Solving Physics Problems with Schrödinger’s Equation

Ricardo López, Physics application developer.

(RowDefault[, (Spacer1), , (Spacer1), ])  #WolframTechConf

Agenda.

SchrodingerPDEComponent

Time independent problems

1D

2D

Axisymmetric case

3D

Quantum dots application.

Time dependent problem.

Schrödinger equation

i ℏ((∂Ψ(t,x))/(∂t))= ∇·(-((ℏ^2)/(2m)) ∇Ψ(t,x)) +V(t,x) Ψ(t,x)

SchrodingerPDEComponent

SchrodingerPDEComponent

N[SchrodingerPDEComponent[{Ψ[x],{x}},<||>]]

SchrodingerPDEComponent[{Ψ[x],{x}},<|"ReducedPlanckConstant"->ℏ,"Mass"->1|>]

%/.{ℏ->N@QuantityMagnitude[(Quantity[, "ℏ", reduced Planck constants, "ReducedPlanckConstant"]),"SIBase"]}

Time independent problems

1D:

harmonic oscillator

ℋ=SchrodingerPDEComponent[{Ψ[x],{x}},<|"ReducedPlanckConstant"->1,"Mass"->1,"SchrodingerPotential"->(x^2)|>]

{en,funs}=NDEigensystem[ℋ,Ψ[x],{x,-3,3},6];

Plot[{Evaluate[(funs^2)+en],(x^2)},{x,-3,3},]

A bit more complicated potential

ℋ=SchrodingerPDEComponent[{Ψ[x],{x}},<|"ReducedPlanckConstant"->1,"Mass"->1,"SchrodingerPotential"->UnitStep[x] (2 (x^3)) + UnitStep[-x] (((x^2)/2)) |>]

{en,funs}=NDEigensystem[ℋ,Ψ[x],{x,-5,5},6];

Plot[{Evaluate[(funs^2)+en],UnitStep[x] (2 (x^3)) + UnitStep[-x] (((x^2)/2))},{x,-5,5},]

3D to 2D Axisymmetric case:

Separation of variables: Ψ(r,ϕ,z)=ψ(r,z)(e^(-i l ϕ))

ℋ=SchrodingerPDEComponent[{ψ[r,z],{r,z}},<|"ReducedPlanckConstant"->1,"Mass"->(1/2),"RegionSymmetry"->"Axisymmetric","AzimuthalQuantumNumber"->l|>]

Ω=DiscretizeRegion[Disk[{4,0},1],"MaxCellMeasure"->0.005];

Graphics3D[{{Opacity[0.1],Torus[{0,0,0},{3,5}]},{MeshRegion[]}},]

bc=DirichletCondition[ψ[r,z]==0,True];

{en,funs}=NDEigensystem[{ℋ,bc}/.{l->3},ψ[r,z],{r,z}∈Ω,6];

Table[DensityPlot3D[#/.{r->(Sqrt[(x^2)+(y^2)]),ϕ->ArcTan[x,y],l->3},{x,-6,6},{y,-6,6},{z,-6,6},]&/@{Re[(E^(-I l ϕ  )) funs[[i]]],Im[(E^(-I l ϕ  )) funs[[i]]]},{i,Length[en]}]

3D:

Research problem:

Introduction to quantum dots:

Semiconductor quantum dots are, in layman’s terms, nanoscale islands made out of semiconductor materials, that can trap charge carriers like electrons.

Growing process

Cross-sectional scanning tunneling microscopy (X-STM)

Mikkelsen et al. 2005. Cross - sectional  scanning  tunneling  microscopy  studies  of  novel  III–V  semiconductor  structures . https://doi.org/10.1016/j.progsurf.2005.10.001

Gajjela, et  al . 2021  "Atomic-Scale Characterization of Droplet Epitaxy Quantum Dots"   https://doi.org/10.3390/nano11010085

Confinement effect

S. M. Sze , Physics of Semiconductor Devices, John Wiley & Sons, Inc. (1981)

b=14;
WL=0.5;
L=30;
h=10;

Ω=Cube[L];
pyramid=Pyramid[{{-((#)/2),-((#)/2),0.},{((
#)/2),-((#)/2),0.},{((#)/2),((#)/2),0.},{-((#)/2),((#)/2),0.},{0.,0.,#2}}]&[b,h];
wetLayer=Cuboid[{-(L/2),-(L/2),-WL},{(L/2),(L/2),0}];
QDot= RegionUnion[pyramid,wetLayer];
Region[QDot,]

Needs["NDSolve`FEM`"];

mesh=ToElementMesh[RegionUnion[RegionDifference[Ω,QDot],wetLayer],MaxCellMeasure->0.1h]; 
Show[mesh["Wireframe"],]

QDotMemberFunction=RegionMember[QDot];

m0=QuantityMagnitude[(Entity["electron", Entity["Particle","Electron"], "Entity[\"Particle\", \"Electron\"]", "particle"])[(EntityProperty["mass", EntityProperty["Particle","Mass"], "EntityProperty[\"Particle\", \"Mass\"]"])],"Kilograms"];

meff[x_,y_,z_]:=Quantity[m0 Piecewise[{{0.04,QDotMemberFunction[{x,y,z}]}},0.067],"Kilograms"];

V[x_,y_,z_]:=Quantity[Piecewise[{{0,QDotMemberFunction[{x,y,z}]}},0.500],"Electronvolts"];

ℋ=SchrodingerPDEComponent[{Ψ[x,y,z],{x,y,z}},<|"Mass"->meff[x,y,z],"SchrodingerPotential"->V[x,y,z],"ScaleUnits"->{"Meters"->"Nanometers"}|>]

bc=DirichletCondition[Ψ[x,y,z]==0,z==(L/2)||z==-(L/2)];

{en,funs}=NDEigensystem[{ℋ,bc},Ψ[x,y,z],{x,y,z}∈mesh,6];

Table[Show[ContourPlot3D[funs[[i]],{x,y,z}∈mesh,],mesh["Edgeframe"]],{i,Length[en]}]

Table[Show[DensityPlot3D[((funs[[i]])^2),{x,y,z}∈mesh,],mesh["Edgeframe"]],{i,Length[en]}]

UnitConvert[Quantity[en,"Kilograms"((("Nanometers")^2)/(("Seconds")^2)) ],"Millielectronvolts"]

Time dependent example:

Ω=DiscretizeRegion[BooleanRegion[#1&&!(#2||#3||#4)&,{Rectangle[],Rectangle[],Rectangle[],Rectangle[]}],MaxCellMeasure->0.4];

RegionPlot[Ω]

ic=((E^(2 I y+(1/18) (-(x^2)-(((10+y))^2))))/((Sqrt[3]) (π^(1/4))));

GraphicsRow[{DensityPlot[Re[ic[[2]]],],DensityPlot[Im[ic[[2]]],]}]

bc=DirichletCondition[Ψ[t,x,y]==0,True];

ℋ=SchrodingerPDEComponent[{Ψ[t,x,y],t,{x,y}},<|"ReducedPlanckConstant"->1|>]

solution=NDSolveValue[{ℋ== 0,Ψ[0,x,y]==ic,bc},Ψ,{t,0,8},{x,y}∈Ω]

animation=Table[DensityPlot[((Abs[solution[t,x,y]])^2),{x,y}∈ Ω,],{t,0,8,0.1} ];

ListAnimate[animation,]

Want to see more?

ref/SchrodingerPDEComponent

PDEModels/tutorial/Physics/ModelCollection/QuantumRing

Thanks
Questions?