(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 7681, 245]*) (*NotebookOutlinePosition[ 8446, 271]*) (* CellTagsIndexPosition[ 8402, 267]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Hints on Problems 8.1, 8.3, 8.4, 8.7 (Engel)", "Title"], Cell["8.1", "Subtitle"], Cell[TextData[{ "Strategy.", StyleBox[" The problem gives you ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\[Nu]\&_\ = \ 2170\ cm\^\(-1\)\)], FontWeight->"Plain"], StyleBox[". It asks for \[Nu], \[Tau], and ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[\(E\_0\), FontWeight->"Plain"], TraditionalForm]], FontWeight->"Plain"], StyleBox[", where:\n\n\[Nu] = c", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[ OverscriptBox[ StyleBox["\[Nu]", FontWeight->"Plain"], "_"], FontWeight->"Plain"], TraditionalForm]]], "\n\n", StyleBox["\[Tau] = ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[\(1\/\[Nu]\), FontWeight->"Plain"], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ StyleBox[\(E\_0\ = \ \(h\[Nu](n + 1\/2)\ = \ h\[Nu]\/2\)\), FontWeight->"Plain"], TraditionalForm]]], "\n" }], "Text", FontWeight->"Bold"], Cell[CellGroupData[{ Cell["8.3", "Subtitle"], Cell[TextData[{ StyleBox["Strategy.", FontWeight->"Bold"], " The problem asks for ratios of state populations. These are obtained \ using the Boltzmann equation. To use this, you need ", Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]\ E\)]], ", ", StyleBox["T", FontSlant->"Italic"], ", and degeneracies of vibrational quantum states.\n\nVibration quantum \ states are generally non-degenerate (CO2 bending modes are the rare example \ of a degenerate state), but you have to look at the molecule and the normal \ mode to decide this. For ", StyleBox["diatomic", FontSlant->"Italic"], " molecules, the states are always non-degenerate.\n\nTo obtain ", Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]\ E\)]], ", you need to know quantum numbers (provided in the problem), and either \ the vibration frequency \[Nu], or a combination of the force constant ", StyleBox["k", FontSlant->"Italic"], " and reduced mass \[Mu]. In this case, the book provides ", StyleBox["k", FontSlant->"Italic"], " and you can calculate \[Mu] from the masses of the atoms in the molecule, \ so you can convert these quantities into a frequency using:\n\n", Cell[BoxData[ \(TraditionalForm\`\[Nu]\ = \ \(1\/\(2\ \[Pi]\)\) \@\(k\/\[Mu]\)\)]], "\n" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["8.4", "Subtitle"], Cell[TextData[{ "Strategy.", StyleBox[" ", FontWeight->"Plain"], "Part A.", StyleBox[" Use ", FontWeight->"Plain"], StyleBox["k", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" (provided) and \[Mu] (obtained from reduced mass formula and \ isotopic exact masses provided on inside back cover) to calculate \[Nu] (see \ hints for problem 8.3 for formula). First four quantum numbers are ", FontWeight->"Plain"], StyleBox["n", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" = 0, 1, 2, 3.\n\n", FontWeight->"Plain"], "Strategy. Part B.", StyleBox[" Fundamental transition is ", FontWeight->"Plain"], StyleBox["n", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" = 0 \[Rule] ", FontWeight->"Plain"], StyleBox["n", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" = 1. The first three overtones of this are ", FontWeight->"Plain"], StyleBox["n", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" = 0 \[Rule] ", FontWeight->"Plain"], StyleBox["n", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" = 2, 3, 4, respectively. The relative error in a quantity is \ always defined as\n\n", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[\(\(\[CapitalDelta]\ X\)\/X\ = \ \(\(|\)\(X\_actual - \ \ X\_estd\)\(|\)\)\/X\_actual\), FontWeight->"Plain"], TraditionalForm]]], "\n\n", StyleBox["The bottom of an anharmonic potential well is the part that is \ closest to being harmonic, i.e., well-described by a quadratic potential (if \ you think about a harmonic potential as being the leading term in a Taylor \ series expansion of the true potential, this assertion will make sense \ automatically).\n\nSo the error in quantum state energy, and the error in \ quantum state transition energy should be smallest for the ground state, and \ the errors should rise as we ascend. This assertion is supported \ quantitatively by the formula (given in the problem) for Morse potential \ energy states. The \"correction term\" grows as ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`n\^2\)], FontWeight->"Plain"], StyleBox[".\n", FontWeight->"Plain"] }], "Text", FontWeight->"Bold"] }, Open ]], Cell[CellGroupData[{ Cell["8.7", "Subtitle"], Cell[TextData[{ "Strategy.", StyleBox[" Compare the formulas in problems 8.7 and 8.4. The formula in 8.7 \ for wave numbers should morph into the formula in 8.4 for energies if you \ multiply the first formula by ", FontWeight->"Plain"], StyleBox["hc", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" (recall ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[\(E\ = \ \(h\ \[Nu]\ = \ h\ c\ \[Nu]\&_\)\), FontWeight->"Plain"], TraditionalForm]]], ")", StyleBox[". Once you transform the frequency formula in 8.7, you can equate \ the coefficient with ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\((n\ + \ 1\/2)\)\^2\)], FontWeight->"Plain"], " ", StyleBox["with ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\((h\ \[Nu])\)\^2\/\(4\ D\_e\)\)], FontWeight->"Plain"], StyleBox[".\n", FontWeight->"Plain"] }], "Text", FontWeight->"Bold"] }, Open ]] }, Open ]] }, FrontEndVersion->"5.2 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 951}}, WindowSize->{729, 651}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, Magnification->1.5, StyleDefinitions -> "ArticleModern.nb" ] (******************************************************************* 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[1776, 53, 61, 0, 208, "Title"], Cell[1840, 55, 23, 0, 56, "Subtitle"], Cell[1866, 57, 1044, 37, 229, "Text"], Cell[CellGroupData[{ Cell[2935, 98, 23, 0, 56, "Subtitle"], Cell[2961, 100, 1308, 31, 430, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[4306, 136, 23, 0, 56, "Subtitle"], Cell[4332, 138, 2286, 65, 543, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[6655, 208, 23, 0, 56, "Subtitle"], Cell[6681, 210, 972, 31, 156, "Text"] }, Open ]] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)