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jrjohansson/sympy-quantum-fieldoperator-tests.ipynb

Last active December 31, 2015 16:59
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Notebook that demonstrates the usage of BosonOperator and FermionOperator from the quantum_field_operators branch in https://github.com/jrjohansson/sympy
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sympy-quantum-fieldoperator-tests.ipynb
{
"metadata": {
"name": ""
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Examples of using BosonOperator and FermionOperator"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"J. R. Johansson, http://jrjohansson.github.com, <robert@riken.jp>"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Note: This notebook requires and experimental branch of sympy available at: https://github.com/jrjohansson/sympy"
]
},
{
"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 *"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 2
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Docstrings"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#help(BosonOperator)"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 3
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#help(FermionOperator)"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 4
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"help(normal_order)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Help on function normal_order in module sympy.physics.quantum.fieldoperator:\n",
"\n",
"normal_order(expr, recursive_limit=10, _recursive_depth=0)\n",
" Normal order an expression with bosonic or fermionic operators.\n",
" \n",
" Parameters\n",
" ==========\n",
" \n",
" expr : expression\n",
" The expression to normal order.\n",
" \n",
" recursive_limit : int (default 10)\n",
" The number of allowed recursive applications of the function.\n",
" \n",
" Examples\n",
" ========\n",
" \n",
" >>> from sympy.physics.quantum import normal_order, BosonOperator, Dagger\n",
" >>> a = BosonOperator(\"a\")\n",
" >>> normal_order(a * Dagger(a))\n",
" Dagger(a)*a\n",
"\n"
]
}
],
"prompt_number": 5
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"help(normal_ordered_form)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Help on function normal_ordered_form in module sympy.physics.quantum.fieldoperator:\n",
"\n",
"normal_ordered_form(expr, independent=False, recursive_limit=10, _recursive_depth=0)\n",
" Write an expression with bosonic or fermionic operators on normal\n",
" ordered form, where each term is normally ordered.\n",
" \n",
" Parameters\n",
" ==========\n",
" \n",
" expr : expression\n",
" The expression write on normal ordered form.\n",
" \n",
" recursive_limit : int (default 10)\n",
" The number of allowed recursive applications of the function.\n",
" \n",
" Examples\n",
" ========\n",
" \n",
" >>> from sympy.physics.quantum import normal_ordered_form\n",
" >>> from sympy.physics.quantum import BosonOperator, Dagger\n",
" >>> a = BosonOperator(\"a\")\n",
" >>> normal_ordered_form(a * Dagger(a))\n",
" 1 + Dagger(a)*a\n",
"\n"
]
}
],
"prompt_number": 6
},
{
"cell_type": "heading",
"level": 3,
"metadata": {},
"source": [
"Test Boson operators"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"a = BosonOperator('a')\n",
"b = BosonOperator('b')"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 7
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"a.is_annihilation, Dagger(a).is_annihilation"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 8,
"text": [
"(True, False)"
]
}
],
"prompt_number": 8
},
{
"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": 9,
"text": [
"1"
]
}
],
"prompt_number": 9
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"Commutator(Dagger(a), a).doit()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$-1$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAPBAMAAAAMihLoAAAAJ1BMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAilU6eAAAADHRSTlMAEM3dMiK7mat272a9noTYAAAACXBI\nWXMAAA7EAAAOxAGVKw4bAAAALklEQVQIHWNggANFOIuBIewggiNWgcRh4CSTI2QMAioMZBsAcxB7\n1+FuGBtMAwDSnw2AS5zPhQAAAABJRU5ErkJggg==\n",
"prompt_number": 10,
"text": [
"-1"
]
}
],
"prompt_number": 10
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"Commutator(a, a).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": 11,
"text": [
"0"
]
}
],
"prompt_number": 11
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"Commutator(Dagger(a), Dagger(a)).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": 12,
"text": [
"0"
]
}
],
"prompt_number": 12
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"Dagger(Dagger(Dagger(a)))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{a}^\\dagger}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAABIAAAATBAMAAABvvEDBAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIquJdjLdEETvu2aZ\nVM0GsGrEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAdklEQVQIHWNggAANKM3AgGCZw8TWrZkLZRaY\nF8AEVWEMhrVYWLpwMZB5jMqu6QwMR4GsTgGuDwwMhgwM7LMZeBYwMLgzMPB8Z2BzYGAAGsy5gKE+\nAKiIgaHegOE+SATIusCwjlcAxOJz4E5iBjEY2FVMhTxADACdGxMZ3sSvXwAAAABJRU5ErkJggg==\n",
"prompt_number": 13,
"text": [
"BosonOperator(a,0)"
]
}
],
"prompt_number": 13
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"Commutator(a, Dagger(Dagger(a))) #.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": 14,
"text": [
"0"
]
}
],
"prompt_number": 14
},
{
"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": 15,
"text": [
"BosonOperator(a,1)\u22c5BosonOperator(b,1) - BosonOperator(b,1)\u22c5BosonOperator(a,1)"
]
}
],
"prompt_number": 15
},
{
"cell_type": "heading",
"level": 3,
"metadata": {},
"source": [
"Test Fermion operators"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"c = FermionOperator('c')\n",
"d = FermionOperator('d')"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 16
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"Commutator(c,Dagger(c))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$- \\left[{{c}^\\dagger},{c}\\right]$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAEQAAAAaBAMAAAD8uecGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMrsiiXZU76uZ\nZkSnMwLyAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5UlEQVQoFWNgwAL2YRGDCLEam0EYSEr4jM2R\n1TPDOIUwBojORubAlATFrEQSxqpEoFCAkBKGLUgqsFvEEEkVJTsJm4IULtg8Xd7hwHAEZgqIg+Hp\n5AlMCgxlUCVgDoaS5QzcExgaoUrAnGwGIWMQUAEJAkOX5yOIAQ05CAfdFK4AkBIogHDQlcgXwOSB\nNISDroT/AgPDBJgqCAddCc8GBhkHkBJ5BaDDwBx0JQzldxtAKhiYQe4GczCUgOVBxDMYC7cSoKMg\nAKcSRsJKeGGGYMYRXAbOwGkRXAWaKbB8hJBn4DO2BgBSmizYW7uajQAAAABJRU5ErkJggg==\n",
"prompt_number": 17,
"text": [
"-[FermionOperator(c,0),FermionOperator(c,1)]"
]
}
],
"prompt_number": 17
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"AntiCommutator(c,Dagger(c))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$\\left\\{{{c}^\\dagger},{c}\\right\\}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAADgAAAAbBAMAAADbkFG1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMolUmc0idhCru+9m\n3URozekSAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABK0lEQVQoFWNgQIB5CCacxagMZaJIPhEAi7om\nQCU1oDSYYosCUew/oGJLl2yGssAUtwKQ4gyACjVoNEBZYIoLJA6XZJiILAcRR0guJltyMj6dKP6E\n2Mm1AKRBqVyAwRGmk0m9EOrQ7gKgWK4BawCDGlSS7TJDKAMDD9D/bCEgoU0MnAYMFVBJoPdB1usK\nMJxNANIs4FBqALJAwL8BTHGEMzDHAFncC8BcKLELQpseYGDIN2Bg6FeAioMojk9gDt9HIAVyLf8B\nBgagGij4w8DADHUtKPhYJjC0CICk+oEcBk8GDqCvwMEKJpTOFIC18YJcxqRzHEgiJMEyIOIZnIVF\n8gAeSQ58kjxwjQzglMC0ASGAzOJ1APEiG0AkBpjbABJiRk0eUFWvDkAZuCgAZBQ3Pr2Io6QAAAAA\nSUVORK5CYII=\n",
"prompt_number": 18,
"text": [
"{FermionOperator(c,0),FermionOperator(c,1)}"
]
}
],
"prompt_number": 18
},
{
"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": 19,
"text": [
"1"
]
}
],
"prompt_number": 19
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"AntiCommutator(Dagger(c), c).doit()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$1$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAzRAiu5mrdu/dZmiL4QAAAAAJcEhZcwAA\nDsQAAA7EAZUrDhsAAAAjSURBVAgdY2BgEGJgYDDZxMCgEgYkGNhJJVgzdmYB9TEwAACPpQrvlUCH\ncAAAAABJRU5ErkJggg==\n",
