Schrodinger's Equation in a Two-Dimensional Stadium

Define the region of interest as a Bunimovich stadium:

In[1]:=

Ω=StadiumShape[{{-0.4,0},{0.4,0}},0.4]

Out[1]=

StadiumShape[{{-0.4,0},{0.4,0}},0.4]

Visualize it:

In[2]:=

Graphics[Ω]

Out[2]=

Define the Schrödinger PDE operator in units such that

ℏ=2m=1

:

In[3]:=

ℋ=SchrodingerPDEComponent{Ψ[x,y],{x,y}},"ReducedPlanckConstant"1,"Mass"

1

2



Out[3]=

∇

{x,y}

·{{-1,0},{0,-1}}.

∇

{x,y}

Ψ[x,y]

Set the boundary condition such that the wavefunctions vanishes at the boundary and find the first 50 eigenstates:

In[4]:=

{en,funs}=NDEigensystem[{ℋ,DirichletCondition[Ψ[x,y]0,True]},Ψ[x,y],{x,y}∈Ω,50];

Plot the wavefunctions for states

1

,

7

,

20and50

:

In[5]:=

ContourPlotfuns#,{x,y}∈Ω,

&/@{1,7,20,50}//Multicolumn

Out[5]=

Wolfram Language Example Repository