Motion on a Ring

SetOptions[Plot,
PlotRange->{{0,7},{0,0.5}},
PlotStyle->{{Blue,Thick},{Red,Thick},{Black,Thick},{Green,Thick}},
AxesLabel->{"ϕ","𝒫(ϕ)"},
ImageSize->400,
BaseStyle->{FontFamily->"Arial",FontSize->16},AxesStyle->Directive[FontFamily->"Arial",FontSize->16,Black,AbsoluteThickness[0.5],Arrowheads[0.04]],TicksStyle->{{Black,AbsoluteThickness[0.5]},{Black,AbsoluteThickness[0.5]}}];
SetOptions[ParametricPlot3D,
Mesh->False,
AxesLabel->{"x","y",None},
ImageSize->400,
AxesStyle->Directive[FontFamily->"Arial",FontSize->16,Black,AbsoluteThickness[0.5]],
ColorFunctionScaling->False,
Axes->True,
BoundaryStyle->None,
BoxRatios->{1,1,0.5},
ViewPoint->{1.1,-2.4,1},
PerformanceGoal->"Quality"];
SetOptions[ParametricPlot,
Mesh->False,
ColorFunctionScaling->False,
Axes->False,
ImageSize->400,
Frame->False,
BoundaryStyle->None,
Epilog->{Line[{{0.85,0},{1.1,0}}],Line[{{0,0.85},{0,1.1}}],Text[Style["ϕ=0",16,FontFamily->"Arial"],{1.3,0},{0,0}],
Text[Style["ϕ=π/2",16,FontFamily->"Arial"],{0,1.2},{0,0}]},
PlotRange->{{-1.1,1.5},{-1.1,1.3}},
PerformanceGoal->"Quality"];
SetOptions[Animate,AnimationRunning->False];
SetOptions[Manipulate,BaseStyle->{FontSize->16}];
SetOptions[Manipulator,Appearance->"Labeled"];
Choose=PopupMenu[Dynamic[WaveFunctionSelector],Table[x->StringForm["ΨT``[ϕ,t]",x],{x,1,11}]];
ChosenWaveFunction[ϕ_,t_]:=ToExpression[ToString[StringForm["ΨT``[ϕ,t]",WaveFunctionSelector]]]

In this worksheet, you will examine wave functions and probability densities for a particle confined to a ring.

Define the normalized spatial eigenfunctions ((Subscript[Φ, m])(ϕ)=(1/(Sqrt[2 π]))(e^(i m ϕ))) for the particle confined to a ring.

Φ[ϕ_,m_]:=(1/Sqrt[2*Pi])*Exp[I*m*ϕ]

Multiply each spatial eigenfunction by its time dependence ((e^((-i (m^2)t)/2))) in order to construct the time-dependent states.

PhiT[ϕ_,m_]:=Exp[-I*m^2*t/2]*Φ[ϕ,m]

Each of these time-dependent wavefunctions consists of an eigenstate or a linear combination of eigenstates on the ring.

ΨT1[ϕ_,t_]:=PhiT[ϕ,1]
ΨT2[ϕ_,t_]:=PhiT[ϕ,-3]

ΨT3[ϕ_,t_]:=1/Sqrt[3]*(PhiT[ϕ,1]+Sqrt[2]*PhiT[ϕ,3])
ΨT4[ϕ_,t_]:=1/Sqrt[3]*(PhiT[ϕ,1]-Sqrt[2]*PhiT[ϕ,3])
ΨT5[ϕ_,t_]:=1/Sqrt[3]*(PhiT[ϕ,1]+I*Sqrt[2]*PhiT[ϕ,3])

ΨT6[ϕ_,t_]:=1/Sqrt[3]*(I*PhiT[ϕ,1]+I*Sqrt[2]*PhiT[ϕ,3])
ΨT7[ϕ_,t_]:=1/Sqrt[3]*(PhiT[ϕ,-1]+Sqrt[2]*PhiT[ϕ,-3])
ΨT8[ϕ_,t_]:=1/Sqrt[3]*(PhiT[ϕ,1]+Sqrt[2]*PhiT[ϕ,-3])
ΨT9[ϕ_,t_]:=1/Sqrt[3]*(PhiT[ϕ,-1]+Sqrt[2]*PhiT[ϕ,3])

ΨT10[ϕ_,t_]:=1/Sqrt[4]*(PhiT[ϕ,1]+PhiT[ϕ,2]+PhiT[ϕ,-1]+PhiT[ϕ,-2])
ΨT11[ϕ_,t_]:=1/Sqrt[6]*(PhiT[ϕ,1]+PhiT[ϕ,2]+PhiT[ϕ,3]+PhiT[ϕ,-1]+PhiT[ϕ,-2]+PhiT[ϕ,-3])

1a) Do the wavefunctions of the energy eigenstates ΨT1 and ΨT2 have time dependence? How do you know?

1b) Do the wavefunctions of the linear combinations of energy eigenstates ΨT3 through and ΨT11 have time dependence? How do you know?

Choose a wavefunction:

Choose

"Square" your chosen wavefunction to get the probability density.

You must evaluate the following line each time you choose a new wave function.

ΨTsq[ϕ_,t_]=FullSimplify[ComplexExpand[Conjugate[ChosenWaveFunction[ϕ,t]]*ChosenWaveFunction[ϕ,t]]];
fmax=Max[Table[ΨTsq[ϕ,0],{ϕ,0,2*π,2π/100}]];
ΨTsq[ϕ,t]

2a) Do the probability densities of the energy eigenstates ΨT1 and ΨT2 have time dependence? Why or why not?

2b) Do the probability densities of the linear combinations of energy eigenstates ΨT3 through and ΨT11 have time dependence? Why or why not?

Plot the probability density

Animate[
myplot1=Plot[ΨTsq[ϕ,t],{ϕ,0,2*Pi},PlotRange->{{0,2π},{0,fmax}}];
myplot2=ParametricPlot[{r*Cos[ϕ],r*Sin[ϕ]},{r,0.9,1},{ϕ,0,2*π},ColorFunction->Function[{x,y,r,ϕ},Hue[0.67*(1-ΨTsq[ϕ,t]/fmax)]]];
myplot3=ParametricPlot3D[{Cos[ϕ],Sin[ϕ],u*ΨTsq[ϕ,t]},{u,0,1},{ϕ,0,2*π},ColorFunction->Function[{x,y,z,u,ϕ},Hue[0.67*(1-ΨTsq[ϕ,t]/fmax)]],PlotRange->{{-1,1},{-1,1},{0,fmax}}];
Grid[{{myplot2,myplot1,myplot3}}],{t,0,Pi},AnimationRate->.1]

3a) When do probability densities rotate? In which direction do they rotate?

3b) When do probability densities not rotate? Explain.

4) What is different between the equations for ΨT3 through ΨT6? How do the plots of their probability densities differ at time t=0?

By Corinne Manogue, Kerry Browne, Mike Vignal, and Ian Founds
Copyright 2010 Corinne Manogue