let u = [];

let K = [];

let r = 0;

let norm = 0;

let N=100;

let dt;

let h;

let t=0;

​

function setup() {

  createCanvas(400, 400);

  h=1/N;

  dt=0.3*pow(h,2);

  for (var i=0;i<N+1;i++){

    u[i]=[];

    K[i]=[];

  for (var j=0;j<N+1;j++){

    u[i][j]=0;    r=pow((i-50)/h,2)+pow((j-50)/h,2)+0.000001;

    K[i][j]=5/sqrt(r);

  }

  }

  for (var m=0;m<20;m++){

  for (var n=0;n<20;n++){

  u[40+m][40+n]=1;

  }

  }

}

​

function draw() {

  background(220);

  t=t+1000*dt;

  noStroke();

  norm = 0;

  for (var i=1;i<N;i++){

  for (var j=1;j<N;j++){

u[i][j]=u[i][j]+0.5*dt*(u[i+1][j]-4*u[i][j]+u[i-1][j]+u[i][j+1]+u[i][j-1])/pow(h,2)+dt*u[i][j]*K[i][j];

  norm = norm + pow(u[i][j]*h,2);

  }

  }

  for (var i=1;i<N;i++){

  for (var j=1;j<N;j++){

  u[i][j]=u[i][j]/sqrt(norm);

  fill(255,0,0,100*u[i][j]);

  square(4*i,4*j,4);

  }

  }

  for (var i=1;i<N-1;i++){

  fill(255,0,0,255);

  ellipse(4*i,200-30*u[i][50],4);

  fill(0,255,0,255);

  ellipse(4*i,200-30*u[i][50]*cos(t),4);

  fill(0,0,255,255);

  ellipse(4*i,200-30*u[i][50]*sin(t),4);

  }

  fill(0);

  text(norm,350,390);

  ellipse(200,200,10);

  text("norm(psi) = ",300,390);

  text("attractive proton kernel",160,215);

  text("2d 1-Electron Atom Ground State Wave Function Psi(x,t) solves",20,20);

  text("i*dPsi/dt + HPsi =0,   H = - 0.5*Laplacian - 1/r, r = 1/sqrt(x^2+y^2).      ",50,40);

//  text("Minimise sum of kinetic + potential energy:",20,60);

//  text("integral (0.5*|grad(u)|^2 - u^2/K(x,y))dxdy ",50,90);

  text("takes form Psi(x,t) = exp(i*E*t)*psi(x) = cos(E*t)*psi(x) + i*sin(E*t)*psi(x)",20,60);

  text("with psi(x) eigenfunction of H satisfying Hpsi = E*psi, E smallest eigenvalue. ",0,80);

  text("Compute psi by time stepping dpsi/dt+Hpsi=0 with normalisation",20,110)

  text("norm(psi) = sqrt(integral psi^2dxdy) = 1     (*)",100,130);

  text("See concentration of electron density psi^2 (red) towards kernel",20,250);

  text("with concentration balanced by Laplacian (kinetic energy)",20,270);

  text("red: eigenfunction psi, density, mid section",20,350);

  text("blue, green: components of Psi, mid section",20,370);

  text("See time dependency of Psi(x,t).",20,290);

  text("Electron oscillates (Psi), electron density (psi) stationary.",20,310);

  stroke(0);

  line(0,200,400,200);

    }