Prof. M. V. Berry is a master of exploring foundations of quantum theory, with simple systems. His scholarly focus centers on the realm of physical asymptotics, particularly delving into the geometric intricacies of waves, with a special emphasis on phase dynamics and chaos. He is also an avid Mathematica user. In this short computational essay, I try to reproduce some results of his paper: M V Berry 2022, Quantum carpets in leaky boxes, Eur. J. Phys. 43 055403. https://doi.org/10.1088/1361-6404/ac7e8d
Leaky boxes, modeled with nonhermitian boundaries, exhibit waves as superpositions of complex eigenmodes. These systems offer valuable pedagogical insights into nonhermiticity and serve as rich sources for advanced student projects.
Leaky boxes, modeled with nonhermitian boundaries, exhibit waves as superpositions of complex eigenmodes. These systems offer valuable pedagogical insights into nonhermiticity and serve as rich sources for advanced student projects.
First and foremost, congratulations to Sir Michael Berry on receiving the 2024 Wolfram Innovator Award – a well-deserved recognition of his contributions.
Refer to Prof. Berry’s detailed and pedagogical paper for more details: “Quantum carpets in leaky boxes”.
Physical system: a particle-in-a-1D-box with the position in the region . For convenience, we set and . The initial wave-function is set as . We considered three boundary conditions: (i) Dirichlet boundary condition, (ii) Neumann boundary condition, and (iii) Robin boundary condition (with Real and Complex as Robin parameter)
-1<=x<=1
ℏ=1
m=1/2
Exp-Cos[qx]
2
x
2
2
L
Berry’s Eq(2.5) notation: u(x)Cu(x)
C can be real (Hermitian case, with impenetrable walls) or complex (antihermitian case, with leaky walls).
∂
x
C can be real (Hermitian case, with impenetrable walls) or complex (antihermitian case, with leaky walls).
Initial wave function:
In[]:=
L=.1;q=20;ψi=Exp-(Exp[qx]+Exp[-qx]);
x^2
2L^2
Wolfram notation on Robin’s boundary condition: u(x)g-qu(x)
A function returning numerical solution given Robin’s boundary condition:
∂
x
A function returning numerical solution given Robin’s boundary condition:
In[]:=
solution[g_,q_]:=NDSolveValue[{D[ψ[x,t],t]==-Laplacian[ψ[x,t],{x}]+NeumannValue[g-qψ[x,t],True],ψ[x,0]==ψi},ψ,{x,-1,1},{t,0,.5}]
The case with :
g,q=0
In[]:=
With[{sol=solution[0,0]},Row[{DensityPlot[Abs[sol[x,t]]^2/4,{x,-1,1},{t,0,.31},PlotRange->{0,1},MaxRecursion->6,ImageSize->Medium,PlotLabel->"Neumann boundary condition, c=0",PlotLegends->Automatic,ColorFunction->(ColorData["Rainbow"][Sqrt@#]&)],DensityPlot[Abs[sol[x,t]]^2/4,{x,-1,1},{t,0,1/40},PlotRange->{0,1},MaxRecursion->6,ImageSize->Medium,PlotLabel->"Neumann boundary condition, c=0",ColorFunction->(ColorData["Rainbow"][Sqrt@#]&)]}]]
Out[]=
The case with :
g=0,q=5
In[]:=
With[{sol=solution[0,5]},Row[{DensityPlot[Abs[sol[x,t]]^2/4,{x,-1,1},{t,0,.31},PlotRange->{0,1},MaxRecursion->6,ImageSize->Medium,PlotLabel->"Neumann boundary condition, c=5",PlotLegends->Automatic,ColorFunction->(ColorData["Rainbow"][Sqrt@#]&)],DensityPlot[Abs[sol[x,t]]^2/4,{x,-1,1},{t,0,1/40},PlotRange->{0,1},MaxRecursion->6,ImageSize->Medium,PlotLabel->"Neumann boundary condition, c=5",ColorFunction->(ColorData["Rainbow"][Sqrt@#]&)]}]]
Out[]=
The case with :
g=0,q=5+3
In[]:=
With[{sol=solution[0,5+3]},Row[{DensityPlot[Abs[sol[x,t]]^2/4,{x,-1,1},{t,0,.31},PlotRange->{0,1},MaxRecursion->6,ImageSize->Medium,PlotLabel->"Robin boundary condition, c=5+3I",PlotLegends->Automatic,ColorFunction->(ColorData["Rainbow"][Sqrt@#]&)],DensityPlot[Abs[sol[x,t]]^2/4,{x,-1,1},{t,0,1/40},PlotRange->{0,1},MaxRecursion->6,ImageSize->Medium,PlotLabel->"Robin boundary condition, c=5+3I",ColorFunction->(ColorData["Rainbow"][Sqrt@#]&)]}]]
Out[]=
Dirichlet boundary condition:
In[]:=
With[{sol=NDSolveValue[{D[ψ[x,t],t]==-Laplacian[ψ[x,t],{x}],ψ[x,0]==ψi,DirichletCondition[ψ[x,y]0,True]},ψ,{x,-1,1},{t,0,.5}]},Row[{DensityPlot[Abs[sol[x,t]]^2/4,{x,-1,1},{t,0,.31},PlotRange->{0,1},MaxRecursion->6,ImageSize->Medium,PlotLabel->"Dirichlet boundary condition",PlotLegends->Automatic,ColorFunction->(ColorData["Rainbow"][Sqrt[#]]&)],DensityPlot[Abs[sol[x,t]]^2/4,{x,-1,1},{t,0,1/40},PlotRange->{0,1},MaxRecursion->6,ImageSize->Medium,PlotLabel->"Dirichlet boundary condition",ColorFunction->(ColorData["Rainbow"][Sqrt[#]]&)]}]]
Out[]=
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Quantum carpets in leaky boxes
by Mads Bahrami
Wolfram Community, STAFF PICKS, October 18, 2024
https://community.wolfram.com/groups/-/m/t/3299973
by Mads Bahrami
Wolfram Community, STAFF PICKS, October 18, 2024
https://community.wolfram.com/groups/-/m/t/3299973