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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Test boson and fermion commutators and anticommutators"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy import *\n",
      "init_printing()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy.physics.quantum import *\n",
      "from sympy.physics.quantum.boson import *\n",
      "from sympy.physics.quantum.fermion import *\n",
      "from sympy.physics.quantum.operatorordering import *"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "These distinguishable single-mode boson/fermion operators, but do not in general need to be independent."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a, b = BosonOp(\"a\"), BosonOp(\"b\")\n",
      "c, d = FermionOp(\"c\"), FermionOp(\"d\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Boson"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Commutator(a, Dagger(a)).doit()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAzRAiu5mrdu/dZmiL4QAAAAAJcEhZcwAA\nDsQAAA7EAZUrDhsAAAAjSURBVAgdY2BgEGJgYDDZxMCgEgYkGNhJJVgzdmYB9TEwAACPpQrvlUCH\ncAAAAABJRU5ErkJggg==\n",
       "prompt_number": 4,
       "text": [
        "1"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "AntiCommutator(a, Dagger(a)).doit()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$${{a}^\\dagger} {a} + {a} {{a}^\\dagger}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFUAAAAVBAMAAADFg78FAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIquJdjLdEETvu2aZ\nVM0GsGrEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABN0lEQVQoFZWRPUvDQByHfybi1ZhI6OIqWMFF\nqKB0DYjoWBc/gAiuh0txalZHV0XpB9DaydnFQRfrJ9BBHBxCQcRJ8F56/97Fy2CH3HP3PHfkUkD/\nVsZj5RDkpP7TtmhXBYTciP7VhUH/GH08DMeGt2ibv2Vhh8QyUQUEbRLXRBUQmVcA3Jalf3ZY7aoj\nk8yZykkwOdf9vr42FxumlrYOgXsiATCtdvX1PY5Y3u0kjUbAGpHdKlc7xQ0Qiiuwc8Q9YJvIarU7\nHuARmOZA/I2ZDJiQ1Wr3xHEgThLrsz1029IThUXxflkUd2blDOxLBeg28SJPhUXmbsp1PpGM5Kmi\nGKKfpC5RK93CD6LXZ9XOZ3P7YYlMq90bbhebqmCNjfpOiUyr3dHmbiNXhfdh/guvLC3WBqUFNf0F\nrsBN1BM5mj4AAAAASUVORK5CYII=\n",
       "prompt_number": 5,
       "text": [
        " \u2020        \u2020\n",
        "a \u22c5a + a\u22c5a "
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "normal_ordered_form(AntiCommutator(a, Dagger(a)).doit())"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1 + 2 {{a}^\\dagger} {a}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAVBAMAAAAeOBUnAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzRAiu5mrdu/dZlSJ\nRDLkM64aAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABHklEQVQoFWNgwAfYLuCThcsRqYxHAa4DD4NF\nJGIDHmmYFBOPEozJwCCEYIJZzKEuD6BCbDAGA4PJJ6gYjFJj4PkLZbPA7VQJgyljMoBIJjEwdGAo\nY2CHKeMugEh2MjDkP4Aw2eCmYSqTV0AouwA1lgFTGVBmvgKD2fGXCgysIFPBLGzKeP8w8DYz+DMw\n8ACdC2FhU8bpwKCxgCEEaJMCA5SFTRlQQbwCw3QGBiagnRAWVBmPoKBko6BgAFCcgT2BgaGFgekf\niA1nYQQIw1YGZqM/DNwfQGYxMEFZGMr4AhjYjX4zsAhsBiuDsjCUWZ45mskgzrAy4QBIGZTFmvEz\nC8xlgMWC/P//PxnUah6nQ8IWwUJVBuHhJHkX4JRiAACNP0xWaf/ChQAAAABJRU5ErkJggg==\n",
       "prompt_number": 6,
       "text": [
        "       \u2020  \n",
