Time Dependent Solutions for Motion on a Ringby Corinne Manogue and Kerry BrowneCopyright 2010 Corinne Manoguerestart: with(plots):
assume(phi, real, t, real, m, real):Define the normalized spatial eigenfunctions (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEmJlBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtRi82JVEibUYnL0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1) for the particle confined to a ring.Phi:=(m)->(1/sqrt(2*Pi))*exp(I*m*phi);Multiply the spatial eigenfunctions by their time dependence to obtain the time dependent eigenfunctions and then make this into a procedure taking the arguement m.PhiT:=(m)->Phi(m)*exp(-I*m^2*t);Here are some sample wavefunctions. Each of these consists of a linear combination of time dependent eigenstates on the ring. PsiT:=PhiT(1);PsiT:=1/sqrt(2)*(PhiT(1)+PhiT(2));PsiT:=1/sqrt(3)*(PhiT(1)+PhiT(2)+PhiT(3));PsiT:=1/sqrt(4)*(PhiT(1)+PhiT(2)+PhiT(-1)+PhiT(-2));PsiT:=1/sqrt(6)*(PhiT(1)+PhiT(2)+PhiT(3)+PhiT(-1)+PhiT(-2)+PhiT(-3));"Square" your chosen wavefunction to get the probability density.
You must execute the commands starting here, each time you choose a new wavefunction. PsiTsq:=simplify(expand(abs(PsiT)^2));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, [PsiTsq, phi=0..2*Pi], t=0..2*Pi, frames=100);Plot an animation of the wavefunction using color to indicate the probability density (red =0, purple =1)animate(cylinderplot, [[r, phi, 0], phi=0..2*Pi, r=0.98..1.02, color=0.9*PsiTsq, axes=boxed, scaling=constrained, style=patchnogrid, orientation = [0, 0]], t=0.0001..2*Pi, frames=50);
Plot an animation of the wavefunction using both height and color to indicate probability density.animate(cylinderplot, [[1,phi,z], phi=0..2*Pi,z=0..PsiTsq, color=0.9*PsiTsq ,numpoints=5000,axes=boxed,style=patchnogrid], t=0.0001..2*Pi,view=0..1,frames=50, orientation=[170,65]);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=