Effective Potential

SetOptions[Plot,PlotStyle->{{Blue,Thick},{Red,Thick},{Black,AbsoluteThickness[4]},{Green,Thick}},AxesLabel->{"r","Energy"},ImageSize->800,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]}},
LabelStyle->Large];
SetOptions[Manipulator,Appearance->{"Labeled"}];
SetOptions[Manipulate,ControlPlacement->Left];

In this worksheet you will look at the effective potential for different values of parameters:
•   k, the potential’s scaling constant,
•   ℓ, the magnitude of the angular momentum,
•   μ, the reduced mass, and
•   E, the total energy.

The four plotted curves are:
1)  Potential energy U(r)=((-k)/r) (blue)
2)  Angular part of the kinetic energy ((ℓ^2)/(2((μr)^2))) (red)
3)  Effective potential (Subscript[V, eff])(r)=U(r)+((ℓ^2)/(2((μr)^2))) (black)
4)  E (green)

U[r_,k_]:=-k/r
AKE[r_,el_,mu_]:=el^2/(2*mu*r^2)
Veff[r_,k_,el_,mu_]:=-k/r+el^2/(2*mu*r^2)

Manipulate[
root1=r/.FindRoot[Veff[r,k,el,mu]==e,{r,.00001},AccuracyGoal->5];
If[e>=0,root2=10,root2=r/.FindRoot[Veff[r,k,el,mu]==e,{r,1000},AccuracyGoal->5]];
Plot[{U[r,k],AKE[r,el,mu],Veff[r,k,el,mu]},{r,0,10},PlotRange->{{0,10},{-1,1}}, Epilog->{Green,Thick,Line[{{root1,e},{root2,e}}]}],{k,1,4},{{el,1,"ℓ"},0.5,5},{{mu,1,"μ"},1,10},{{e,-0.3,"E"},-1.1,1.1}]

1) How does each parameter k, ℓ, μ, E affect the plot?  What is the physical/geometric meaning of each of these parameters?

2) Which term in the effective potential (((-k)/r) or ((ℓ^2)/(2((μr)^2)))) dominates for small values of r? For large values of r? Explain in terms of both the equation and the graph.

3) What are the classical turning points? Under what conditions will the particle be bound? Unbound?

4) How do your answers for the last question change (at all) if you consider a repulsive potential? Hint: Figure out what you must change in this notebook and investigate.

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