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Therefore, the eigenfunctions of the Hamiltonian operator must \ also be eigenfunctions of the momentum operator, ", Cell[BoxData[ \(TraditionalForm\`\(p\&^\)\)]], "." }], "Text"], Cell[CellGroupData[{ Cell["Solution", "Subtitle"], Cell[TextData[{ StyleBox["Tricky problem. Remember, the theorem concerning commuting \ operators is this: they ", FontWeight->"Plain"], StyleBox["can", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" share the same eigenfunctions, not they ", FontWeight->"Plain"], StyleBox["must ", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["share the same eigenfunctions. When does ", FontWeight->"Plain"], StyleBox["can", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" dominate over ", FontWeight->"Plain"], StyleBox["must", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["?\n\nIn this problem, \[Psi](", FontWeight->"Plain"], StyleBox["x", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[") = A sin(", FontWeight->"Plain"], StyleBox["n", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" \[Pi] ", FontWeight->"Plain"], StyleBox["x", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" / ", FontWeight->"Plain"], StyleBox["a", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["), is an eigenfunction of ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\(H\&^\)\)], FontWeight->"Plain"], StyleBox[", but not an eigenfunction of ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\(p\&^\)\)], FontWeight->"Plain"], StyleBox[". The eigenfunctions of ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\(p\&^\)\)], FontWeight->"Plain"], StyleBox[", on the other hand, look", FontWeight->"Plain"], StyleBox[" like ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`e\^\(\(-i\)\ k\ x\)\)], FontWeight->"Plain"], StyleBox[" with eigenvalues of ", FontWeight->"Plain"], StyleBox["hbar k.", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" They do not obey the boundary conditions for the particle in a \ box and are not eigenfunctions of ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[\(H\&^\), FontWeight->"Plain"], TraditionalForm]]], StyleBox[".\n\nOne more thing: notice", FontWeight->"Plain"], StyleBox[" \[Psi] can be written as a combination of momentum \ eigenfunctions: \[Psi] = ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\((A/2 i)\) \((e\^\(\(-i\)\ \((\(-k\))\)\ x\)\ + \ e\^\(\(-i\)\ k\ x\))\)\)], FontWeight->"Plain"], StyleBox[", where the two eigenfunctions have eigenvalues of -(hbar ", FontWeight->"Plain"], StyleBox["k", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[") and +(hbar ", FontWeight->"Plain"], StyleBox["k", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["), respectively.\n\nOK, now the answer. The fact that ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\(p\&^\)\)], FontWeight->"Plain"], StyleBox[" and ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\(\(p\^2\)\&^\)\)], FontWeight->"Plain"], StyleBox[" commute tells that we ", FontWeight->"Plain"], StyleBox["can", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" choose a set of functions that will be eigenfunctions of both \ operators. The two momentum eigenfunctions that combine to make \[Psi] are \ eigenfunctions of ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\(\(p\^2\)\&^\)\)], FontWeight->"Plain"], StyleBox[". The problem is this: they are ", FontWeight->"Plain"], StyleBox["degenerate", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" eigenfunctions of ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\(\(p\^2\)\&^\)\)], FontWeight->"Plain"], "; ", StyleBox["their eigenvalues are ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\((hbar\ k)\)\^2\)], FontWeight->"Plain"], StyleBox[".", FontWeight->"Plain"], StyleBox["\n\nWhenever we have degenerate eigenfunctions of an operator, \ any combination of these eigenfunctions will also be an eigenfunction of the \ operator, but different", FontWeight->"Plain"], StyleBox[" combinations may have different mathematical properties and \ satisfy different theorems", FontWeight->"Plain"], StyleBox[". For example, o", FontWeight->"Plain"], StyleBox["ne way of selecting ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\(\(p\^2\)\&^\)\)], FontWeight->"Plain"], StyleBox[" eigenfunctions ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\((e\^\(\(-i\)\ k\ x\))\)\)], FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], StyleBox["gives functions that are also eigenfunctions of ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\(p\&^\)\)], FontWeight->"Plain"], StyleBox[", but this selection does not produce functions that obey the \ boundary conditions for a particle in a box. Another way of selecting \ eigenfunctions (\[Psi]) gives functions that are eigenfunctions of ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\(H\&^\)\)], FontWeight->"Plain"], StyleBox[", but not eigenfunctions of ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\(p\&^\)\)], FontWeight->"Plain"], StyleBox[".\n\nBottom line: you must always be on the lookout for \ degenerate eigenfunctions. There is no unique way of choosing them.", 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, 37, 0, 129, "Title"], Cell[1816, 55, 561, 16, 145, "Text"], Cell[CellGroupData[{ Cell[2402, 75, 28, 0, 56, "Subtitle"], Cell[2433, 77, 5597, 177, 752, "Text"] }, Open ]] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)