"prompt_number": 20,
"text": [
"1"
]
}
],
"prompt_number": 20
},
{
"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": 21,
"text": [
"FermionOperator(c,1)\u22c5FermionOperator(d,1) + FermionOperator(d,1)\u22c5FermionOperat\n",
"or(c,1)"
]
}
],
"prompt_number": 21
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Simplifying operator expressions"
]
},
{
"cell_type": "heading",
"level": 3,
"metadata": {},
"source": [
"Test operator ordering of bosons"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(Dagger(a) * a)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{a}^\\dagger} {a}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAB8AAAATBAMAAACaQstxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIquJdjLdEETvu2aZ\nVM0GsGrEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAjElEQVQYGWNggAANKA2nMATM4VIQxro1c1FF\nCswLUAUYVNH4DGsJCuiiq0ByB6OyazoDw1GgCgiLoVOA6wMDgyFQAMJin83As4CBwZ2BAcri+c7A\n5sDAAHQXlMW5gKE+AKiegQHKqjdguA+SZ2CAsuovMKzjFQALQFh8DtxJzCA+A5TFrmIq5AEWgLAA\nNNUfGy2S/ZoAAAAASUVORK5CYII=\n",
"prompt_number": 22,
"text": [
"BosonOperator(a,0)\u22c5BosonOperator(a,1)"
]
}
],
"prompt_number": 22
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(a * Dagger(a))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$1 + {{a}^\\dagger} {a}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAEEAAAAVBAMAAADrxp6XAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzRAiu5mrdu/dZlSJ\nRDLkM64aAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3UlEQVQoFWNgwAFycYgjhAmrOIRQjJ3l6NqN\nXQIuqnBIAc5mEEIwkVh5CLbJJwQbieUNZ6uEwVQwGcAFgQyECgZ2mAruAmQV2QgODhVI4YGhgjmn\nbhIDQxzQDAgL05YbBiwfGBgOA1VAWBgqmLoYWB0YGIoYGKAsDBWsnxg4gW5WYGCAsqAqeAQFJRsF\nBQMYGDgcGPQfAK1AsNBdqn+AYT/IBAYGGAtDxQYGR24DsAooC10FYwHXNB6QAgYoizXjZxaYzwAN\nU6b0c2a1YBEEC6IApgLKw0bxLsAmygAAFlQ3VdFpO00AAAAASUVORK5CYII=\n",
"prompt_number": 23,
"text": [
"1 + BosonOperator(a,0)\u22c5BosonOperator(a,1)"
]
}
],
"prompt_number": 23
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(a * Dagger(a) * a)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{a}^\\dagger} \\left({a}\\right)^{2} + {a}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAGAAAAAbBAMAAABxU5cJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n+v3h3NfZf6AL3ykjYEJ4nXYawcz+AxjGXnyLG4MSAAAAAElFTkSuQmCC\n",
"prompt_number": 24,
"text": [
" 2 \n",
"BosonOperator(a,0)\u22c5BosonOperator(a,1) + BosonOperator(a,1)"
]
}
],
"prompt_number": 24
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(a * a * Dagger(a))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{a}^\\dagger} \\left({a}\\right)^{2} + 2 {a}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAGkAAAAbBAMAAACNRrzDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
"prompt_number": 25,
"text": [
" 2 \n",
"BosonOperator(a,0)\u22c5BosonOperator(a,1) + 2\u22c5BosonOperator(a,1)"
]
}
],
"prompt_number": 25
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(a * a * a * Dagger(a))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{a}^\\dagger} \\left({a}\\right)^{3} + 3 \\left({a}\\right)^{2}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAIYAAAAbBAMAAABLr+o/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIquJdjLdEETvu2aZ\nVM0GsGrEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACVUlEQVQ4EY2UP2gTURzHv/FMXpK7C2dXHQpW\ncBFijRRKh0AR3YwILg7+QXARPMSliwmIQrd2EGlFiZNLbW5yUSSLSykYFzcxQxGXhoh/KkiNv3fv\n/e7O3rvoG+73/f2+n/e73929BMhctWk/04uNubUxlBM47RjNUpa3r5/lAU7VHma77BS8/BfWR1kk\notNPJBmy1Mr9ZMvUY6HB7rjo7rA7wyKOt7diPUYV6tpcX3ucxv7nnQJzvNGfUV/ooirY4VPkd9ne\nE5OU3Y3MI6HKtXWB0oVG7ldkK/fEbE+KJIW7uMfU81BU1DRABSh49kN2VTwEN+yapPLbH54xpXrM\nc7q/inxt2lOp0PFsDz9kJUmVRqMhbzoWimVOxRIrim5dJYst8VsqMwWE56N4WcF03Y5U3IOUfBYD\nlTt86jrwRu6xunSZqF3wE1+MKjwHvSYJGKhFr0xPdJxMlAO6yzI6JF/LXK+ox/kbspKmxCPIs3Ra\nuoUefdIAGyTfy1yvqAesB1RKU84O5Gml8YFSC3jr4xrJSzLXK+6BFeLSVKmNZkPD0l2F+E5p1MMa\nDD49GQy6VDsAdGjSNNWs4qOaIpxSfIU7FMB92sOL5xj56AQmqtnDuuspnN6W2EW5/878TleBm0Sm\nqUrdvmrpO1qTwBZeTFaBTR6CIs/xEu43StOUmDo5cUZvkKfn1vy5qRbwWZdk4B7FzQ1ywjNmoCQp\n1xUV6Ke5xIoi9+CSmWL3FQurzopiMUgkJM0UM2VPqztcMcXxVPTv8tS0l2v/oA4qzuZ5eNvfcQ/1\nBxDVkbPIPgSCAAAAAElFTkSuQmCC\n",
"prompt_number": 26,
"text": [
" 3 2\n",
"BosonOperator(a,0)\u22c5BosonOperator(a,1) + 3\u22c5BosonOperator(a,1) "
]
}
],
"prompt_number": 26
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(a * Dagger(a) * a * Dagger(a))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$1 + 3 {{a}^\\dagger} {a} + \\left({{a}^\\dagger}\\right)^{2} \\left({a}\\right)^{2}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAALsAAAAhBAMAAABzQ8d8AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nqdTqAAAAAElFTkSuQmCC\n",
"prompt_number": 27,
"text": [
" 2 \n",
"1 + 3\u22c5BosonOperator(a,0)\u22c5BosonOperator(a,1) + BosonOperator(a,0) \u22c5BosonOperato\n",
"\n",
" 2\n",
"r(a,1) "
]
}
],
"prompt_number": 27
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(a * Dagger(a) * Dagger(a) * Dagger(a))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$3 \\left({{a}^\\dagger}\\right)^{2} + \\left({{a}^\\dagger}\\right)^{3} {a}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAJYAAAAhBAMAAADJ4E8cAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
"prompt_number": 28,
"text": [
" 2 3 \n",
"3\u22c5BosonOperator(a,0) + BosonOperator(a,0) \u22c5BosonOperator(a,1)"
]
}
],
"prompt_number": 28
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(a * Dagger(b))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{b}^\\dagger} {a} + \\left[{a},{{b}^\\dagger}\\right]$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAGoAAAAaBAMAAACtLdRlAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
"prompt_number": 29,
"text": [
"BosonOperator(b,0)\u22c5BosonOperator(a,1) + [BosonOperator(a,1),BosonOperator(b,0)\n",
"]"
]
}
],
"prompt_number": 29
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(a * Dagger(b)).doit()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${a} {{b}^\\dagger}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAABsAAAATBAMAAACTqWsLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIquJdjLdEETvu2aZ\nVM0GsGrEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAsklEQVQYGWNgQACmBgQbyELjMhcgy3K9OnkB\nic/OXA7jVU0D6Q2AcRkeAVlcCLV/Ubgc30GK4bLMCSBuA5BgVHZNZ+BT8bnAwAMyqlOA6wND/XOG\njwzMAgwM7LMZeBYw3C9g+MzAUsDAwPOdgc2B4RwD+2cGdqBazgUM9QEMbxg4QFYxMNQbMNxnKPgN\nUgXmXmBYxyvwg6FyA5jL58CdxMygzbAGzGNgVzEV8mCocYJIAgB3IikhotO3vgAAAABJRU5ErkJg\ngg==\n",
"prompt_number": 30,
"text": [
"BosonOperator(a,1)\u22c5BosonOperator(b,0)"
]
}
],
"prompt_number": 30
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(a * Dagger(b), independent=True)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{b}^\\dagger} {a}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAB0AAAATBAMAAACetxtMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAELvv3c2rVESJdpki\nZjI6QXVuAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAuUlEQVQYGWNgAIMMCAUn0fkT4TJghqq2OYqA\nwEQBGJ9R2QHIzINxGRhYE4BsdQSfvwCV3w/Smo6QnwJiItkftmo7A8M+oBhP5jpHBgbGrwL8Bxjm\nAPnNB5g+MDCwf2Tgn8CwEChhy8CtwMDAHcAgP4EBaCb3RwaWBUDrNzD0g2xkYFZgkH/AwMB/gWE/\nyEaQqnqgOv4GoEYwv4BBlf0AA9sFpg1gPusCXjdOoLG7csFcBsasecdWMAAA89EirNLptQcAAAAA\nSUVORK5CYII=\n",
"prompt_number": 31,
"text": [
"BosonOperator(b,0)\u22c5BosonOperator(a,1)"
]
}
],
"prompt_number": 31
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(a * Dagger(b) * Dagger(a))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{b}^\\dagger} + {{b}^\\dagger} {{a}^\\dagger} {a} + \\left[{a},{{b}^\\dagger}\\right] {{a}^\\dagger}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAMEAAAAaBAMAAAD8lSVDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nLQAAAABJRU5ErkJggg==\n",
"prompt_number": 32,
"text": [
"BosonOperator(b,0) + BosonOperator(b,0)\u22c5BosonOperator(a,0)\u22c5BosonOperator(a,1) \n",
"+ [BosonOperator(a,1),BosonOperator(b,0)]\u22c5BosonOperator(a,0)"
]
}
],
"prompt_number": 32
