ORIGINAL ARTICLE: H.S. Lima, M. M. A. Paixão, and C. Tsallis, de Broglie-Bohm analysis of a nonlinear membrane: From quantum to classical chaos. Chaos 34, 023125 (2024). DOI: https://doi.org/10.1063/5.0175044
Quantum physics presents unique phenomena that are completely distinct from classical physics. Yet, a distinct interpretation exists that bridges these two realms, preserving classical structures while introducing a quantum potential. The concept of chaos, so complex and challenging to study at the quantum scale, can be easily explored by introducing the idea of quantum trajectories. In our paper, recently selected as Editor’s Pick by Chaos: An Interdisciplinary Journal of Nonlinear Science, we developed a numerical procedure using the Finite Element Method (FEM) to solve the Schrödinger equation with great accuracy and precision while maintaining the desired stability. After this analysis, we demonstrate that features typically associated with classical systems, such as unpredictability, sensitivity to initial conditions, and exponential deviation of trajectories, can also be used to characterize chaos at the quantum level.

Introduction

The de Broglie-Bohm theory [1,2,3] is an alternative interpretation of Quantum Mechanics that shares close proximity with classical physics, since it is based in the concept of quantum trajectories. In this formalism, the motion of the quantum particles is guided by the wave function
Ψ
through the guidance equations ​​
m
dx
dt
 Im
∇Ψ
Ψ

, (1)​which, when integrated, provide us the particles positions. The pilar of this interpretation is based in the Hamilton-Jacobi equation obtained from the original Schrödinger equation when
Ψ
is substituted for its polar form. The main difference is the presence of a supplementary quantum potential
Q≡-
2
ℏ
2m
2
∇
R
R
, which is responsible for conferring the quantum properties to the system. In the limit when
Q0
, werecover precisely the classical Hamilton-Jacobi equation, enabling us to study the quantum-to-classical transition from quantum to classical systems.​ Such similarity with Classical Mechanics allow us to address very complex subjects, like the quantum chaos, in the same grounds as its classical version, that is, by computing the degree of deviation between a pair of neighboring quantum trajectories. Such measurement can be provided by the Lyapunov exponent, defined as ​​
λ(t)
lim
t∞
1
t
ln

ξ
(t)

ξ
(0)
, (2)​where

ξ

∑
i

2
(δ
x
i
)
+
2
(δ
p
i
)

is the trajectories distance in the phase space. ​ In our article [4], which is a natural extension of our previous work [5], we consider a generic two-dimensional anharmonic oscillator with Hamiltonian​
H
2
p
x
2m
+
2
p
y
2m
+
1
2

2
ω
x
2
x
+
2
ω
y
2
y
-κxy+
α
3

3
x
+
3
y
+
β
4

4
x
+
4
y

. (3) By using the Finite Element Methods (FEM), we solve the associated 2D-Schrödinger equation
-
1
2
2
∇
Ψ+
1
2
2
ω

2
x
+
2
y
Ψ+
α
3

3
x
+
3
y
Ψ+
β
4

4
x
+
4
y
Ψ-κxyΨi
∂Ψ
∂t
. (4) After that, we numerically compute the velocity field given by equation (1) and using the Range-Kutta method we obtained the corresponding quantum trajectories. So, for very close initial positions, we calculate the Lyapunov exponents via equation (2). The complete details regarding the methods can be founded at https://doi.org/10.1063/5.0175044.

Part I: Solving the Schrödinger equation with FEM

In[]:=
$HistoryLength=0;​​PacletObject
Name: FEMAddOns
Version: 1.4.7
;​​Needs["NDSolve`FEM`"];​​Needs["NumericalCalculus`"];​​Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"];
Initial definitions
In[]:=
hbar=1;(*---hbar---*)​​L=5;(*---Boundary:L=25forhbar=1.0,L=10forhbar=0.1,L=8forhbar=0.05---*)​​time=20;​​Nstates=1;(*---Numberosstates---*)​​anorm=1/((Pi*hbar)^(1/4.0));(*---normalizationconstant---*)​​κ=0.1;(*---couplingconstant---*)​​α=0.05;(*---cubicinteractionconstant---*)​​β=0.04;(*---quarticinteractionconstant---*)​​VF[x_,y_]:=0.5*x^2+0.5*y^2-κ*x*y+α*(x^3+y^3)+β*(x^4+y^4);(*---anharmonicpotential+coupling---*)​​psi[x_,n_]:=(1/Sqrt[(2^n)*n!])*anorm*E^(-x^2/(2*hbar))*HermiteH[n,(x/hbar^0.5)];(*---1Deigenstates---*)​​Psi[x_,y_,n1_,n2_]:=psi[x,n1]*psi[y,n2];(*---2Deigenstates---*)​​cond[x_,y_,n1_,n2_]:=(1/(Sqrt[n2+1]*Sqrt[n1+1]))*Sum[Psi[x,y,i,j],{i,0,n1},{j,0,n2}];(*---Ψ(x,y,t=0)---*)
Initial wave function
Ψ(x,y,t0)
In[]:=
cond[x,y,Nstates,Nstates]
Out[]=
1
2
0.56419
-
2
x
2
-
2
y
2