        "1 + 2\u22c5a \u22c5a"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In general, different boson operators do not commute, and we do not know anything about the commutation and anticommutation relations:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Commutator(a, b).doit()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$${a} {b} - {b} {a}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEEAAAAOBAMAAACC12zPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIquJdjLdEETvu2aZ\nVM0GsGrEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAxUlEQVQYGWNgAIOqaRAaTmIIMDyCy0EZGAJ/\n0VWgC3B8R1OBIcCcgKYCWYBR2TWdgU/F5wKKGrAAWIqBoVOA6wND/XOGjygqwAJgKQb22Qw8Cxju\nFzB8RlEBEoBIMfB8Z2BzYDjHwA5SUZcGAk+BLJAARIqBcwFDfQDDGwYOVO+BBCBSDPUGDPcZCn6D\nzEIGIAGIFEP9BYZ1vAI/GCo3ICtgAAlApBj4HLiTmBm0GdagKAALQKQY2FVMhTwYapxQjQALQKQA\nvhFEEKnjnJwAAAAASUVORK5CYII=\n",
       "prompt_number": 7,
       "text": [
        "a\u22c5b - b\u22c5a"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "AntiCommutator(a, b).doit()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$${a} {b} + {b} {a}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEEAAAAQBAMAAAC7Cw8kAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIquJdjLdEETvu2aZ\nVM0GsGrEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA6UlEQVQoFWNgAIOqaRAaTmIIMDyCy0EZGAJ/\nYSoKoQy4AJTP8R2m4gqEgRCASjAnQBkMUBUIAQYGRmXXdAY+FZ8LEDVQFWABsBQDQ6cA1weG+ucM\nH1FUgAXAUgzssxl4FjDcL2D4jKICJACRYuD5zsDmwHCOgR1VBUgAIsXAuYChPoDhDQMHyHt2795l\nvnv3BsgCCUCkGOoNGO4zFPwGmQUGUJeCBCBSDPUXGNbxCvxgqNyAogIkAJFi4HPgTmJm0GZYA1EA\nCw+QAESKgV3FVMiDocYJagRMBUgAIgXViaCg7kAIYLC60UUAYUFQEoaexGAAAAAASUVORK5CYII=\n",
       "prompt_number": 8,
       "text": [
        "a\u22c5b + b\u22c5a"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "But often different bosons are independent and do commute. We can use the `independent=True` hint to `doit` method in this situation:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Commutator(a, b).doit(independent=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB\n5wID+0cG/gsMfN8Z5BUY+L4wzDdgYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxy\nlQEA9FISlFfRJtkAAAAASUVORK5CYII=\n",
       "prompt_number": 9,
       "text": [
        "0"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "AntiCommutator(a, b).doit(independent=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$2 {a} {b}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAB0AAAAOBAMAAAAh/woJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAwklEQVQIHWNgVDJ2YAADVssFQDqMgf0HhM/A\nnQBkpDIwzITy+S8AGbMYGPIdIAL9AUB6fwCcXwURZlgfwCBW4xXAsO2MLkiE4zsDx0QGewbW3wH8\nAkA+lwFD5AMGZQberwz8BUC+MgODfgDDGgamDQzxQD5PAgPDZAbWLwzcCgz9FxgYLjEwin5n4P3A\nyv8ApI5FgYFH9C8D84aL/A2s8xgYpMqLshj2MLxKKGB7wKwAdM///78Ywo475jSw6uQxMAAAszgt\nzYyZsNMAAAAASUVORK5CYII=\n",
       "prompt_number": 10,
       "text": [
        "2\u22c5a\u22c5b"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Fermion"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "AntiCommutator(c, Dagger(c)).doit()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAzRAiu5mrdu/dZmiL4QAAAAAJcEhZcwAA\nDsQAAA7EAZUrDhsAAAAjSURBVAgdY2BgEGJgYDDZxMCgEgYkGNhJJVgzdmYB9TEwAACPpQrvlUCH\ncAAAAABJRU5ErkJggg==\n",
       "prompt_number": 11,
       "text": [