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(a * Dagger(b) * Dagger(a), independent=True)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{b}^\\dagger} + {{b}^\\dagger} {{a}^\\dagger} {a}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAFsAAAAVBAMAAADbSo+2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nJllfNwAAAABJRU5ErkJggg==\n",
"prompt_number": 33,
"text": [
"BosonOperator(b,0) + BosonOperator(b,0)\u22c5BosonOperator(a,0)\u22c5BosonOperator(a,1)"
]
}
],
"prompt_number": 33
},
{
"cell_type": "heading",
"level": 4,
"metadata": {},
"source": [
"Normal ordering"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_order(Dagger(a) * a)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{a}^\\dagger} {a}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAB8AAAATBAMAAACaQstxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIquJdjLdEETvu2aZ\nVM0GsGrEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAjElEQVQYGWNggAANKA2nMATM4VIQxro1c1FF\nCswLUAUYVNH4DGsJCuiiq0ByB6OyazoDw1GgCgiLoVOA6wMDgyFQAMJin83As4CBwZ2BAcri+c7A\n5sDAAHQXlMW5gKE+AKiegQHKqjdguA+SZ2CAsuovMKzjFQALQFh8DtxJzCA+A5TFrmIq5AEWgLAA\nNNUfGy2S/ZoAAAAASUVORK5CYII=\n",
"prompt_number": 34,
"text": [
"BosonOperator(a,0)\u22c5BosonOperator(a,1)"
]
}
],
"prompt_number": 34
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_order(a * Dagger(a))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{a}^\\dagger} {a}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAB8AAAATBAMAAACaQstxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIquJdjLdEETvu2aZ\nVM0GsGrEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAjElEQVQYGWNggAANKA2nMATM4VIQxro1c1FF\nCswLUAUYVNH4DGsJCuiiq0ByB6OyazoDw1GgCgiLoVOA6wMDgyFQAMJin83As4CBwZ2BAcri+c7A\n5sDAAHQXlMW5gKE+AKiegQHKqjdguA+SZ2CAsuovMKzjFQALQFh8DtxJzCA+A5TFrmIq5AEWgLAA\nNNUfGy2S/ZoAAAAASUVORK5CYII=\n",
"prompt_number": 35,
"text": [
"BosonOperator(a,0)\u22c5BosonOperator(a,1)"
]
}
],
"prompt_number": 35
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_order(a * a * a * Dagger(a))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{a}^\\dagger} \\left({a}\\right)^{3}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAADoAAAAbBAMAAADfZYGIAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n",
"prompt_number": 36,
"text": [
" 3\n",
"BosonOperator(a,0)\u22c5BosonOperator(a,1) "
]
}
],
"prompt_number": 36
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_order(a * Dagger(a) * a * Dagger(a) * a)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$\\left({{a}^\\dagger}\\right)^{2} \\left({a}\\right)^{3}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAFMAAAAhBAMAAABXdor5AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
"prompt_number": 37,
"text": [
" 2 3\n",
"BosonOperator(a,0) \u22c5BosonOperator(a,1) "
]
}
],
"prompt_number": 37
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Wick's contraction:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"a * Dagger(a) - normal_order(a * Dagger(a))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$- {{a}^\\dagger} {a} + {a} {{a}^\\dagger}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAGQAAAAVBAMAAABCslYDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMiKriXZE77tm\nmVQzv5s6AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABS0lEQVQ4EZ2RvUvEQBDF3yomaswRBS1sIsFC\nEAsRRKwCYp/qagvBs5E7UbA8QRC7q8XWSpHjej/Af8FKC9MIdp4g2ImbWfeyO9nKbfLeb+bNJFnA\nOl7Hsg6zy9k/In7Eh3DfYiBYfM4ZYrZ3e2MT4U/aoOKiFn8PL600MbDNPIKcE+7vOGCRaV4HKhEv\nt5reLUdmjyN2L46IeZX1o40IYSqHkKJhOkJkZOdkGXgqt4xd4h7wM0ApM6LIeRb0gcMyMvuAF2BU\n/nWlzAgRcY2wCxwDUyvFmcdrhDVAyE6lzAiR8BPDbUDO1OcK4ktprfaTZD1JliQkMt5FnOru4im+\nUesXOwwFqM9XtbiJD3MHxA+CuTeKDNQgQiTO0atlRYc+C3jcapIp1d8WEBlqT6z6upueM6ebjQ5T\nOkI10Tion1kRp9FX6Sy64YUL/wJuV1aA6E8ryQAAAABJRU5ErkJggg==\n",
"prompt_number": 38,
"text": [
"-BosonOperator(a,0)\u22c5BosonOperator(a,1) + BosonOperator(a,1)\u22c5BosonOperator(a,0)"
]
}
],
"prompt_number": 38
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(a * Dagger(a) - normal_order(a * Dagger(a)))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$1$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAzRAiu5mrdu/dZmiL4QAAAAAJcEhZcwAA\nDsQAAA7EAZUrDhsAAAAjSURBVAgdY2BgEGJgYDDZxMCgEgYkGNhJJVgzdmYB9TEwAACPpQrvlUCH\ncAAAAABJRU5ErkJggg==\n",
"prompt_number": 39,
"text": [
"1"
]
}
],
"prompt_number": 39
},
{
"cell_type": "heading",
"level": 3,
"metadata": {},
"source": [
"Test operator ordering of fermions"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(Dagger(c) * c)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{c}^\\dagger} {c}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAABsAAAATBAMAAACTqWsLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXYyIt1Uze+rmRC7\nZkTTotXTAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfUlEQVQYGWNgAIN5EApGonGlYeJg+vipNmT+\nBukNyFyGySg8hmN4udNQZZHsFVQyYFgMkgUxGFwT2AsYRIA8MIOhkYEzgUEbyAUzuH6D1AFdAWFw\nHABxgQDC2C8A4TFAGPwXGBgSQCIQBtcEhl1A84F6IQzBuwogHtBaIAMAohMbe/npeTMAAAAASUVO\nRK5CYII=\n",
"prompt_number": 40,
"text": [
"FermionOperator(c,0)\u22c5FermionOperator(c,1)"
]
}
],
"prompt_number": 40
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(c * Dagger(c))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$1 - {{c}^\\dagger} {c}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAD0AAAAUBAMAAADM7ygkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzRAiu5mrdu/dZjKJ\nVEQJ2sCdAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAs0lEQVQoFWNgwA7ysAvDRQnJH4KrxMpwdO3F\nKg4TVDikAGMyCMFZSIxsONvkE5yJxPCFsVXC8MszsGOVz4XpxyGP8D+m/jPlGxiiQfpBDEz9lhcY\nBRgOA6XBDEz5Hga2CwxFQHkwAyrPJDkTCCYCzWMFu1cBxsDQz+EA1AoCUAa6+/QPgGUZGKAMdHn+\nBQwMF0BKoAx0edYEBlWgM4D2gxmsGT+zQDwEOLOqAMKBMxByGCwA394vqiwGFSwAAAAASUVORK5C\nYII=\n",
"prompt_number": 41,
"text": [
"1 - FermionOperator(c,0)\u22c5FermionOperator(c,1)"
]
}
],
"prompt_number": 41
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(c * c * Dagger(c))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{c}^\\dagger} \\left({c}\\right)^{2}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAADYAAAAbBAMAAADFWWEGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXYyIt1Uze+rmRC7\nZkTTotXTAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABRklEQVQoFXWRvUvDUBTFT2xi0qbRVJxahKij\nS/4BMYuj4CDOWXTuVDepIE4OBcFRCk662E1wsUNRqIu4CIJQEXcFN9H63r33lVhJhnfO75x83JcH\n5FzVfiengZUU63mdkxQ+pTsZv8cb2D95HTD5JV1t/DnAiTi77R/8K6uSdGo88DJxECsJ2tLhiIzd\nZdb0glcG3JC6sl0XKDS27/5080ITIbzh8F3omHRfyG+KIaG9T62bqGFMZSHGqQarrZeZRcDMv5q6\ndczqrnwP2JvYAR41qmsPxRRL2jkp4IVQ397SCJTot9Hw3gC4ILdCFbwuq1p1d0gk3XU46tQ7/Q+i\nN86m1QApWz3Ltzqc0SylFq5i7qwWcAZf7eecA1SeI3F679bcg6JdSTKyId5uZkKxT6JWJCYj5YTh\nMpMZa862Z4KsrhEECWe/nnM8h0OEP9sAAAAASUVORK5CYII=\n",
"prompt_number": 42,
"text": [
" 2\n",
"FermionOperator(c,0)\u22c5FermionOperator(c,1) "
]
}
],
"prompt_number": 42
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(c * c * c * Dagger(c))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$- {{c}^\\dagger} \\left({c}\\right)^{3} + \\left({c}\\right)^{2}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAIEAAAAbBAMAAACpc/FGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
"prompt_number": 43,
"text": [
" 3 2\n",
"- FermionOperator(c,0)\u22c5FermionOperator(c,1) + FermionOperator(c,1) "
]
}
],
"prompt_number": 43
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(c * Dagger(c) * c * Dagger(c))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$1 - {{c}^\\dagger} {c} - \\left({{c}^\\dagger}\\right)^{2} \\left({c}\\right)^{2}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAKkAAAAhBAMAAABQGJapAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
"prompt_number": 44,
"text": [
" 2 \n",
"1 - FermionOperator(c,0)\u22c5FermionOperator(c,1) - FermionOperator(c,0) \u22c5FermionO\n",
"\n",
" 2\n",
"perator(c,1) "
]
}
],
"prompt_number": 44
