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The paper presents all the results of \[OpenCurlyDoubleQuote]Exact \
time-domain solution of the Schr\[CapitalODoubleDot]dinger equation for a new \
scattering model\[CloseCurlyDoubleQuote]\
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The paper presents a new method for solving a time-dependent Schrodinger \
equation with an ultrashort barrier. So far, there are no barriers other than \
the Dirac delta barrier has an solution. THis is   a first time a barrier \
with a short range has an exact solution\
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Let us visualize the result of Fei sun;
THe derived expression matches with that of the Fei sun. A small variation in \
the magnitudes can be due to numerical inaccuracy of his code\
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This section presents the space-time dynamics of the wave packet interacting \
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                 GridBoxDividers -> {
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                    ImageSize -> {40, 10}, PlotRangePadding -> None, 
                    ImagePadding -> Automatic, 
                    BaselinePosition -> (Scaled[0.1] -> Baseline)], #}}, 
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                  GridBoxDividers -> {
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                    ImagePadding -> Automatic, 
                    BaselinePosition -> (Scaled[0.1] -> Baseline)], #}}, 
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                    GrayLevel[0]]], 
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                    Opacity[1.], 
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                    AbsoluteThickness[1.6], 
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                    GrayLevel[0]]], 
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                    AbsoluteThickness[1.6], 
                    RGBColor[1, 1, 0], 
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                    ImagePadding -> Automatic, 
                    BaselinePosition -> (Scaled[0.1] -> Baseline)], #}}, 
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                  GridBoxDividers -> {
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                  GridBoxItemSize -> {
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            ImageMargins -> {{5, 5}, {5, 5}}, ImageSizeAction -> 
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                    RectangleBox[{0, 0}]}, {
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                    RectangleBox[{0, -1}, {2, 1}]}}, AspectRatio -> 1, Frame -> 
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                    FrontEnd`Private`$ColorSelectorInitialColor = 
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                    FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
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                    GrayLevel[0]]], 
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                    Opacity[1.], 
                    CapForm["Butt"], 
                    AbsoluteThickness[1.6], 
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                    GrayLevel[0]]], 
                    PointSize[0.5], 
                    Opacity[1.], 
                    CapForm["Butt"], 
                    AbsoluteThickness[1.6], 
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                    ImageSize -> {40, 10}, PlotRangePadding -> None, 
                    ImagePadding -> Automatic, 
                    BaselinePosition -> (Scaled[0.1] -> Baseline)], #}}, 
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                  GridBoxDividers -> {
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                    RectangleBox[{0, 0}]}, {
                    GrayLevel[0], 
                    RectangleBox[{1, -1}]}, {
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                    RectangleBox[{0, -1}, {2, 1}]}}, AspectRatio -> 1, Frame -> 
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                    FrontEnd`Private`$ColorSelectorInitialColor = 
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                    FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
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                    FrontEndResource["RGBColorValueSelector"], {
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             RowBox[{"LegendMarkers", "\[Rule]", "None"}], ",", 
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                    GrayLevel[0]]], 
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                    Opacity[1.], 
                    CapForm["Butt"], 
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In this section, we calculate various properties of the wavepacket and the \
barrier. First we calculated the time-dependent dispersion of the wavepacket. \
Next, we calculate the trajectory of the  mean-position of the wavepacket.  \
Next the dwell probability inside the barrier, reflection probability and \
transmission probability are calculated together and is presented.\
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Computing the probability properties and the statistical properties of the \
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                    FrontEnd`Private`$ColorSelectorInitialColor = 
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This section presents miscellaneous trials;  and all these results are not \
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the plots for the dynamics of an exponential wavepacket and delta function \
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