H]]) * dt;\n",
    "schirmer2"
   ]
  },
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   "source": [
    "state2 = Matrix([[Ψ], [0], [0]])"
   ]
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     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}e^{- i H dt} & 0 & 0\\\\- i dt μ_{1} e^{- i H dt} & e^{- i H dt} & 0\\\\- i dt μ_{2} e^{- i H dt} & 0 & e^{- i H dt}\\end{matrix}\\right]$"
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      "text/plain": [
       "Matrix([\n",
       "[          exp(-I*H*dt),            0,            0],\n",
       "[-I*dt*μ_1*exp(-I*H*dt), exp(-I*H*dt),            0],\n",
       "[-I*dt*μ_2*exp(-I*H*dt),            0, exp(-I*H*dt)]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
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   ],
   "source": [
    "sympy.exp(schirmer2)"
   ]
  },
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     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\Psi e^{- i H dt}\\\\- i \\Psi dt μ_{1} e^{- i H dt}\\\\- i \\Psi dt μ_{2} e^{- i H dt}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[          Psi*exp(-I*H*dt)],\n",
       "[-I*Psi*dt*μ_1*exp(-I*H*dt)],\n",
       "[-I*Psi*dt*μ_2*exp(-I*H*dt)]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
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   ],
   "source": [
    "sympy.exp(schirmer2) * state2"
   ]
  }
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