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(c * Dagger(d))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$\\left\\{{c},{{d}^\\dagger}\\right\\} - {{d}^\\dagger} {c}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAHIAAAAbBAMAAABSCMbcAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMolUmc0idhCru+9m\n3URozekSAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACGElEQVQ4Ea1UPUwUQRT+3DuOueXuoLL1wEQo\nJNlGW86Cglh4BYZAINmWwnDYaGW2sNBQHD8FFYREqCwkaCwMxXY2kBAIBQXJFcQCC39oKIjxvdnZ\nmd2bXSsn2Xnf+7733vy8yQJZYzOLVFy5o8VbdzWMAWdeDMRe2iYyx/20RN4IfaUZi5aEE8R873WM\ntN3Z/ki4r66JBHBnHnvKLTcTfASDkYCAawvECudNFAR0Z9beN7DCYrfA3OUnlPX5rYB+D+/yMnEF\nN96sXfpHkJ9Zu/5X5iiwlremc4Ry/pqTgHwJ1jGoXE8Dpp/uFi8w9CI6eGl18Qp4xJSdOT98x0OR\nA53hRWCpReiZ19PkaHz3XervPYaV7kbXPuNDAMenV3KCpyjR5oA9tX2xjyKVeMkcRrl6YiyFGAd6\nA/lI1nDok1aMyxOo1gESaYgpnubOaZxykScBjomIEFCYJdS3RRMPAu2ORDQ9CGMU2S8Qf2LEdsED\n2vWIYTCmVkTtlyKVEb9R+MkrghAPvtv+EKACVMLDTsFnZN+toPfTfCilG9prFFBcxuuBdpP61aqc\nOVK0MzGFw+W6FCcgqHGybUMHLVTpmsT64PyrvMy3z2+vd6To3P9K1jT8m0pRxghpXnsmINScBEZI\n89rTASLUnARaSNPG0wEVw0mU/U9IBDm7CScJq42kl4WngywW2MjhTXRB/juMr9BlaFH/ifgLXcto\nxkcUjs8AAAAASUVORK5CYII=\n",
"prompt_number": 45,
"text": [
"{FermionOperator(c,1),FermionOperator(d,0)} - FermionOperator(d,0)\u22c5FermionOper\n",
"ator(c,1)"
]
}
],
"prompt_number": 45
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(c * Dagger(d), independent=True)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$- {{d}^\\dagger} {c}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAACwAAAATBAMAAAAZhvJKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMkS775lUiSKr\ndmb9QEdWAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAvElEQVQYGWNgQAE8AShcGAeHMJcATAEyzabW\nPgGZD2UzcgliioZVMPA4YAoz/GRgw2IG60eswlwGDDxYVDM1MGC6e89a/wkMHCArudZegVvMWs2Q\nL8DAtYGBgduSQQkuLP6AoYmBgVeAgYF9AcNJBiFjEFBh6BdgsGJgYAQqA7IQoIyB8SuEV4YQZGD8\nxcD5AaQWxEIARqAPFWaD+f8YGDjhEroMjw8sAPN6GBhb4MIyt7zPBoB5XOveMgAArhUjIYzwxGYA\nAAAASUVORK5CYII=\n",
"prompt_number": 46,
"text": [
"-FermionOperator(d,0)\u22c5FermionOperator(c,1)"
]
}
],
"prompt_number": 46
},
{
"cell_type": "heading",
"level": 4,
"metadata": {},
"source": [
"Normal ordering"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_order(Dagger(c) * c)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{c}^\\dagger} {c}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAABsAAAATBAMAAACTqWsLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXYyIt1Uze+rmRC7\nZkTTotXTAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfUlEQVQYGWNgAIN5EApGonGlYeJg+vipNmT+\nBukNyFyGySg8hmN4udNQZZHsFVQyYFgMkgUxGFwT2AsYRIA8MIOhkYEzgUEbyAUzuH6D1AFdAWFw\nHABxgQDC2C8A4TFAGPwXGBgSQCIQBtcEhl1A84F6IQzBuwogHtBaIAMAohMbe/npeTMAAAAASUVO\nRK5CYII=\n",
"prompt_number": 47,
"text": [
"FermionOperator(c,0)\u22c5FermionOperator(c,1)"
]
}
],
"prompt_number": 47
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_order(c * Dagger(c))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$- {{c}^\\dagger} {c}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAACoAAAATBAMAAAAUmIINAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMol2IlTvq5m7\nZkT3Gyx5AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAiklEQVQYGWNgQAZcD5B5MDZ2UVYBmDwSzaOz\nswGJC2UysgpiCjIwcDlgE+VpIF6UC6taTPeyhqUwcDsAzQUxYIDdgkGJgbWAgQHMgIlyBjCsYGAD\n+g3MEDIGARWG/SDPMgIxmAFTOweDAVL1FSoKZ4D5fxkYOFAZIN4uBsYtYFE4A8Rjjb0GFgQzAO18\nGLDF/XLYAAAAAElFTkSuQmCC\n",
"prompt_number": 48,
"text": [
"-FermionOperator(c,0)\u22c5FermionOperator(c,1)"
]
}
],
"prompt_number": 48
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_order(c * Dagger(c) * c * Dagger(c) * c)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$- \\left({{c}^\\dagger}\\right)^{2} \\left({c}\\right)^{3}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAGUAAAAhBAMAAAAym3iGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMkS7mXYiie9U\nZqsqREkJAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACJUlEQVQ4Ea2TvWtUQRTFz375srvZuNiksHAF\nCysRLDSg8BBsbDbgH+BioQgKq0gQQYhWsjYLgo3FbipNQEmRNEHwVVYBtxBEg7iKCoJF/AvivXNn\n7sxOHmvjwJs553fefLyZN8CUUlu6NSXNj1bxLT+YQhfQ7k+Jc6NNDBajoNb7GpF99l0zQvexFJF9\n9kxM7uFLjMT/cLjWdcq3t7006oJ46jNj1IcoJ1vYmGTljvgVoDImWRlWUmrCUni4HFqgKF985dpr\n4CVF57ffxnuAeA/uyhDNFXqzRM+bvT1DyiPTvDd1OzONrap/rPhO7UzXR09FzqX8NQMZwKa6SzcI\nVHd9n6NWtoAF7DR9ANQz67gPzllDUw6t7AEXe7+Us2i7EX6z8+dd77OncliasN5xxpzpY+dwyali\n5pS25oTXHqW4w+iymxXP2TLXD2Yg5So1T7JiC6vs2ynXXG7SY3ijxXainCC3hbkMDxj7a3DW8dmT\nHACHTnE5xvI0kJgjMqsaZMy4bDpe3hUQ1DRPvaPez0N9hOf3OTLUPv5a09qEV+za9B3gJ3BwBGSC\n/L7RHgjP2QM6kWSM+VT66GnhleNJV6Kg5pHXPi9b4v8DPlPDi4vB2yLrE+i45qW+lc8UqUiGKukO\nu4sBNLqWvwhyK8vrAWuMvbkustDySJX5n60rpYph79wBt0af0D8RmE+Bro6M+RgglbNu4XRNO0r/\nIbY1T5oq/4v4C3WYaoa3V/8gAAAAAElFTkSuQmCC\n",
"prompt_number": 49,
"text": [
" 2 3\n",
"-FermionOperator(c,0) \u22c5FermionOperator(c,1) "
]
}
],
"prompt_number": 49
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Wick's contraction"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"c * Dagger(c) - normal_order(c * Dagger(c))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{c}^\\dagger} {c} + {c} {{c}^\\dagger}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAE0AAAAVBAMAAADx+n4ZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXYyIt1Uze+rmRC7\nZkTTotXTAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABGElEQVQoFWNgAIN5EAoHyfkAJkGsOmmYBqw0\n4wao8PFTbVgVQAR5alYkQFgbpGE6sCnnZtwNE54MY2ClOQ1gwsdgDKw0TwJMGEUdtwNMGEoj1E1D\nlmFTQOYB2ZwJMAGU8MNU9wCkTlDJgGExjAGi4epAMoxCygxcIH+4JrAXMIjAGMjqQDIsYQzlDIwO\nQOFGkPXaMAayOpAMhwDDNAYmYOhy/QZJwRlI6sAy64EyDNxAzHEAJAUEUAZjeXlpe3n5AqhAD1gO\nSOwXgLLgDJg/QALcH2Hq+C8wMCSAOHAGTB1Y4C/Q+2ClXBMYdhmAWHAGTB1YYCUDNzjMgMF3VwGs\nAcGAhR9IhlH2EkQWCwlTh0UKRYgP6GR0AACsszrw459K0gAAAABJRU5ErkJggg==\n",
"prompt_number": 50,
"text": [
"FermionOperator(c,0)\u22c5FermionOperator(c,1) + FermionOperator(c,1)\u22c5FermionOperat\n",
"or(c,0)"
]
}
],
"prompt_number": 50
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(c * Dagger(c) - normal_order(c * Dagger(c)))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$1$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAzRAiu5mrdu/dZmiL4QAAAAAJcEhZcwAA\nDsQAAA7EAZUrDhsAAAAjSURBVAgdY2BgEGJgYDDZxMCgEgYkGNhJJVgzdmYB9TEwAACPpQrvlUCH\ncAAAAABJRU5ErkJggg==\n",
"prompt_number": 51,
"text": [
"1"
]
}
],
"prompt_number": 51
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"States"
]
},
{
"cell_type": "heading",
"level": 3,
"metadata": {},
"source": [
"Boson Fock states"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"ket = BosonFockKet(2)\n",
"\n",
"ket"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${\\left|2\\right\\rangle }$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAWBAMAAADQs7pMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVCKJRHYymbsQq+/d\nzWZtGdnkAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAoUlEQVQIHWMQYgCDB0DSBMIsATOZyjsSGLLA\nzFkMzF8YeCaAFKxkYNjNwPoAxNzDwLA+gfMCiHl/ApDJ0AQ14fwEBmEIk+UzAwMPhMnTwMDA7gC2\nogJoKNsCEJN1AZDJaABiijMwKTBwKACZbAUMrAoMGiBtOsbmaxgYDEHM+////2FgOABiggFLAZzJ\nlQBnzgXKQRU8BTLDwEoZOiEUlAQAd+Ag08QZt+sAAAAASUVORK5CYII=\n",
"prompt_number": 52,
"text": [
"\u27582\u27e9"
]
}
],
"prompt_number": 52
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"bra = BosonFockBra(2)\n",
"\n",
"bra"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${\\left\\langle 2\\right|}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAAWBAMAAAAyb6E1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZjKrmUTdVIm7Is12\nEO9EeINyAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAApUlEQVQIHT3MLQ7CQBCG4RfatNCmDYfA4HoE\nEoLAVeCphBsQVE0THASJQqBQPQI/DlWP4Qz4ZpmdTTrJzjyZzH7ACK0d9DPHCvyc6+IEwoAk45Bb\nbvEa0q/lmbTA+wmjmrh1HGhW0Mo20KhVKVwqNzZsbukfLR+WL3kVH+nhlJtweIExvIVhSbTezwoh\nBbExRqkR+o2JDFtyEDcde/eOSe34dEP7H+J5JnA4wY8wAAAAAElFTkSuQmCC\n",
"prompt_number": 53,
"text": [
"\u27e82\u2758"
]
}
],
"prompt_number": 53
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"type(bra), type(ket)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 54,
"text": [
"(sympy.physics.quantum.fieldoperator.BosonFockBra,\n",
