Dynamic (Diffusion) (Green's function) Monte-Carlo method in quantum physics.

Imaginary time Schrödinger equation. The Schrödinger equation i^∂ψ/_∂t = Hψ has a formal solution ψ = e^-iHtψ_init. With imaginary time τ=it the solution ψ = e^-Hτψ_init will decay exponentially and only the state with the lowest energy, the ground state, will survive. So the trick is: start with an arbitrary function ψ_init and propagate it in imaginary time until the ground state filters out.

Propagation in imaginary time in small steps Δτ. For a small time Δτ one can approximate e^-iHΔτ ≈ e^-iTΔτe^-iVