Time Dependent Solutions for Motion on a Ring
           By Corinne Manogue and Kerry Browne
          Copyright 2010 Corinne Manogue

SetOptions[Plot,PlotRange->{{0,5},{-0.6,0.3}},PlotStyle->{{Blue,Thick},{Red,Thick},{Black,Thick},{Green,Thick}},AxesLabel->{"ϕ","𝒫(ϕ)"},ImageSize->500,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]}}];

Define the normalized spatial eigenfunctions (Subscript[Φ, 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 in order to construct the time-evolved states.

PhiT[ϕ_,m_]:=Exp[-I*m^2/2*t]*Φ[ϕ,m]

Construct some time-dependent wavefunctions. Each of these consists of a linear combination of eigenstates on the ring.  Choose one.

ΨT[ϕ_,t_]:= PhiT[1]

ΨT[ϕ_,t_]:=1/Sqrt[2]*(PhiT[ϕ,1]+PhiT[ϕ,2])

ΨT[ϕ_,t_]:=1/Sqrt[3]*(PhiT[ϕ,1]+I*Sqrt[2]*PhiT[ϕ,3])

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

ΨT[ϕ_,t_]:=1/Sqrt[3]*(PhiT[ϕ,1]+PhiT[ϕ,2]+PhiT[ϕ,3])

ΨT[ϕ_,t_]:=1/Sqrt[4]*(PhiT[ϕ,1]+PhiT[ϕ,2]+PhiT[ϕ,-1]+PhiT[ϕ,-2])

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

"Square" your chosen wavefunction to get the probability density. You must execute the commands starting here, each time you choose a new wavefunction.

ΨTsq[ϕ_,t_]=ComplexExpand[Conjugate[ΨT[ϕ,t]]*ΨT[ϕ,t]]

Here is the above expression in it's simplest form. What identity was used to make the simplification? Make sure that you can expand the expression on your own!

FullSimplify[%]

Plot an animation of the wavefunction on a standard cartesian plot. In this plot the probability density is indicated by the height of the function.

Animate[Plot[ΨTsq[ϕ,t],{ϕ,0,2*Pi},PlotRange->{{0,7},{0,0.5}}],{t,0,2*Pi},AnimationRunning->False]

Color Plot: Red is high, blue is low.

fmax=Max[Table[ΨTsq[ϕ,0],{ϕ,0,2*π,π/100}]]

ParametricPlot[{r*Cos[ϕ],r*Sin[ϕ]},{r,0.9,1},{ϕ,0,2*π},PlotRange->{{-1.1,1.5},{-1.1,1.3}},Mesh->False,ColorFunction->Function[{x,y,r,ϕ},Hue[0.67*(1-ΨTsq[ϕ,0]/fmax)]],ColorFunctionScaling->False,Axes->False,Frame->False,BoundaryStyle->None,Epilog->{Line[{{0.85,0},{1.1,0}}],Line[{{0,0.85},{0,1.1}}],
Text[Style["ϕ=0",14,FontFamily->"Arial"],{1.3,0},{0,0}],
Text[Style["ϕ=π/2",14,FontFamily->"Arial"],{0,1.2},{0,0}]}]

Color and height plot

SetOptions[ParametricPlot3D,AxesLabel->{"x","y",None},AxesStyle->Directive[FontFamily->"Arial",FontSize->14,Black,AbsoluteThickness[0.5]]];

ParametricPlot3D[{Cos[ϕ],Sin[ϕ],u*ΨTsq[ϕ,0]},{u,0,1},{ϕ,0,2*π},Mesh->False,ColorFunction->Function[{x,y,z,u,ϕ},Hue[0.67*(1-ΨTsq[ϕ,0]/fmax)]],ColorFunctionScaling->False,Axes->True,BoundaryStyle->None,BoxRatios->{1,1,0.5},ViewPoint->{1.1,-2.4,1}]