" sympy.physics.quantum.fieldoperator.BosonFockKet)"
]
}
],
"prompt_number": 54
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"a * ket, qapply(a * ket)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$\\begin{pmatrix}{a} {\\left|2\\right\\rangle }, & \\sqrt{2} {\\left|1\\right\\rangle }\\end{pmatrix}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAIAAAAAaBAMAAACN7UndAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nbJl6vPh0jB2LavL52PM4M/5oMvkHt0um6w93xAoAAAAASUVORK5CYII=\n",
"prompt_number": 55,
"text": [
"\u239b ___ \u239e\n",
"\u239dBosonOperator(a,1)\u22c5\u27582\u27e9, \u2572\u2571 2 \u22c5\u27581\u27e9\u23a0"
]
}
],
"prompt_number": 55
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"bra * a, qapply(bra * a, dagger=True)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$\\begin{pmatrix}{\\left\\langle 2\\right|} {a}, & \\sqrt{3} {\\left\\langle 3\\right|}\\end{pmatrix}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAIAAAAAaBAMAAACN7UndAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrL+0XfkL3EGg4raQe50AAAAASUVORK5CYII=\n",
"prompt_number": 56,
"text": [
"\u239b ___ \u239e\n",
"\u239d\u27e82\u2758\u22c5BosonOperator(a,1), \u2572\u2571 3 \u22c5\u27e83\u2758\u23a0"
]
}
],
"prompt_number": 56
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"qapply(Dagger(ket) * ket)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$\\left\\langle 2 \\right. {\\left|2\\right\\rangle }$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAC0AAAAWBAMAAACmiQjHAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZjKrmUTdVIm7Is12\nEO9EeINyAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABIElEQVQoFWXQIU/DQBQH8D8r60rbLfsAiCZg\ncFU4QhNAzE3gqQSFQSAnSJiD4CAhqSAkoPYNBsGhLkFiUEgECQSzHP/37g7EXtp77355174WQB8h\nOo2rVFplYCRDV+5IWhziZnChQE+3LwusyC5Gt8SZdtJfkHwhNvRjRAa9N2mhbwAfyPvcXKFXI/r0\n/g5sFt0JkE6Qzf59aujsRSJnEM98P9Oh4bP5Wom9kaw6Z/tHcVcAB7qqx+c6u/sG1gz1AQt+66PA\nkyzO85JFZ4RXSRXugx8hbbCkF1aBZ++tCnmDOz2S7o+3au+345N14JS7Gpm1NvjU2m8xQAdiZug8\nzO2KyxpvH8GjgpCZwH/9MiMWHuZ8WUR+qo+8cMV1gPn8C/ofPWSWdkkOAAAAAElFTkSuQmCC\n",
"prompt_number": 57,
"text": [
"\u27e82\u27582\u27e9"
]
}
],
"prompt_number": 57
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"(Dagger(ket) * ket).doit()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$1$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAzRAiu5mrdu/dZmiL4QAAAAAJcEhZcwAA\nDsQAAA7EAZUrDhsAAAAjSURBVAgdY2BgEGJgYDDZxMCgEgYkGNhJJVgzdmYB9TEwAACPpQrvlUCH\ncAAAAABJRU5ErkJggg==\n",
"prompt_number": 58,
"text": [
"1"
]
}
],
"prompt_number": 58
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"(bra * Dagger(bra)).doit()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$1$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAzRAiu5mrdu/dZmiL4QAAAAAJcEhZcwAA\nDsQAAA7EAZUrDhsAAAAjSURBVAgdY2BgEGJgYDDZxMCgEgYkGNhJJVgzdmYB9TEwAACPpQrvlUCH\ncAAAAABJRU5ErkJggg==\n",
"prompt_number": 59,
"text": [
"1"
]
}
],
"prompt_number": 59
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"Dagger(ket) * Dagger(a) * ket, qapply(Dagger(ket) * Dagger(a) * ket).doit()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$\\begin{pmatrix}{\\left\\langle 2\\right|} {{a}^\\dagger} {\\left|2\\right\\rangle }, & 0\\end{pmatrix}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAHYAAAAaBAMAAACQv7UDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\njV3ToqfTSHF2KtVE9A8hO56Tt5hb0gAAAABJRU5ErkJggg==\n",
"prompt_number": 60,
"text": [
"(\u27e82\u2758\u22c5BosonOperator(a,0)\u22c5\u27582\u27e9, 0)"
]
}
],
"prompt_number": 60
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"(bra * Dagger(a) * Dagger(a) * a * a * ket).doit()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${\\left\\langle 2\\right|} \\left({{a}^\\dagger}\\right)^{2} \\left({a}\\right)^{2} {\\left|2\\right\\rangle }$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAIsAAAAhBAMAAAAbsEVEAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
"prompt_number": 61,
"text": [
" 2 2 \n",
"\u27e82\u2758\u22c5BosonOperator(a,0) \u22c5BosonOperator(a,1) \u22c5\u27582\u27e9"
]
}
],
"prompt_number": 61
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"qapply(Dagger(ket) * Dagger(a) * a * ket).doit()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$2$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOBAMAAAAPuiubAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAIpm7MhCriUTv3c12VLge\nopIAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABOSURBVAgdY2BUMnZgYAhjYH/BwJDKwDCTgWEWA0Oe\nA8O+ABAJBOsCgATHcxCTywBEKoMIngQQeYmBUYCBRYGBR4BBqrwoi4Fh37t3rxgAK1UOqPfNIGYA\nAAAASUVORK5CYII=\n",
"prompt_number": 62,
"text": [
"2"
]
}
],
"prompt_number": 62
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"qapply(bra * Dagger(a) * a * Dagger(bra)).doit()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$2$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOBAMAAAAPuiubAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAIpm7MhCriUTv3c12VLge\nopIAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABOSURBVAgdY2BUMnZgYAhjYH/BwJDKwDCTgWEWA0Oe\nA8O+ABAJBOsCgATHcxCTywBEKoMIngQQeYmBUYCBRYGBR4BBqrwoi4Fh37t3rxgAK1UOqPfNIGYA\nAAAASUVORK5CYII=\n",
"prompt_number": 63,
"text": [
"2"
]
}
],
"prompt_number": 63
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"ket = BosonFockKet(4)\n",
"bra = BosonFockBra(2)\n",
"\n",
"bra * a * a * ket, qapply(bra * a * a * ket).doit()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$\\begin{pmatrix}{\\left\\langle 2\\right|} \\left({a}\\right)^{2} {\\left|4\\right\\rangle }, & 2 \\sqrt{3}\\end{pmatrix}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAKoAAAAdBAMAAADIl+p8AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYAAAAABJRU5ErkJggg==\n",
"prompt_number": 64,
"text": [
"\u239b 2 ___\u239e\n",
"\u239d\u27e82\u2758\u22c5BosonOperator(a,1) \u22c5\u27584\u27e9, 2\u22c5\u2572\u2571 3 \u23a0"
]
}
],
"prompt_number": 64
},
{
"cell_type": "heading",
"level": 3,
"metadata": {},
"source": [
"Coherent states"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"alpha, beta = symbols(\"alpha, beta\")"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 65
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"ket = BosonCoherentKet(alpha)\n",
"\n",
"ket"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${\\left|\\alpha\\right\\rangle }$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAABcAAAAWBAMAAADZWBo2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVCKJRHYymd0Q77vN\nZqvYY74YAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAApUlEQVQYGWMQYoCCC0DaBMYpQebkIXM4JiAp\nYwVqguvhPIDEYdgE4ZiHOwB1C4M5Gg6ci5gEGDjAnMUMDNtZJzCwOwANYP/JwFDDzMDA9gDI4drA\nwGBlycDAaADkzDdgYLAXYGDgVgBy+IG22QNN0wYZwP6AgbeebQKDIYjDYHz3ArspA0MDmANUAgQs\nBUgcvgQkzhygLNzVV4GcMCAGg50wBpQGAPqnG1rZsH6bAAAAAElFTkSuQmCC\n",
"prompt_number": 66,
"text": [
"\u2758\u03b1\u27e9"
]
}
],
"prompt_number": 66
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"bra = BosonCoherentBra(beta)\n",
"\n",
"bra"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${\\left\\langle \\beta\\right|}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAABcAAAAWBAMAAADZWBo2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZjKrmUTdVIm7Is12\nEO9EeINyAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAt0lEQVQYGU3OMQrCQBCF4T9qTIwRUlhYqp1d\nvIFNCkFERKxtvIOkSmlhYW2VA4ikso43SC+CeAKP4Ew2osMuOx9vd1ggwJSTQi2s4C6gIZtzb4Ki\nKb2X0C2xE7iwLnESxDBVeJkgx34p3EAGxrNrqtD31mM/7ytWAl9WpogM7LfiJhjKzafiLsihHipa\nKRzhUk5zEhhT26IJG+zBYV8YRDpZS5ORfvOLdvGXWPnyl3Qy0+P3q8YcH2kBIXvHGaUQAAAAAElF\nTkSuQmCC\n",
"prompt_number": 67,
"text": [
"\u27e8\u03b2\u2758"
]
}
],
"prompt_number": 67
