Diffusion Quantum Monte Carlo
The time dependent Schrödinger equation
- i h dU(X)/dt = h²/2m d²/dx² U(X) - V U(X) .
Replacing with
t = it /
h yields an equation similar in
form to the diffusion equation.
d U(X) / dt = h²/2m d²/dx² U(X) - V U(X) .
Interpret this as a diffusion problem, and use Einstein's relationship between diffusion and random walks to simulate
h²/2m = 1/2 (delta x)²/(delta t)
Excited states decay faster than the ground state, so after enough iterations, only the ground state is left. Add a constant to V (the potential energy at a position X) to bring the ground state to roughly zero, and the ground state will no longer decay or grow.
The video demonstrates growth,
splitting, and destruction as particles fall off the box. Follow the
purple psip in the lower left corner.
