Free Fall

Maria Daghofer,  Wolfgang von der Linden
Institut fuer Theoretische Physik
Technische Universitaet Graz
A - 8010 Graz, 
    Austria
Projekt Multimediale Lehre des
Zentralen Informatikdienstes der TU Graz

2001 - 12 - 13
Null

Parameters

Begin["fallQM`"]; 
    Clear["fallQM`*"]; 
Needs["Graphics`MultipleListPlot`"];

Explanation

Exact solution

Block[{z, y}, 
      DSolve[-y''[z] + z y[z] == 0, y[z], z]]

Block[{z, airygraph}, 
      airygraph = 
        Plot[{AiryAi[z], AiryBi[z]}, {z, -3, 2}, 
          DisplayFunction -> Identity]; 
      Show[airygraph , 
        Graphics[{Text["Ai(z)", {-2.1, 0.4}, 
              TextStyle -> {FontColor -> Hue[0.6]}], 
            Text["Bi(z)", {0.8, 1.4}, 
              TextStyle -> {FontColor -> Hue[1]}]}], 
        PlotRange -> {{-3, 2}, {-0.5, 2}}, 
        DisplayFunction -> $DisplayFunction, 
        AxesLabel -> {"z", ""}];
]

Block[{z, airygraph},airygraph  = 
            Plot[AiryAi[z], {z, -13, 10}, 
              PlotStyle -> {Thickness[0.01], Hue[0.6]}, 
              DisplayFunction -> Identity];
airygraph  = 
            Show[airygraph , 
              Plot[(Sqrt[(1/Pi)]) 
                  Cos[2/3  (-z)^(3/2) - Pi/4]/
                    Abs[z]^0.25, {z, -13, 0}, 
                PlotStyle -> {Hue[1]}, 
                DisplayFunction -> Identity]];
Show[airygraph ,Graphics[{Text["Ai(z)", {2.5, 0.2}, 
                      TextStyle -> {FontColor -> Hue[0.6]}],Text["(Sqrt[(1/π)]) cos((2/3) (((-z))^((3/2))) - (π/4))/|z\!\(\( | \^\(1/4\)\)\)",{6, 0.75},TextStyle -> {FontSize -> 8, 
                            FontColor -> Hue[1]}]}],PlotRange -> {{-13, 10}, {-0.5, 
                    1}},DisplayFunction -> $DisplayFunction,AxesLabel -> {"z", ""}];
]

Definition:

\(\(ExactEigenvalue[m_] := \ 
        Block[{x0, xy}, 
							x0\  = \ 
            N[\((\((2  m - 1\/2)\) \(π\ 3\)\/4)\)\^\(2/3\)]; 
							Abs[xy] /. 
            FindRoot[
              AiryAi[xy] == 
                0, {xy, \(-x0\)}]
						];\)
\)
\(\(density[m_]\  := \ 
        Block[{}, 
		ϵ\  = \ 
            ExactEigenvalue[
              m]; 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ NormA\  = \ 
            NIntegrate[
              AiryAi[x - ϵ]\^2, {x, 
                0, ∞}]; 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(g[m] = 
              Plot[AiryAi[x - ϵ]\^2/NormA, {x, 0, 
                  2  ϵ}];\)];\)\)

Groundstate and low-lying excited states

Table[ExactEigenvalue[m], {m, 1, 5}]

Table[density[m], {m, 1, 5}];