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"bra * ket"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$\\left\\langle \\beta \\right. {\\left|\\alpha\\right\\rangle }$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAADIAAAAWBAMAAABwLNKiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZjKrmUTdVIm7Is12\nEO9EeINyAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABMElEQVQoFW2RL0/DUBTFzyst60oHJUEgByhc\n4RNgJggEFkLQM0iSyWWqAlExYAI1VYUiZApdEj5APSEQFEHxBUg49/WupMDrn3vu/d33et4rgAjV\naGQqjUQn1ozB76p2RdjX7eqO1Cpi2+dZCRKs1AjOmA6kFzipkwemEz5DYLdOnrjQlKUc3ludNDP4\nEe0N9++zGTl/FF+NBGLAPKcHbSU33dZH0Ad6OGYl5C2Liut34DosgA4vS7zPkrhfwB6tYgNib50L\nvpZkcQxcXlAPQHu0hrm4JC8JcNWnnoD2wL47BvnOUkRCFYzFHrbgnAogcWOYQ6eAz3k9eGujtFCC\ndDlyR7Cb6YhpHdVZ281synn+IdusLBT/zGlNSUx+NJvy8+dMzprFysK2imbpSLNf4RvgwDoJs5Ja\nSgAAAABJRU5ErkJggg==\n",
"prompt_number": 68,
"text": [
"\u27e8\u03b2\u2758\u03b1\u27e9"
]
}
],
"prompt_number": 68
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"(bra * ket).doit()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$e^{\\alpha \\overline{\\beta} - \\frac{\\left\\lvert{\\alpha}\\right\\rvert^{2}}{2} - \\frac{\\left\\lvert{\\beta}\\right\\rvert^{2}}{2}}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAHYAAAAcBAMAAABG5lYeAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAB4klEQVQ4EZVTvy8EQRh9e7d77nBuw3UiNlTX\ncL90ignRiGL9B5cQRKXSqolCQmhERCRKxD9wIrqTENGoTqGVqJRndnZnZmduNs5XzPe+931vZmd3\nH/BXXEz5hpH+iU8Dq1PtzJZO0XoI6wZWp56tb52idR7jroHWqZzpXOBEn5N1dEmaxiIY9BzAas2w\n9CZndURCgsC6jbVsII0V0DTqxmgUOkFwhoSAoLjscQ6BaBOLNFkLw5LVEUFulrgg2OlsyR7VVu4I\n1fZ1fiTLUG1JEARHqJapVgmqLeM2OF6PlNsnbkHwjquGQdvEp0n7iGxM+4IrdGszTadt0no4FI9C\ncNlarXRr06cPrqpNVYkF5/i1HdMySATBgE0/EQ3lvtu4SyHnoyRGSYiixGkbgwGMa+02ntaQAup8\nCF6IosRpOhJElBjOHmCDBI8zh7MvGi7YfyIW5AL2I/p9OM2kKPi4djEC7IV1tCaYVHV0voFpC6uw\nPUWbYFLV0c58yy5hpiZfFdsjwaTS0dZk3Q9PKyuHssJsUuHodd/xkrSW2aTc0c4uiudM298It4it\nqkl5Qzh6YL92z0k9J5hUOLpwoytk3W1S1hOOLnhy9r8oS7WGF9zTNulnFKNv1NO8MlSfbSp178Uv\nPMpycCRab9IAAAAASUVORK5CYII=\n",
"prompt_number": 69,
"text": [
" 2 2\n",
" _ \u2502\u03b1\u2502 \u2502\u03b2\u2502 \n",
" \u03b1\u22c5\u03b2 - \u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\n",
" 2 2 \n",
"\u212f "
]
}
],
"prompt_number": 69
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"qapply((bra * qapply(a * ket))).doit()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$\\alpha e^{\\alpha \\overline{\\beta}} e^{- \\frac{\\left\\lvert{\\alpha}\\right\\rvert^{2}}{2}} e^{- \\frac{\\left\\lvert{\\beta}\\right\\rvert^{2}}{2}}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAJkAAAAcBAMAAACADwDiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nVP2vbLKfuNPwrvRFg6ZFqJ7l7C+9QpQ26gjm+QAAAABJRU5ErkJggg==\n",
"prompt_number": 70,
"text": [
" 2 2 \n",
" -\u2502\u03b1\u2502 -\u2502\u03b2\u2502 \n",
" _ \u2500\u2500\u2500\u2500\u2500\u2500 \u2500\u2500\u2500\u2500\u2500\u2500\n",
" \u03b1\u22c5\u03b2 2 2 \n",
"\u03b1\u22c5\u212f \u22c5\u212f \u22c5\u212f "
]
}
],
"prompt_number": 70
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"qapply(Dagger(ket) * Dagger(a) * a * ket, dagger=True)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$\\alpha \\alpha^{\\dagger}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAACIAAAATBAMAAAAHT8L5AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiZl2RCLdEO9Uu81m\nqzKufkATAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAsklEQVQYGWNgQAVXULlAHqaIOLqa1Ys60YQK\nxAvQRBjuoQswrCRC5CaGGrB7hIwcgBKMyiogaT0gDnNgb2WdwJrCUBEQycAgChRpZGBYxFzAIcDA\npWDOwABETN8ZGLQZGfQLGJg2CDAwAB3IsYCBQUKcoQtoUiPEvfVACfkJ7B+BIj+BZgAB/wGgCAPD\nH6DIB7AAA9MFBh59vgJdBoZDH55DhATPHGASZmCUOcOgaAASAQBNXiGRv1lbYQAAAABJRU5ErkJg\ngg==\n",
"prompt_number": 71,
"text": [
" \u2020\n",
"\u03b1\u22c5\u03b1 "
]
}
],
"prompt_number": 71
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"normal_ordered_form(a * Dagger(a))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$1 + {{a}^\\dagger} {a}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAEEAAAAVBAMAAADrxp6XAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzRAiu5mrdu/dZlSJ\nRDLkM64aAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3UlEQVQoFWNgwAFycYgjhAmrOIRQjJ3l6NqN\nXQIuqnBIAc5mEEIwkVh5CLbJJwQbieUNZ6uEwVQwGcAFgQyECgZ2mAruAmQV2QgODhVI4YGhgjmn\nbhIDQxzQDAgL05YbBiwfGBgOA1VAWBgqmLoYWB0YGIoYGKAsDBWsnxg4gW5WYGCAsqAqeAQFJRsF\nBQMYGDgcGPQfAK1AsNBdqn+AYT/IBAYGGAtDxQYGR24DsAooC10FYwHXNB6QAgYoizXjZxaYzwAN\nU6b0c2a1YBEEC6IApgLKw0bxLsAmygAAFlQ3VdFpO00AAAAASUVORK5CYII=\n",
"prompt_number": 72,
"text": [
"1 + BosonOperator(a,0)\u22c5BosonOperator(a,1)"
]
}
],
"prompt_number": 72
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"qapply(bra * a * Dagger(a) * ket, dagger=True)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${\\left\\langle \\beta\\right|} {a} {{a}^\\dagger} {\\left|\\alpha\\right\\rangle }$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAFEAAAAZBAMAAAC7qt8EAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
"prompt_number": 73,
"text": [
"\u27e8\u03b2\u2758\u22c5BosonOperator(a,1)\u22c5BosonOperator(a,0)\u22c5\u2758\u03b1\u27e9"
]
}
],
"prompt_number": 73
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"qapply(bra * normal_ordered_form(a * Dagger(a)) * ket, dagger=True)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$\\alpha e^{- \\frac{\\left\\lvert{\\alpha}\\right\\rvert^{2}}{2}} e^{- \\frac{\\left\\lvert{\\beta}\\right\\rvert^{2}}{2}} \\beta^{\\dagger} \\operatorname{transpose}{\\left (e^{\\alpha \\overline{\\beta}} \\right )} + e^{\\alpha \\overline{\\beta}} e^{- \\frac{\\left\\lvert{\\alpha}\\right\\rvert^{2}}{2}} e^{- \\frac{\\left\\lvert{\\beta}\\right\\rvert^{2}}{2}}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAcwAAAAqBAMAAADMn3aeAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
"prompt_number": 74,
"text": [
" 2 2 2 2 \n",
" -\u2502\u03b1\u2502 -\u2502\u03b2\u2502 -\u2502\u03b1\u2502 -\u2502\u03b2\u2502 \n",
" \u2500\u2500\u2500\u2500\u2500\u2500 \u2500\u2500\u2500\u2500\u2500\u2500 \u239b _\u239e _ \u2500\u2500\u2500\u2500\u2500\u2500 \u2500\u2500\u2500\u2500\u2500\u2500\n",
" 2 2 \u2020 \u239c \u03b1\u22c5\u03b2\u239f \u03b1\u22c5\u03b2 2 2 \n",
"\u03b1\u22c5\u212f \u22c5\u212f \u22c5\u03b2 \u22c5transpose\u239d\u212f \u23a0 + \u212f \u22c5\u212f \u22c5\u212f "
]
}
],
"prompt_number": 74
},
{
"cell_type": "heading",
"level": 3,
"metadata": {},
"source": [
"Fermion Fock states"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"ket = FermionFockKet(1)\n",
"\n",
"ket"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${\\left|1\\right\\rangle }$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAWBAMAAADQs7pMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVCKJRHYyzRC7mavv\n3WYouEVCAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAd0lEQVQIHWMQYgCDB0DSBMJcAmVWMDBkQ5ia\nPxgYeBtACjpXA5msD8BqWYFMjgI4k2ESgimMYPIimFwOICtA2hjYNsCZjAZwJqcCnKmJ0GYIYnLt\n+buPgeECWBSon4FlAZzJnQBntgNloL54CmSGATEQzIBQUBIAx4Yejz3P57MAAAAASUVORK5CYII=\n",
"prompt_number": 75,
"text": [
"\u27581\u27e9"
]
}
],
"prompt_number": 75
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"bra = FermionFockBra(0)\n",
"\n",
"bra"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${\\left\\langle 0\\right|}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAAWBAMAAAAyb6E1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZjKrmUTdVIm7Is12\nEO9EeINyAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAqElEQVQIHUXOOwrCQBCA4T8hMYkPkiPY2gVP\nEAR7Cw9g5xXEKo2dRWqr3MDtbNXawkJEEExuoYWCs7uQTLHzMcxjgQQTa3BTyw14M27bCQg7cOBe\naq5wakKluSNKCX7CriJO8d/CMKHK8L9CmdrnuB/hvOUUqtw2nCDOCPTYU44UOHpZVNKv8fSJIIcl\nr6GQBTwuY/MdWWFCqqOWvWtTdY4NB8rybJN5/6RlLJx2z3ytAAAAAElFTkSuQmCC\n",
"prompt_number": 76,
"text": [
"\u27e80\u2758"
]
}
],
"prompt_number": 76
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"(bra * ket).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": 77,
"text": [
"0"
]
}
],
"prompt_number": 77
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"c * ket"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${c} {\\left|1\\right\\rangle }$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAB8AAAAWBAMAAADKj1rCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXYyIt1Uze+rmRC7\nZkTTotXTAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAtUlEQVQYGWNgAAE2BzDFcAFCMTAwPoCw1qEI\nVDMwWCIL+PxmYGDdABUBatmzEijABDMEZAYTUIC7AKECLMBwBCQgqGQAU8GQCOS7JrAXwAVYgQKN\nDJwJcAGuBwxcQLPADgMZysA8gYHjALIAmwDDfgFkAR4HBn6gWxBm+DAwcE1g2IWwVgyoXPCuAshQ\nrhlf5zIwBAAFQABkLQjwLQBTCAFOAzSB7VA+XMslmAATVOlpmAASDQAX7DTp3RaMeQAAAABJRU5E\nrkJggg==\n",