+0.797885
-
2
x
2
-
2
y
2

x+0.797885
-
2
x
2
-
2
y
2

y+1.12838
-
2
x
2
-
2
y
2

xy
Element Mesh for FEM
In[]:=
reg=Rectangle[{-L,-L},{L,L}];​​reginner=Rectangle[{{-5,5},{5,5}}];​​regdisk=Disk[{0,0},{L,L}];​​Omega=ToElementMesh[reg,"MaxCellMeasure"->0.025];​​Omega["Wireframe"]
Out[]=
Time control variables to solve the PDE
In[]:=
$HistoryLength=0;​​showStatus[status_]:=LinkWrite[$ParentLink,SetNotebookStatusLine[FrontEnd`EvaluationNotebook[],ToString[status]]];​​clearStatus[]:=showStatus[""];​​clearStatus[]
Numerical method for solve the PDE
-
1
2
2
∇
Ψ+
1
2
2
ω

2
x
+
2
y
Ψ+
α
3

3
x
+
3
y
Ψ+
β
4

4
x
+
4
y
Ψ-κxyΨi
∂Ψ
∂t
In[]:=
Γ=DirichletCondition[Ψ[t,x,y]==0,True];​​op=-((hbar^2)/2)*Laplacian[Ψ[t,x,y],{x,y}]+VF[x,y]*Ψ[t,x,y]-I*hbar*D[Ψ[t,x,y],t];​​method1={"MethodOfLines","TemporalVariable"->t,"SpatialDiscretization"->{"FiniteElement","IntegrationOrder"->5,"MeshOptions"->{"ComplexMesh"->True},"PDESolveOptions"->{"LinearSolver"->{Automatic,Method->{"Krylov",Method->"BIGSTAB",Tolerance->10^-8,"Preconditioner"->"ILU0"}}}},"DifferentiateBoundaryConditions"->{True,"ScaleFactor"->10^4}};
Solution of the Schrödinger equation
In[]:=
$HistoryLength=0;​​sol=NDSolveValue[{op==0,Γ,Ψ[0,x,y]cond[x,y,Nstates,Nstates]+I*0},Ψ,{t,0,time},{x,y}∈Omega,Method->method1,options,EvaluationMonitor:>showStatus["t = "<>ToString[CForm[t]]]];
Normalization Condition
testTime=10;​​NumInt=NIntegrate[Abs[sol[testTime,x,y]]^2,{x,-L,L},{y,-L,L},Method->"GlobalAdaptive",PrecisionGoal->6,WorkingPrecision->6];(*---Theintegrationmethodcanbechangedtobetterprecision---*)​​{"I[|ψ(x,t="<>ToString[testTime]<>")|^2] = "<>ToString[NumInt],"Error(t="<>ToString[testTime]<>")="<>ToString[Abs[1-NumInt]]}​​
Out[]=
{I[|ψ(x,t=10)|^2] = 0.985505,Error(t=10)=0.014495}
To export the solution...
Export["path\\filename.mx",sol];
Probability Density Plot
In[]:=
strx0[t_]:="t = "<>ToString@NumberForm[t,{Infinity,1}];​​strx1[κ_]:="κ = "<>ToString@NumberForm[κ,{Infinity,1}];​​legend[t_]:=Style[strx0[t],15,FontFamily->"Arial",White];​​legend1[κ_]:=Style[strx1[κ],15,FontFamily->"Arial",White];​​ContourPlotψ[t_]:=ContourPlot[Abs[sol[t,x,y]]^2,{x,-4,4},{y,-4,4},PlotPoints->30,Contours->10,Frame->True,FrameLabel{Style["x",Black,Italic,FontFamily"Arial",18],Style["y",Black,Italic,FontFamily"Arial",18]},FrameStyle->Directive[Black],LabelStyle{FontFamily"Arial",16},ColorFunction->ColorData["TemperatureMap"],ColorFunctionScaling->True,AspectRatio1,PlotRange->All,AspectRatio->1];​​plotc[t_,κ_]:=Show[ContourPlotψ[t],Epilog->{Inset[legend[t],Scaled[{0.14,0.91}]],Inset[legend1[κ],Scaled[{0.85,0.91}]]}];
To export the plots at t=taux
End of Part I

Part II: Computing the velocity field

Import a previous solution of the PDE
Velocity Field Plot
End of Part II

Part III: Trajectories

Sensitivity to initial conditions

Part IV: Lyapunov Exponent

Lyapunov Exponent Calculation
In computing the Lyapunov exponents, it’s essential to determine the phase space distance between each trajectory pairs. As the degree of chaos, indicated by the Lyapunov exponent, is a local property, it varies across different initial positions. Hence, the approach involves considering a significant number of initial conditions and computing their average value to characterize the system’s overall dynamic behavior.
Phase space distance
Export normalized phase space distance
Import normalized phase space distance

References

[1] L . de Broglie, Interference and Corpuscular Light, Nature 118, 441 (1926) .
[2] D. Bohm, A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables I, Phys. Rev. 85, 166 (1952).
[3] D. Bohm, A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables II, Phys. Rev. 85, 180 (1952).
[4] H. S. Lima and M. M. A. Paixão, de Broglie-Bohm analysis of a nonlinear membrane: From quantum to classical chaos. Chaos 34, 023125 (2024). https://doi.org/10.1063/5.0175044
[5] M. M. A. Paixão and H. S. Lima, Numerical validation of Ehrenfest theorem in a Bohmian perspective for non-conservative systems, Eur. Phys. J. Plus 138, 549 (2023). https://doi.org/10.1140/epjp/s13360-023-04023-z

CITE THIS NOTEBOOK

de Broglie–Bohm analysis of a nonlinear membrane: from quantum to classical chaos​
by Henrique Santos Lima, Matheus Paixão & Constantino Tsallis
Wolfram Community, STAFF PICKS, March 11, 2024
​https://community.wolfram.com/groups/-/m/t/3139108