(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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In the \ presence of an electric field along the z-axis, the Hamiltonian is:\ \>", "Text", FontSize->14], Cell[BoxData[ RowBox[{"H1", "=", RowBox[{"(", GridBox[{ {"0", "dE"}, {"dE", "\[CapitalDelta]"} }], ")"}]}]], "Input"], Cell["\<\ We can find the eigenvalues for this system:\ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[\(-\[Lambda]\)\ \((\[CapitalDelta] - \[Lambda])\) - \((dE)\)\^2 \ \[Equal] 0, \[Lambda]]\)], "Input"], Cell[BoxData[ \({{\[Lambda] \[Rule] 1\/2\ \((\[CapitalDelta] - \@\(4\ dE\^2 + \[CapitalDelta]\^2\))\)}, \ {\[Lambda] \[Rule] 1\/2\ \((\[CapitalDelta] + \@\(4\ dE\^2 + \[CapitalDelta]\^2\))\)}}\ \)], "Output"] }, Open ]], Cell[TextData[{ "Of course, we have just reproduced the familiar ", StyleBox["quadratic Stark effect", FontColor->RGBColor[1, 0, 0]], ". Note that when |dE|>>|\[CapitalDelta]|, we can neglect \[CapitalDelta], \ and the Stark shifts are linear in dE. Naturally, this has nothing to do with \ T-violation that forbids linear Stark shifts for non-degenerate systems. \ (There are handbooks that list \"permanent electric dipole moments\" of \ molecules. In fact these are not so, but the condition \ |dE|>>|\[CapitalDelta]| is satisfied for these molecules at very low electric \ fields because of the small intervals between states of opposite parity.) \n\n\ Let us now see how the presence of various interactions changes the \ Hamiltonian. Let us start with P-odd,T-even weak interaction. This \ interaction mixes levels of opposite parity, thus the non-zero matrix element \ corresponding to it should appear off the diagonal. For reasons that will \ become clear very shortly, the matrix element should be pure imaginary, \ therefore, we have:" }], "Text", FontSize->14], Cell[BoxData[ RowBox[{"H2", "=", RowBox[{"(", GridBox[{ {"0", \(dE + \[ImaginaryI]\ \[Delta]\)}, {\(dE - \[ImaginaryI]\ \[Delta]\), "\[CapitalDelta]"} }], ")"}]}]], "Input"], Cell["\<\ We write -\[ImaginaryI] \[Delta] in the lower left element, so the resulting \ matrix is Hermitian (so the eigenvalues, i.e. energies, come out real):\ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[\(-\[Lambda]\)\ \((\[CapitalDelta] - \[Lambda])\) - \((dE + \ \[ImaginaryI]\ \[Delta])\)\ \((dE - \[ImaginaryI]\ \[Delta])\) \[Equal] 0, \[Lambda]]\)], "Input"], Cell[BoxData[ \({{\[Lambda] \[Rule] 1\/2\ \((\[CapitalDelta] - \@\(4\ dE\^2 + 4\ \[Delta]\^2 + \ \[CapitalDelta]\^2\))\)}, {\[Lambda] \[Rule] 1\/2\ \((\[CapitalDelta] + \@\(4\ dE\^2 + 4\ \[Delta]\^2 + \ \[CapitalDelta]\^2\))\)}}\)], "Output"] }, Open ]], Cell[TextData[{ "We see that the ", StyleBox["P-odd,T-even interaction", FontColor->RGBColor[1, 0, 0]], ", although it mixes the two oppsite parity states, does not lead to linear \ (i.e. first-order in dE) Stark shifts. For completeness, let us also add \ terms that describe decay of the states A and B. Note that in this case, the \ resulting effective Hamiltonian is non-Hermitian." }], "Text", FontSize->14], Cell[BoxData[ RowBox[{"H3", "=", RowBox[{"(", GridBox[{ {\(\(-\[ImaginaryI]\)\ \[CapitalGamma]a/ 2\), \(dE + \[ImaginaryI]\ \[Delta]\)}, {\(dE - \[ImaginaryI]\ \[Delta]\), \(\[CapitalDelta] - \ \[ImaginaryI]\ \[CapitalGamma]b/2\)} }], ")"}]}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[\((\(-\[ImaginaryI]\)\ \[CapitalGamma]a/ 2 - \[Lambda])\)\ \((\[CapitalDelta] - \[ImaginaryI]\ \ \[CapitalGamma]b/ 2 - \[Lambda])\) - \((dE + \[ImaginaryI]\ \[Delta])\)\ \ \((dE - \[ImaginaryI]\ \[Delta])\) \[Equal] 0, \[Lambda]]\)], "Input"], Cell[BoxData[ \({{\[Lambda] \[Rule] 1\/4\ \((\(-\[ImaginaryI]\)\ \[CapitalGamma]a - \[ImaginaryI]\ \ \[CapitalGamma]b + 2\ \[CapitalDelta] - \@\(16\ dE\^2 - \[CapitalGamma]a\^2 + 2\ \ \[CapitalGamma]a\ \[CapitalGamma]b - \[CapitalGamma]b\^2 + 16\ \[Delta]\^2 + \ 4\ \[ImaginaryI]\ \[CapitalGamma]a\ \[CapitalDelta] - 4\ \[ImaginaryI]\ \ \[CapitalGamma]b\ \[CapitalDelta] + 4\ \[CapitalDelta]\^2\))\)}, {\[Lambda] \ \[Rule] 1\/4\ \((\(-\[ImaginaryI]\)\ \[CapitalGamma]a - \[ImaginaryI]\ \ \[CapitalGamma]b + 2\ \[CapitalDelta] + \@\(16\ dE\^2 - \[CapitalGamma]a\^2 + 2\ \ \[CapitalGamma]a\ \[CapitalGamma]b - \[CapitalGamma]b\^2 + 16\ \[Delta]\^2 + \ 4\ \[ImaginaryI]\ \[CapitalGamma]a\ \[CapitalDelta] - 4\ \[ImaginaryI]\ \ \[CapitalGamma]b\ \[CapitalDelta] + 4\ \[CapitalDelta]\^2\))\)}}\)], "Output"] }, Open ]], Cell[TextData[{ "Once again, there are no linear Stark shifts. Finally, the way to \ introduce the ", StyleBox["P-odd,T-odd interaction", FontColor->RGBColor[1, 0, 0]], " leading to a permanent electric dipole moment (EDM) is:" }], "Text", FontSize->14], Cell[BoxData[ RowBox[{"H4", "=", RowBox[{"(", GridBox[{ {"0", \(dE + \[Epsilon]\)}, {\(dE + \[Epsilon]\), "\[CapitalDelta]"} }], ")"}]}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[\(-\[Lambda]\)\ \((\[CapitalDelta] - \[Lambda])\) - \((dE + \ \[Epsilon])\)\^2 \[Equal] 0, \[Lambda]]\)], "Input"], Cell[BoxData[ \({{\[Lambda] \[Rule] 1\/2\ \((\[CapitalDelta] - \@\(4\ dE\^2 + \[CapitalDelta]\^2 + 8\ \ dE\ \[Epsilon] + 4\ \[Epsilon]\^2\))\)}, {\[Lambda] \[Rule] 1\/2\ \((\[CapitalDelta] + \@\(4\ dE\^2 + \[CapitalDelta]\^2 + 8\ \ dE\ \[Epsilon] + 4\ \[Epsilon]\^2\))\)}}\)], "Output"] }, Open ]], Cell["\<\ These eigenvalues correspond to linear Stark shifts. Indded,\ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(Series[\@\(4\ dE\^2 + \[CapitalDelta]\^2 + 8\ dE\ \[Epsilon] + 4\ \ \[Epsilon]\^2\), {\[Epsilon], 0, 1}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\(\@\(4\ dE\^2 + \[CapitalDelta]\^2\)\), "+", \(\(4\ dE\ \[Epsilon]\)\/\@\(4\ dE\^2 + \[CapitalDelta]\^2\)\), "+", InterpretationBox[\(O[\[Epsilon]]\^2\), SeriesData[ \[Epsilon], 0, {}, 0, 2, 1]]}], SeriesData[ \[Epsilon], 0, { Power[ Plus[ Times[ 4, Power[ dE, 2]], Power[ \[CapitalDelta], 2]], Rational[ 1, 2]], Times[ 4, dE, Power[ Plus[ Times[ 4, Power[ dE, 2]], Power[ \[CapitalDelta], 2]], Rational[ -1, 2]]]}, 0, 2, 1]]], "Output"] }, Open ]], Cell["\<\ Now it is clear why we wrote the P-odd,T-even weak interaction matrix \ element as pure imaginary. This is exactly to avoid the appearance of an EDM \ and T-violation.\ \>", "Text", FontSize->14] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, WindowSize->{811, 653}, WindowMargins->{{10, Automatic}, {6, Automatic}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic} ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1739, 51, 49, 1, 103, "Title"], Cell[1791, 54, 259, 5, 54, "Text"], Cell[2053, 61, 164, 5, 41, "Input"], Cell[2220, 68, 84, 3, 34, "Text"], Cell[CellGroupData[{ Cell[2329, 75, 127, 2, 31, "Input"], Cell[2459, 79, 234, 5, 42, "Output"] }, Open ]], Cell[2708, 87, 1084, 18, 174, "Text"], Cell[3795, 107, 220, 5, 43, "Input"], Cell[4018, 114, 190, 4, 34, "Text"], Cell[CellGroupData[{ Cell[4233, 122, 195, 3, 30, "Input"], Cell[4431, 127, 268, 5, 42, "Output"] }, Open ]], Cell[4714, 135, 425, 9, 74, "Text"], Cell[5142, 146, 318, 7, 43, "Input"], Cell[CellGroupData[{ Cell[5485, 157, 308, 5, 30, "Input"], Cell[5796, 164, 851, 13, 76, "Output"] }, Open ]], Cell[6662, 180, 265, 7, 54, "Text"], Cell[6930, 189, 194, 5, 41, "Input"], Cell[CellGroupData[{ Cell[7149, 198, 140, 2, 31, "Input"], Cell[7292, 202, 312, 5, 42, "Output"] }, Open ]], Cell[7619, 210, 100, 3, 34, "Text"], Cell[CellGroupData[{ Cell[7744, 217, 137, 2, 34, "Input"], Cell[7884, 221, 696, 20, 46, "Output"] }, Open ]], Cell[8595, 244, 208, 5, 54, "Text"] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)