"prompt_number": 78,
"text": [
"FermionOperator(c,1)\u22c5\u27581\u27e9"
]
}
],
"prompt_number": 78
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"qapply(c * ket)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${\\left|0\\right\\rangle }$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAWBAMAAADQs7pMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVCKJRHYyEJnN3Wbv\nu6uKc90MAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAqElEQVQIHWMQYgCDDUDSBMJsATPZmyMSGPLA\nTE4B9kUMXAUgBXsYGCwZWIGKTRjuMTDIF7BPADH/MTD4CzBcAjLZfwCZAQzCQCbTZwaGfAMGLgST\nwwGoACgKVMD2AKgNqFZegIHRAMhcy8Cwv4CBVwHIjGFg6GRg0AVqY+ASYD/GwGAIYrKbxiUwMCwA\nMcGApQHO5EuAMyuBclAFW4HMMCAGgusQCkoCAOVMIAjdmSrdAAAAAElFTkSuQmCC\n",
"prompt_number": 79,
"text": [
"\u27580\u27e9"
]
}
],
"prompt_number": 79
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"qapply(c * c * ket)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$0$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB\n5wID+0cG/gsMfN8Z5BUY+L4wzDdgYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxy\nlQEA9FISlFfRJtkAAAAASUVORK5CYII=\n",
"prompt_number": 80,
"text": [
"0"
]
}
],
"prompt_number": 80
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"qapply(Dagger(c) * ket)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$0$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB\n5wID+0cG/gsMfN8Z5BUY+L4wzDdgYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxy\nlQEA9FISlFfRJtkAAAAASUVORK5CYII=\n",
"prompt_number": 81,
"text": [
"0"
]
}
],
"prompt_number": 81
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"qapply(Dagger(c) * c * ket)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${\\left|1\\right\\rangle }$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAWBAMAAADQs7pMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVCKJRHYyzRC7mavv\n3WYouEVCAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAd0lEQVQIHWMQYgCDB0DSBMJcAmVWMDBkQ5ia\nPxgYeBtACjpXA5msD8BqWYFMjgI4k2ESgimMYPIimFwOICtA2hjYNsCZjAZwJqcCnKmJ0GYIYnLt\n+buPgeECWBSon4FlAZzJnQBntgNloL54CmSGATEQzIBQUBIAx4Yejz3P57MAAAAASUVORK5CYII=\n",
"prompt_number": 82,
"text": [
"\u27581\u27e9"
]
}
],
"prompt_number": 82
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"Dagger(c) * c"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{c}^\\dagger} {c}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAABsAAAATBAMAAACTqWsLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXYyIt1Uze+rmRC7\nZkTTotXTAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfUlEQVQYGWNgAIN5EApGonGlYeJg+vipNmT+\nBukNyFyGySg8hmN4udNQZZHsFVQyYFgMkgUxGFwT2AsYRIA8MIOhkYEzgUEbyAUzuH6D1AFdAWFw\nHABxgQDC2C8A4TFAGPwXGBgSQCIQBtcEhl1A84F6IQzBuwogHtBaIAMAohMbe/npeTMAAAAASUVO\nRK5CYII=\n",
"prompt_number": 83,
"text": [
"FermionOperator(c,0)\u22c5FermionOperator(c,1)"
]
}
],
"prompt_number": 83
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"qapply(Dagger(ket) * Dagger(c) * c * ket).doit()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$1$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAzRAiu5mrdu/dZmiL4QAAAAAJcEhZcwAA\nDsQAAA7EAZUrDhsAAAAjSURBVAgdY2BgEGJgYDDZxMCgEgYkGNhJJVgzdmYB9TEwAACPpQrvlUCH\ncAAAAABJRU5ErkJggg==\n",
"prompt_number": 84,
"text": [
"1"
]
}
],
"prompt_number": 84
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"qapply(bra * Dagger(c) * c * Dagger(bra)).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": 85,
"text": [
"0"
]
}
],
"prompt_number": 85
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"TensorProduct spaces"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"I2 = TensorProduct(IdentityOperator(), IdentityOperator())\n",
"\n",
"I2"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{\\mathcal{I}}}\\otimes {{\\mathcal{I}}}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAADUAAAARBAMAAACP9fljAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZrvd780yIhBEmVSr\ndokAK0XoAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA7ElEQVQYGV3Pr04DQRDH8a+AdLttEAiQKDwC\nVYnrE4AEQlJXzh1BXZBNGrBUbTBHqu4R0DwCBhQJIdQVHMvsn4EeK2ZnPr/s7h2wue/9ktZSOt0a\n7YyqVqRkhjBoJfxS55jeV8rM0+IjXLBC9PdSNqsw76lVYs1F6B7KdtXEPhOcp7lvC4y7j1kmeK7i\nvE3NhGnsM8FuHHHYC8dGHDLBd8peYE7OMtH91HNmXmDDoITNv+DktTplSlw3yBfCbUdeK07COaXy\nriwfA9hLKa8PUpTOvPcHNwKYIynrcsMfBU5r0tCrdfi/jxdvq/QD4c9F+IJPnbcAAAAASUVORK5C\nYII=\n",
"prompt_number": 86,
"text": [
"I\u2a02 I"
]
}
],
"prompt_number": 86
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"psi = TensorProduct(BosonFockKet(1),BosonFockKet(2))\n",
"\n",
"psi"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{\\left|1\\right\\rangle }}\\otimes {{\\left|2\\right\\rangle }}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAEUAAAAWBAMAAABkuUxDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n8yumlo+HmCAyidOh/D/HH3EAUUloLL0NAAAAAElFTkSuQmCC\n",
"prompt_number": 87,
"text": [
"\u27581\u27e9\u2a02 \u27582\u27e9"
]
}
],
"prompt_number": 87
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"I2 * psi"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{\\mathcal{I}}}\\otimes {{\\mathcal{I}}} {{\\left|1\\right\\rangle }}\\otimes {{\\left|2\\right\\rangle }}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAHwAAAAWBAMAAADwX+WxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nn0XpMFBYwXN1GvItSoeBIh187fwMyOO//AkPUOQAAAAASUVORK5CYII=\n",
"prompt_number": 88,
"text": [
"I\u2a02 I\u22c5\u27581\u27e9\u2a02 \u27582\u27e9"
]
}
],
"prompt_number": 88
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"qapply(I2 * psi)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{\\left|1\\right\\rangle }}\\otimes {{\\left|2\\right\\rangle }}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAEUAAAAWBAMAAABkuUxDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n8yumlo+HmCAyidOh/D/HH3EAUUloLL0NAAAAAElFTkSuQmCC\n",
"prompt_number": 89,
"text": [
"\u27581\u27e9\u2a02 \u27582\u27e9"
]
}
],
"prompt_number": 89
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"a1 = TensorProduct(BosonOperator(\"a\"), IdentityOperator())\n",
"a2 = TensorProduct(IdentityOperator(), BosonOperator(\"a\"))\n",
"\n",
"a1, a2"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$\\begin{pmatrix}{{a}}\\otimes {{\\mathcal{I}}}, & {{\\mathcal{I}}}\\otimes {{a}}\\end{pmatrix}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAIoAAAAaBAMAAACaz9kUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg0AAAAASUVORK5CYII=\n",
"prompt_number": 90,
"text": [
"(BosonOperator(a,1)\u2a02 I, I\u2a02 BosonOperator(a,1))"
]
}
],
"prompt_number": 90
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"a1 * psi"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{a}}\\otimes {{\\mathcal{I}}} {{\\left|1\\right\\rangle }}\\otimes {{\\left|2\\right\\rangle }}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAHkAAAAWBAMAAAAWdi71AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n0m4AAAAASUVORK5CYII=\n",
"prompt_number": 91,
"text": [
"BosonOperator(a,1)\u2a02 I\u22c5\u27581\u27e9\u2a02 \u27582\u27e9"
]
}
],
"prompt_number": 91
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"psi1 = qapply(a1 * psi)\n",
"\n",
"psi1"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{\\left|0\\right\\rangle }}\\otimes {{\\left|2\\right\\rangle }}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAEUAAAAWBAMAAABkuUxDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nuaXhGx0AAAAASUVORK5CYII=\n",
"prompt_number": 92,
"text": [
"\u27580\u27e9\u2a02 \u27582\u27e9"
]
}
],
"prompt_number": 92
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"psi2 = qapply(a2 * psi)\n",
"\n",
"psi2"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$\\sqrt{2} {{\\left|1\\right\\rangle }}\\otimes {{\\left|1\\right\\rangle }}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAGIAAAAaBAMAAAC++pSRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nhfkzpm96qKJl7qQAAAAASUVORK5CYII=\n",
