Phys 425, Homework 8, Finite Square Well

Problem 1, Odd Parity Bound States

Show[
Plot[
{-Cot[z],(Sqrt[((((1/z)))^2)-1]),(Sqrt[((((3/z)))^2)-1]),(Sqrt[((((6/z)))^2)-1])},{z,0,9},
PlotRange->{{0,9},{-0.5,2.5}},
PlotStyle->{{Orange,Thickness[0.004]},{Blue,Thickness[0.004]},{Blue,Thickness[0.004]},{Blue,Thickness[0.004]}},
AxesLabel->{z,Style["\!\(\*SqrtBox[
RowBox[{SuperscriptBox[
RowBox[{"(", FractionBox[SubscriptBox[
StyleBox["z",
FontSlant->"Italic"], "0"], 
StyleBox["z",
FontSlant->"Italic"]], ")"}], "2"], "-", "1"}]]\),-Cot[z]",12]},
Exclusions->Table[n π, {n,1,3,1}],
ImageSize->800,
LabelStyle->({FontSize->16,FontFamily->"Times"}),
Epilog->{
{Rotate[Text[Style["↙(Subscript[z, 0])=1",{FontSize->16,FontFamily->"Times"}],{0.87,1.5}],0°],
Rotate[Text[Style["↙(Subscript[z, 0])=3",{FontSize->16,FontFamily->"Times"}],{2.0,1.5}],0°],
Rotate[Text[Style["↙(Subscript[z, 0])=6",{FontSize->16,FontFamily->"Times"}],{3.7,1.5}],0°]}
}
],
ListPlot[
{{2.27886266,-Cot[2.27886266]},{2.67879,-Cot[2.67878311]},{5.22596353,-Cot[5.22596353]}},
PlotMarkers->{"•",Large},
PlotStyle->{Red}
]
]

Problem 2, Scattering States

Manipulate[
Plot[
(((1+(1/(4 EV(EV+1)))(((Sin[β(Sqrt[(EV+1)])]))^2)))^(-1)),{EV,0,4},
PlotRange->{{0,4},{0,1.01}},
PlotStyle->{Orange,Thickness[0.003]},
AxesLabel->{Style["(E/(Subscript[V, 0]))",Italic],"T"},
ImageSize->800,
LabelStyle->({FontSize->16,FontFamily->"Times"})
],
{{β,20,"(Subscript[z, 0])"},0,40,0.1,Appearance->"Labeled"}
]

Problem 3, Half Finite Square Well

Show[
Plot[
{Tan[π (Sqrt[EE])]+(Sqrt[(EE/(10-EE))])},{EE,0,10},
PlotRange->{{0,10},{-5,5}},
PlotStyle->{Orange,Thickness[0.003]},
AxesLabel->{Style["(E/Ĕ)",Italic],"f(E)"},
Exclusions->Table[((n^2)/4), {n,1,100,1}],
ImageSize->800,
LabelStyle->({FontSize->16,FontFamily->"Times"})
],
ListLinePlot[
{{0.823310,-2},{0.823310,+2}},
PlotStyle->{Blue,Thickness[0.002],Dotted}
],
ListLinePlot[
{{3.26329,-2},{3.26329,+2}},
PlotStyle->{Blue,Thickness[0.002],Dotted}
],
ListLinePlot[
{{7.17390,-2},{7.17390,+2}},
PlotStyle->{Blue,Thickness[0.002],Dotted}
]
]

Show[
Plot[
{Tan[π (Sqrt[EE])]+(Sqrt[(EE/(10-EE))])},{EE,0.82330,0.82332},
PlotRange->{{0.82330,0.82332},{-.00001,.00001}},
PlotStyle->{Orange,Thickness[0.003]},
Axes->{True,False},
ImageSize->800,
LabelStyle->({FontSize->16,FontFamily->"Times"})
],
ListLinePlot[
{{0.8233095261,-0.000005},{0.8233095261,+0.000005}},
PlotStyle->{Blue,Thickness[0.002],Dotted}
]
]

