schroedinger_hydrogen.C File Reference

Detailed Description

Plot the Amplitude of a Hydrogen Atom.

Visualize the Amplitude of a Hydrogen Atom in the n = 2, l = 0, m = 0 state. Demonstrates how TH2F can be used in Quantum Mechanics.

The formula for Hydrogen in this energy state is \( \psi_{200} = \frac{1}{4\sqrt{2\pi}a_0 ^{\frac{3}{2}}}(2-\frac{\sqrt{x^2+y^2}}{a_0})e^{-\frac{\sqrt{x^2+y^2}}{2a_0}} \)

 
#include <cmath>
 
double WaveFunction(double x, double y)
{
   double r = sqrt(x * x + y * y);
 
   double w = (1 / pow((4 * sqrt(2 * TMath::Pi()) * 1), 1.5)) *
              (2 - (r / 1) * pow(TMath::E(), (-1 * r) / 2)); // Wavefunction formula for psi 2,0,0
 
   return w * w; // Amplitude
}
 
void schroedinger_hydrogen()
{
   TH2F *h2D = new TH2F("Hydrogen Atom",
                        "Hydrogen in n = 2, l = 0, m = 0 state; Position in x direction; Position in y direction", 200,
                        -10, 10, 200, -10, 10);
 
   for (float i = -10; i < 10; i += 0.01) {
      for (float j = -10; j < 10; j += 0.01) {
         h2D->Fill(i, j, WaveFunction(i, j));
      }
   }
 
   gStyle->SetPalette(kCividis);
   gStyle->SetOptStat(0);
 
   TCanvas *c1 = new TCanvas("c1", "Schroedinger's Hydrogen Atom", 750, 1500);
   c1->Divide(1, 2);
 
   auto c1_1 = c1->cd(1);
   c1_1->SetRightMargin(0.14);
   h2D->GetXaxis()->SetLabelSize(0.03);
   h2D->GetYaxis()->SetLabelSize(0.03);
   h2D->GetZaxis()->SetLabelSize(0.03);
   h2D->SetContour(50);
   h2D->Draw("colz");
 
   TLatex *l = new TLatex(
      -10, -12.43,
      "The Electron is more likely to be found in the yellow areas and less likely to be found in the blue areas.");
   l->SetTextFont(42);
   l->SetTextSize(0.02);
   l->Draw();
 
   auto c1_2 = c1->cd(2);
   c1_2->SetTheta(42.);
 
   TH2D *h2Dc = (TH2D *)h2D->Clone();
   h2Dc->SetTitle("3D view of probability amplitude;;");
   h2Dc->Draw("surf2");
}

Author

Advait Dhingra

Definition in file schroedinger_hydrogen.C.