"prompt_number": 93,
"text": [
" ___ \n",
"\u2572\u2571 2 \u22c5\u27581\u27e9\u2a02 \u27581\u27e9"
]
}
],
"prompt_number": 93
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"qapply(Dagger(psi1) * psi2).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": 94,
"text": [
"0"
]
}
],
"prompt_number": 94
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"qapply(Dagger(psi2) * psi2).doit()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$2 \\cdot {1}\\otimes {1}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAEUAAAAQBAMAAACy4K9eAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/klEQVQoFWNgVDJ2YIAD7gA4E8oAiYQxsP+A\nizPao6sBi6QyMMyEqWE1yoeoYT1UXgNmQURmMTDkO8AUMZyHqGkKYGCthQiCRPYHYKrh2AiUD74A\nVgTVtR6iGSQEEWFjEmBgVVBFUsPxHcxBEnnIoMgQxBCDJMJlAOJAzIKYo8DAJKrAwI6kRhnEZv2N\nJFLAwKDEgKyGJwEsuwxJjQIDq5IAAxNC5BIDowCYByJgdgUBXYRQw6LAwIOupoMR6BoBP7BOkC6p\n8qIsMAcuwsDAJATklCTARfb///8LzAERavOXPQBSrFuBBDfYdKgIkI8Ogi4w8CqiC6Lz3csrEEIA\nRXM+O2NZdEEAAAAASUVORK5CYII=\n",
"prompt_number": 95,
"text": [
"2\u22c51\u2a02 1"
]
}
],
"prompt_number": 95
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"e = TensorProduct(IdentityOperator(), BosonFockKet(1))\n",
"\n",
"e"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${{\\mathcal{I}}}\\otimes {{\\left|1\\right\\rangle }}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAD0AAAAWBAMAAACBJ4kvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZrvd780yIhBEmVSr\ndokAK0XoAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABTElEQVQoFW2QvUvDUBTFD1bty2sq4qCTOImO\nGZwc3dzc7FhF6CIaukScQsElILrq9HCpFodsbtKCk3+Ag4udhCJ2ET8m473v5aU+6B3uuef8kptL\ngHF1Z8JZLXNrWfbpPjUw9oZld76x1IhNYDvzGrBPIjaBdZtbJd78AUoxUK6j8m1y8Tx8p4RqgNYt\n8ao5wA90iMsY4i3nqBIXW9pNKi3eNkkr5Zn2M8cKGxzqDL4MIdQ1J5bfs8FLrGUBbSQ45dnykgbL\nukNBHinM/OfynN2v4X2gA5dPdwl5X9SoFEQnhOTR7q/0yciAI+YJXeDwiZDysxR0OXBRpq+HO/yo\nfb9Jc3QVRY8cymNqr11qBae/tZdl2YY+U9SITPEm4vLhowcEbIpKUlTa2tF+Lk9pKdrBMAe5+PUC\nuUPOT9x05HK+OErc6cnYVTcd5/4AEJ5dESW0s2YAAAAASUVORK5CYII=\n",
"prompt_number": 96,
"text": [
"I\u2a02 \u27581\u27e9"
]
}
],
"prompt_number": 96
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"qapply(Dagger(psi2) * e).doit()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$\\sqrt{2} {{\\left\\langle 1\\right|}}\\otimes {1}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAFUAAAAaBAMAAAA01Q3QAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nf027dT818k/smmFTt61f8V/1BedpaAa5IiX4AAAAAElFTkSuQmCC\n",
"prompt_number": 97,
"text": [
" ___ \n",
"\u2572\u2571 2 \u22c5\u27e81\u2758\u2a02 1"
]
}
],
"prompt_number": 97
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"qapply(qapply(Dagger(psi2) * e) * TensorProduct(BosonFockKet(1), IdentityOperator())).doit()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$\\sqrt{2} \\cdot {1}\\otimes {{\\mathcal{I}}}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAFsAAAAWBAMAAABd3v0YAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n3QAAAABJRU5ErkJggg==\n",
"prompt_number": 98,
"text": [
" ___ \n",
"\u2572\u2571 2 \u22c51\u2a02 I"
]
}
],
"prompt_number": 98
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"bra = TensorProduct(BosonFockBra(1),BosonFockBra(2))\n",
"ket = TensorProduct(BosonFockKet(1),BosonFockKet(2))"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 99
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"qapply(bra * ket).doit()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$${1}\\otimes {1}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAACsAAAAQBAMAAAB9zpudAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzRAiu5mrdu/dZolE\nVDLjuNgcAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAs0lEQVQYGWNgYBBiQAOMCkABk09oosz+QGGV\nMKiwevTuIpACJtd8oDADO0SYJ4iBQfsCSJyhXgFIQIUPGgAV+oJEUYQTghkYVFkV0IS5DFgNmAKY\nCtCEWRkYglUZGDagCXMzMLAGYAoDVasCTQfKoFjJysAUwGqAIcxzwJSBYTPfAjTVDMFA/dzPQKJg\nd7Nm/MwCsRUVgIQwEDOE9U9bAKLBgCmWgeGqAYyHoPl27z4A5wEAqfAqOS1nfgAAAAAASUVORK5C\nYII=\n",
"prompt_number": 100,
"text": [
"1\u2a02 1"
]
}
],
"prompt_number": 100
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"e = qapply(qapply(TensorProduct(IdentityOperator(), BosonCoherentBra(2)) * \n",
" TensorProduct(a, normal_ordered_form(Dagger(b)*b*Dagger(b)*b)) * \n",
" TensorProduct(IdentityOperator(), BosonCoherentKet(2))))\n",
"\n",
"e"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$2 {{\\mathcal{I}}}\\otimes {{\\left\\langle 2\\right|}} {{a}}\\otimes \\left({{{b}^\\dagger} {\\left|2\\right\\rangle }}\\right) + 4 {{\\mathcal{I}}}\\otimes {{\\left\\langle 2\\right|}} {{a}}\\otimes \\left({\\left({{b}^\\dagger}\\right)^{2} {\\left|2\\right\\rangle }}\\right)$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAAY8AAAAnCAMAAADAfDTiAAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\ns3+SIv/Hd59h1PEJ4L9TnDorlQnuQAAAAABJRU5ErkJggg==\n",
"prompt_number": 101,
"text": [
" \n",
"2\u22c5I\u2a02 \u27e82\u2758\u22c5BosonOperator(a,1)\u2a02 (BosonOperator(b,0)\u22c5\u27582\u27e9) + 4\u22c5I\u2a02 \u27e82\u2758\u22c5BosonOperator\n",
"\n",
" \u239b 2 \u239e\n",
"(a,1)\u2a02 \u239dBosonOperator(b,0) \u22c5\u27582\u27e9\u23a0"
]
}
],
"prompt_number": 101
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"ket = TensorProduct(IdentityOperator(), BosonFockKet(2))\n",
"bra = Dagger(ket)\n",
"\n",
"qapply(qapply(bra * TensorProduct(a, normal_ordered_form(Dagger(b)*b*Dagger(b)*b)) * ket)).doit()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"latex": [
"$$4 {{a}}\\otimes {1}$$"
],
"metadata": {},
"output_type": "pyout",
"png": "iVBORw0KGgoAAAANSUhEUgAAADgAAAAQBAMAAACxV6F2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMpndu3bvImbNiRBU\nq0Qb3U6NAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABD0lEQVQYGWNgYGAQAmIsYDZYzASLDANDyW+Q\nMGsKkuRWrVXRYO4WXbBkxRSEJJ82A8OuA2A+K1hyAZLkywIGBm4fhCRHAVCS/VagNUjogjoDwyam\nDSAmWOdWBqDkswKWD0ABjgKmAm4F7gC45AKgJHcGA5MDUISJgUF9EwPDAiATrJNHACjJ9JuBGaSc\nCyivgCS5iwEoyebAsF8AKAnUuQloK1AeovPs3bvfru5fwHCeYQNIkluBqQAhCVSTwrD/AIMnTwED\nA9+DIgaG5YwNMJ1A+gsDZwCHLdA+oHMUgBbLglgQrzAs+X+D++rC2jiQyE6g2QzTQSyma59vgGgk\nwK3DwHAIaD52wLhq1QO4DACvBkXDDCHzEAAAAABJRU5ErkJggg==\n",
"prompt_number": 102,
"text": [
"4\u22c5BosonOperator(a,1)\u2a02 1"
]
}
],
"prompt_number": 102
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Versions"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#%install_ext http://raw.github.com/jrjohansson/version_information/master/version_information.py\n",
"%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>3.3.2+ (default, Oct 9 2013, 14:50:09) [GCC 4.8.1]</td></tr><tr><td>IPython</td><td>1.1.0</td></tr><tr><td>OS</td><td>posix [linux]</td></tr><tr><td>sympy</td><td>0.7.4.1-git</td></tr><tr><td colspan='2'>Fri Jan 10 11:53:51 2014 CET</td></tr></table>"
],
"json": [
"{ \"Software versions\" : [{ \"module\" : \"Python\", \"version\" : \"3.3.2+ (default, Oct 9 2013, 14:50:09) [GCC 4.8.1]\" }, { \"module\" : \"IPython\", \"version\" : \"1.1.0\" }, { \"module\" : \"OS\", \"version\" : \"posix [linux]\" }, { \"module\" : \"sympy\", \"version\" : \"0.7.4.1-git\" } ] }"
],
"latex": [
"\\begin{table}\n",
"\\begin{tabular}{|l|l|}\\hline\n",
"{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n",
"Python & 3.3.2+ (default, Oct 9 2013, 14:50:09) [GCC 4.8.1] \\\\ \\hline\n",
"IPython & 1.1.0 \\\\ \\hline\n",
"OS & posix [linux] \\\\ \\hline\n",
"sympy & 0.7.4.1-git \\\\ \\hline\n",
"\\hline \\multicolumn{2}{|l|}{Fri Jan 10 11:53:51 2014 CET} \\\\ \\hline\n",
"\\end{tabular}\n",
"\\end{table}\n"
],
"metadata": {},
"output_type": "pyout",
"prompt_number": 103,
"text": [
"Software versions\n",
"Python 3.3.2+ (default, Oct 9 2013, 14:50:09) [GCC 4.8.1]\n",
"IPython 1.1.0\n",
"OS posix [linux]\n",
"sympy 0.7.4.1-git\n",
"\n",
"Fri Jan 10 11:53:51 2014 CET"
]
}
],
"prompt_number": 103
}
],
"metadata": {}
}
]
}
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