        "1"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Commutator(c, Dagger(c)).doit().expand()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- {{c}^\\dagger} {c} + {c} {{c}^\\dagger}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFwAAAAVBAMAAAA5lpTPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMol2IlTvq5m7\nZkT3Gyx5AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABJUlEQVQoFWNgQAZcD5B5GOx1aCIkKmcVQNOP\nyg1E5fLo7GxAFUHhHTo9A4XPyCqIwkfjCASiWc7lgKYClbsElcvA04AmgMo9gcpFUy6OJsuArpyr\nAVlFGzIHxF6BJoAa7hjKkaOJNSyFgdsBqB/EAAOYcrBAaJoDwxaIOIhkt2BQYmAtgDKQlYNlihuY\nFBjCEco5A4BOYwMGLJiBrBwsMJUB6K9EBgYhYxBQYdgPigJGIAYzkJWDBLg/gkRASqBgDhojSEnJ\nUklJGygKkuE8AJOG0IxfoXw4g4EB4lWwgHwAqnKGvwwMHGAhOAOqHCzDf4GBoQFZxy4GRkg4wRkw\n5SAB7gUM0g7Iylljr0G4cAZMOVgg9G4CsmpsbFg0YZPDIvYWUwwA8Io7hSRiaCIAAAAASUVORK5C\nYII=\n",
       "prompt_number": 12,
       "text": [
        "   \u2020        \u2020\n",
        "- c \u22c5c + c\u22c5c "
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In general different fermion operators do not commute (if they are dependent):"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Commutator(c, d).doit()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$${c} {d} - {d} {c}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEEAAAAOBAMAAACC12zPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXYyIt1Uze+rmRC7\nZkTTotXTAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAyElEQVQYGWNggII3XTAWjMYQ+QaTgdNoIny/\n4TJQBroIYwC6CnQR9gXoKhAigkoGDJ5C9gmoKiAiIDkG1wT2Ar4WhvMbUFRAREByDAyNDJwJOy4w\nLEdRwAARAckxcIH8sH4DQyBIxfZQECiGiYDlGDgOAPk9DNz/gBQSAIuA5Rj2CzAwcH9kYPvAjSQP\nFQHJMTDwX2BgyP7GwFOQgqICLAKSA7pjAsMugyKGqxPA6uGqwCJgOQYGwbsKDDsVTSY9gMuCGBAR\nkBwAN28+4jo7MJcAAAAASUVORK5CYII=\n",
       "prompt_number": 13,
       "text": [
        "c\u22c5d - d\u22c5c"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "And we do not know their anticommutator:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "AntiCommutator(c, d).doit()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$${c} {d} + {d} {c}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEEAAAAQBAMAAAC7Cw8kAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXYyIt1Uze+rmRC7\nZkTTotXTAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5UlEQVQoFWWQPw/BQBiHfxZFpf4sVnyCDnaT\nwcZmbMRmaSyX2CQmMTQ+gc2ogwEL30AXo8QHIGHpIhF97/pW2t5y1+d5cr07IByPBa94ThGfTTQn\niPFRZhcFTBhkemo1ZAAmDLR1smACVJomutW2FS8UIYeOpdnGHBc3VihCDpghb+09bFSA8ByKkEOB\n7rB10Y8XkkiH3DlQS+hfKmpCDISYMpEOpzKgv5F96ZQg/Isi5ICSB4x9FO0RfUWFJOSCczg4mhNc\nHdlzAUmkC57jVseh0Vrd5RZ8F0XIpcb/1VMqBM+k+AHB0ESrsAQx6QAAAABJRU5ErkJggg==\n",
       "prompt_number": 14,
       "text": [
        "c\u22c5d + d\u22c5c"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "But if we know that they are independent we can indicate this with the `independent=True` hint: "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Commutator(c, d).doit(independent=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB\n5wID+0cG/gsMfN8Z5BUY+L4wzDdgYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxy\nlQEA9FISlFfRJtkAAAAASUVORK5CYII=\n",
       "prompt_number": 15,
       "text": [
        "0"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "and in this case the anticommutator is:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "AntiCommutator(c, d).doit(independent=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$2 {c} {d}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAB0AAAAOBAMAAAAh/woJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAuUlEQVQIHWNgVDJ2YIACDksFhjAG9h8wPgP/\nBYZUBoaZcH5/AMMsBoZ8B5hAFQPD/gAk/iawxPoABvaaIwyM6Uf+gvgc3xkY1zLsZugUYP4K4nMZ\nMHAWMOSyzmNg2gDiKzMw6AcwMDB9ZeAqAHJ5EhgYJgNpTgOG+AYgfYmBUfQ7kI4vYLjPEMDAosDA\nI/CPgYE3/gKDNa8Ag1R5URaDGgOrCvcBvmXsQPf8//+Lgb3uDQNrTonYCQYAy3spH8FnWFYAAAAA\nSUVORK5CYII=\n",
