Find Aharonov–Bohm Eigenvalues: New in Wolfram Language 11

Find Aharonov–Bohm Eigenvalues

The Aharonov–Bohm effect is a quantum mechanical phenomenon where a charged particle detects an electromagnetic gauge potential even though the magnetic field inside the region where the particle can move is zero. In this example, the Schrödinger equation for a fixed magnetic field concentrated at a line perpendicular to the domain of the particle is considered.

Specify the region as sector in the plane centered about the axis.

In[1]:=

\[CapitalOmega] = Disk[{0, 0}, 1, {-\[Pi]/8, \[Pi]/8}];

Specify the Aharonov–Bohm operator. The magnetic field is restricted to the axis.

In[2]:=

 A = 1/2 {-(y - b), x - a}/((x - a)^2 + (y - b)^2) /. {a -> 0.2, 
    b -> 0.4};
\[ScriptCapitalL] = -Laplacian[u[x, y], {x, y}] + 
   I Div[A, {x, y}] u[x, y] + 2 I A.D[u[x, y], {{x, y}}] + 
   A.A u[x, y];

Specify Dirichlet boundary conditions on the entire boundary.

In[3]:=

\[ScriptCapitalB] = DirichletCondition[u[x, y] == 0, True];

Compute six eigenvalues and eigenfunctions of the operator.

In[4]:=

{vals, funs} = 
  NDEigensystem[{\[ScriptCapitalL], \[ScriptCapitalB]}, 
   u[x, y], {x, y} \[Element] \[CapitalOmega], 6];

Inspect the eigenvalues.

In[5]:=

vals

Out[5]=

Visualize the real part of the eigenfunctions.

show complete Wolfram Language input

In[6]:=

Grid[Partition[
   Plot3D[Re[#], {x, y} \[Element] \[CapitalOmega], 
      ColorFunction -> "TemperatureMap", Boxed -> False, Axes -> None,
       PlotRange -> All, PlotStyle -> {Specularity[White, 20]}, 
      Method -> {"ShrinkWrap" -> True}] & /@ funs, 3], 
  Background -> Transparent] // Quiet

Out[6]=