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Re: Commutators with boson operators

  • To: mathgroup at smc.vnet.net
  • Subject: [mg97878] Re: Commutators with boson operators
  • From: Sotirios Bonanos <sbonano at inp.demokritos.gr>
  • Date: Tue, 24 Mar 2009 05:33:29 -0500 (EST)

Hi there, 

This is precisely the type of problems that the symbolic matrix capabilities of my EDC package (http://www.inp.demokritos.gr/~sbonano/EDC/) can handle. It is rather slow but does what you want. Download the code (matrixEDC.m), put it in an appropriate directory, and evaluate the attached notebook. 

Sotirios Bonanos 


> 
Hello, 
> I need some help. 
> 
> I want to write code that Mathematica would calculate commutators [A,B], [A,[B,C]] and so on..., 
> 
> where A,B,C,D,... are functions like 
> 
> A = a+ b*creat + c*annih + d*creat**annih + e*annih**creat +  f*creat^2+g*annih^2+... 
> 
> where a,b,c,d,... are usual complex numbers; 
> 
> and annih & creat are boson noncommutative operators 
> that satisfy commutation relation: 
> [annih, creat] = 1. 
> 
> Thank you in advance! 

CreatAnnih.nb

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Cell[BoxData[
    \(<< matrixEDC.m\)], "Input"],

Cell[BoxData[
    \(DeclareMatrixForms[{0, annih[_], annih[_]}, {0, creat[_], 
        creat[_]}]\)], "Input"],

Cell[BoxData[
    \(annih[n]\ and\ creat[
          n]\ will\ denote\ the\ nth\ powers\ of\ your\ operators . \ \ \
\[IndentingNewLine]Wedge \((\[Wedge])\)\ is\ used\ to\ denote\ non - 
      commutative\ \ multiplication\)], "Text",
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    \(Wedge[creat[1], annih[1]]\)], "Input"],

Cell["\<\
Give rules for simplifying Wedge products of powers of creat, annih\
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Cell[BoxData[
    \(creat[0] = 1; annih[0] = 1;\)], "Input"],

Cell[BoxData[
    \(Wedge[creat[m_], creat[n_]] := creat[m + n]\)], "Input"],

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    \(Wedge[annih[m_], annih[n_]] := annih[m + n]\)], "Input"],

Cell[BoxData[
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Cell["\<\
The last rule implements the commutator \
creat[1]\[Wedge]annih[1]-annih[1]\[Wedge]creat[1] = -h0; (h0 = 1 in your \
case) to put annih[] operators before creat[] ones. If you prefer to have \
creat[] before annih[] use:
        Wedge[annih[m_],creat[n_]]:=Wedge[annih[m-1], \
creat[1]\[Wedge]annih[1]+h0, creat[n-1]]/;m>0&&n>0\
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Cell["\<\
The function reWrite collects identical annih, creat powers / \
products\
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      reWrite[Sum[
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    \(\[IndentingNewLine]\)], "Input"],

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    \(comm[x_, y_] := reWrite[x\[Wedge]y - y\[Wedge]x]\)], "Input"],

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    \(comm[A, CC]\)], "Input"],

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