Show[
Plot[
{Tan[π (Sqrt[EE])]+(Sqrt[(EE/(10-EE))])},{EE,3.263285,3.263295},
PlotRange->{{3.26327,3.26331},{-.00001,.00001}},
PlotStyle->{Orange,Thickness[0.003]},
Axes->{True,False},
ImageSize->800,
LabelStyle->({FontSize->16,FontFamily->"Times"})
],
ListLinePlot[
{{3.26329,-0.000005},{3.26329,+0.000005}},
PlotStyle->{Blue,Thickness[0.002],Dotted}
]
]

Show[
Plot[
{Tan[π (Sqrt[EE])]+(Sqrt[(EE/(10-EE))])},{EE,7.17388,7.17392},
PlotRange->{{7.17388,7.17392},{-.00001,.00001}},
PlotStyle->{Orange,Thickness[0.003]},
Axes->{True,False},
ImageSize->800,
LabelStyle->({FontSize->16,FontFamily->"Times"})
],
ListLinePlot[
{{7.17389711,-0.000005},{7.17389711,+0.000005}},
PlotStyle->{Blue,Thickness[0.002],Dotted}
]
]

FindRoot[Tan[π (Sqrt[e])]+(Sqrt[(e/(10-e))])==0,{e,1}]

FindRoot[Tan[π (Sqrt[e])]+(Sqrt[(e/(10-e))])==0,{e,4}]

FindRoot[Tan[π (Sqrt[e])]+(Sqrt[(e/(10-e))])==0,{e,9}]

ee={0.82331,3.26329,7.1739}

CovrB={Sin[π(Sqrt[ee[[1]]])](E^(π(Sqrt[10-ee[[1]]]))),Sin[π(Sqrt[ee[[2]]])](E^(π(Sqrt[10-ee[[2]]]))),Sin[π(Sqrt[ee[[3]]])](E^(π(Sqrt[10-ee[[3]]])))}

psi1[n_,x_]:=If[x<1, Sin[π(Sqrt[ee[[n]]])x],CovrB[[n]]*(E^(-(π(Sqrt[10-ee[[n]]])*x)))];

Solve[((∫_0^∞)(A psi1[1,x]*A psi1[1,x])\[DifferentialD]x)==1,A]

Solve[((∫_0^∞)(A psi1[2,x]*A psi1[2,x])\[DifferentialD]x)==1,A]

Solve[((∫_0^∞)(A psi1[3,x]*A psi1[3,x])\[DifferentialD]x)==1,A]

B={1.3453,1.33474,1.29676}

Plot[{B[[1]]psi1[1,x],B[[2]]psi1[2,x],B[[3]]psi1[3,x]},{x,0,2},
PlotRange->{{0,2},{-1.5,1.5}},
Exclusions->None,
Ticks->{{{1.0,"a"},{2.0,"2a"}},{ {-1.4,"-1.4"},{1.4,"1.4"}}},
AxesLabel->{x,"ψ"},
PlotStyle->{{Orange,Thickness[0.005]},{Blue,Thickness[0.005]},{Darker[Darker[Green]],Thickness[0.005]}},
GridLines->{{1},{ -1.4,-1.2,-1,-.8,-.6,-.4,-.2,.2,.4,.6,.8,1,1.2,1.4}},
GridLinesStyle->Directive[LightBlue,AbsoluteThickness[1.0]],
ImageSize->800,
LabelStyle->({FontSize->16,FontFamily->"Times"}),
Epilog->{
{Rotate[Text[Style["(Subscript[ψ, 1])↗",{FontSize->16,FontFamily->"Times"}],{0.7,1.1}],0°],
Rotate[Text[Style["↙(Subscript[ψ, 2])",{FontSize->16,FontFamily->"Times"}],{0.435,1.1}],0°],
Rotate[Text[Style["↙(Subscript[ψ, 3])",{FontSize->16,FontFamily->"Times"}],{1.05,1.1}],0°]}
}
]