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Notebook[{
Cell[BoxData[{
    \(\("\<Perturbation Theory\>";\)\[IndentingNewLine]\[IndentingNewLine]\), \
"\[IndentingNewLine]", 
    \(\("\<H=H0+H1\>";\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(H0 = {{\[CapitalDelta], 
            0}, {0, \(-\[CapitalDelta]\)}};\)\), "\[IndentingNewLine]", 
    \(\(H1 = {{0, 1}, {1, 0}};\)\)}], "Input"],

Cell[BoxData[{
    \(\(H = H0 + \[Lambda]\ H1;\)\), "\[IndentingNewLine]", 
    \(MatrixForm[H]\)}], "Input"],

Cell[BoxData[
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Cell[BoxData[{
    \(eigenvalue0 = sol[\([1, 1]\)]\), "\[IndentingNewLine]", 
    \(eigenvalue1 = sol[\([1, 2]\)]\)}], "Input"],

Cell[BoxData[
    \(\("\<Lets define  Tan[\[Theta]]=\[Lambda]/\[CapitalDelta]\>";\)\)], \
"Input"],

Cell[BoxData[{
    \(\(eigenstate0 = {Cot[\[Theta]] - 1/Sin[\[Theta]], 
          1};\)\), "\[IndentingNewLine]", 
    \(\(eigenstate1 = {1, 
          Tan[\[Theta]]/\((1 + Sqrt[1 + Tan[\[Theta]]^2])\)};\)\)}], "Input"],

Cell[BoxData[{
    \(MatrixForm[eigenstate0]\), "\[IndentingNewLine]", 
    \(MatrixForm[eigenstate1]\)}], "Input"],

Cell[BoxData[
    \(\("\< Normalizations ?\>";\)\)], "Input"],

Cell[BoxData[{
    \(\[IndentingNewLine]\(normeigenstate0 = 
        Simplify[
          FullSimplify[
              eigenstate0/
                Sqrt[eigenstate0 . 
                    eigenstate0]]\ \  /. \ {\@\(\(Sec[\[Theta]/2]\^2\)\(\ \
\)\) \[Rule] Sec[\[Theta]/2], 
              1\/\@Sec[\[Theta]\/2]\^2 \[Rule] 
                1/Sec[\[Theta]/2]}];\)\), "\[IndentingNewLine]", 
    \(\(normeigenstate1 = 
        Simplify[
          Simplify[
              FullSimplify[
                  eigenstate1/
                    Sqrt[eigenstate1 . 
                        eigenstate1]]\  /. \ \@Sec[\[Theta]]\^2 \[Rule] 
                  Sec[\[Theta]]]\  /. \ {\@\(\(Sec[\[Theta]/2]\^2\)\(\ \)\) \
\[Rule] Sec[\[Theta]/2], 
              1\/\@Sec[\[Theta]\/2]\^2 \[Rule] 
                1/Sec[\[Theta]/2]}];\)\)}], "Input"],

Cell[BoxData[{
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    \(MatrixForm[normeigenstate1]\)}], "Input"],

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[{
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\[Theta]]\), "\n", 
    \(FullSimplify[\((H . eigenstate0 \[Equal] 
            eigenvalue0\ eigenstate0)\)\  /. \ \[Lambda] \[Rule] \ \
\[CapitalDelta]\ Tan[\[Theta]]]\), "\[IndentingNewLine]", 
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\[CapitalDelta]\ Sec[\[Theta]]]\)}], "Input"],

Cell[BoxData[{
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            eigenvalue1\ eigenstate1)\)\ \  /. \ \[Lambda] \[Rule] \
\[CapitalDelta]\ Tan[\[Theta]]]\), "\[IndentingNewLine]", 
    \(FullSimplify[%\  /. \ {\@\(\[CapitalDelta]\^2\ Sec[\[Theta]]\^2\) \
\[Rule] \[CapitalDelta]\ Sec[\[Theta]], \@Sec[\[Theta]]\^2 \[Rule] 
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Cell[BoxData[
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  Editable->False],

Cell[BoxData[
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\(\(\[Alpha]\)\(>\)\)\)\/\((E\_\[Alpha] - E\_\[Beta])\)\)\>\""\)], "Input"],

Cell[BoxData[{
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    \(\("\< There is only one |\[Beta]> state =beta\>";\)\), "\
\[IndentingNewLine]", 
    \(\("\<with\>";\)\), "\[IndentingNewLine]", 
    \(\ \ \(E0alpha = \(-\[CapitalDelta]\)\ ;\)\), "\[IndentingNewLine]", 
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Cell[BoxData[{
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Cell[BoxData[{
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Cell[BoxData[{
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    \(MatrixForm[psi1]\)}], "Input"],

Cell[BoxData[
    \(\("\< Lets compare this with exact soln as \[Lambda]<<1 \>";\)\)], \
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Cell[BoxData[{
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    \(\("\<Since Tan[\[Theta]]=\[Lambda]/\[CapitalDelta] or \
\[Theta]=\[Lambda]/\[CapitalDelta], if \[Lambda]<<\[CapitalDelta]\>";\)\), "\
\[IndentingNewLine]", 
    \(MatrixForm[
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\[IndentingNewLine]", 
    \(\)}], "Input"],

Cell[BoxData[{
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\[Theta]=\[Lambda]/\[CapitalDelta], if \[Lambda]<<\[CapitalDelta]\>";\)\), "\
\[IndentingNewLine]", 
    \(MatrixForm[
      Normal[%%]\  /. \ \[Theta] \[Rule] \[Lambda]/\[CapitalDelta]]\), "\
\[IndentingNewLine]", 
    \(\)}], "Input"],

Cell[BoxData[
    \(\(\("\< For the energy\>";\)\(\[IndentingNewLine]\)
    \)\)], "Input"],

Cell[BoxData[
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\[Lambda] <\[Alpha]|H1|\[Alpha]> + \[Lambda]^2 \!\(\(\[Sum] | \)\+\[Beta]\)<\
\[Alpha]|H1|\[Beta]>|^2/(\!\(E\_\[Alpha]\)-\!\(E\_\[Beta]\))\>\"";\)\)\)], \
"Input"],

Cell[BoxData[{
    \(e0perturb = 
      e0alpha + \ \[Lambda]\ alpha . H1 . 
            alpha\  + \ \[Lambda]^2\ \((\ alpha . H1 . beta)\)^2/\((e0alpha - 
                e0beta)\)\), "\[IndentingNewLine]", 
    \(e1perturb = 
      e0beta + \ \[Lambda]\ beta . H1 . 
            beta\  + \ \[Lambda]^2\ \((\ beta . H1 . alpha)\)^2/\((e0beta - 
                e0alpha)\)\[IndentingNewLine]\[IndentingNewLine]\
\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\("\< Note that the first order shift vanishes; Why?\>";\)\
\[IndentingNewLine]\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\)}], "Input"],

Cell[BoxData[{
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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        1}]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[{
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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