       "prompt_number": 16,
       "text": [
        "2\u22c5c\u22c5d"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Mix of fermion and boson operators"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Boson and fermion operators commute:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Commutator(a, c).doit()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB\n5wID+0cG/gsMfN8Z5BUY+L4wzDdgYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxy\nlQEA9FISlFfRJtkAAAAASUVORK5CYII=\n",
       "prompt_number": 17,
       "text": [
        "0"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So their anticommutator is this one:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "AntiCommutator(a, c).doit()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$2 {c} {a}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAB0AAAAOBAMAAAAh/woJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAArUlEQVQIHWNgVDJ2YEACYQzsP5C4DKkMDDOR\n+bMYGPIdkAT2B6DygVLrAxjYa44wMKYfWQpSyPGdgXEtw26GTgHmDyA+lwEDZwFDLus8BiYDEF+Z\ngUE/gIGB6SsD1wEglyeBgWEykOY0YIh3ANKXGBhFvwPp+AKG+wwBDCwKDDwC/xgYeOMvMFjzCjBI\nlRdlMagxsKpwH+Bbxs7AsP///18M7HVvGFhzSsROMAAAL3UlB3AAQt0AAAAASUVORK5CYII=\n",
       "prompt_number": 18,
       "text": [
        "2\u22c5c\u22c5a"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Versions"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%reload_ext version_information\n",
      "\n",
      "%version_information sympy"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<table><tr><th>Software</th><th>Version</th></tr><tr><td>Python</td><td>3.4.1 (default, May 21 2014, 01:39:38) [GCC 4.2.1 Compatible Apple LLVM 5.1 (clang-503.0.40)]</td></tr><tr><td>IPython</td><td>2.1.0</td></tr><tr><td>OS</td><td>posix [darwin]</td></tr><tr><td>sympy</td><td>0.7.4.1-git</td></tr><tr><td colspan='2'>Thu Jun 05 15:27:54 2014 JST</td></tr></table>"
       ],
       "json": [
        "{ \"Software versions\" : [{ \"module\" : \"Python\", \"version\" : \"3.4.1 (default, May 21 2014, 01:39:38) [GCC 4.2.1 Compatible Apple LLVM 5.1 (clang-503.0.40)]\" }, { \"module\" : \"IPython\", \"version\" : \"2.1.0\" }, { \"module\" : \"OS\", \"version\" : \"posix [darwin]\" }, { \"module\" : \"sympy\", \"version\" : \"0.7.4.1-git\" } ] }"
       ],
       "latex": [
        "\\begin{tabular}{|l|l|}\\hline\n",
        "{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n",
        "Python & 3.4.1 (default, May 21 2014, 01:39:38) [GCC 4.2.1 Compatible Apple LLVM 5.1 (clang-503.0.40)] \\\\ \\hline\n",
        "IPython & 2.1.0 \\\\ \\hline\n",
        "OS & posix [darwin] \\\\ \\hline\n",
        "sympy & 0.7.4.1-git \\\\ \\hline\n",
        "\\hline \\multicolumn{2}{|l|}{Thu Jun 05 15:27:54 2014 JST} \\\\ \\hline\n",
        "\\end{tabular}\n"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 19,
       "text": [
        "Software versions\n",
        "Python 3.4.1 (default, May 21 2014, 01:39:38) [GCC 4.2.1 Compatible Apple LLVM 5.1 (clang-503.0.40)]\n",
        "IPython 2.1.0\n",
        "OS posix [darwin]\n",
        "sympy 0.7.4.1-git\n",
        "\n",
        "Thu Jun 05 15:27:54 2014 JST"
       ]
      }
     ],
     "prompt_number": 19
    }
   ],
   "metadata": {}
  }
 ]
}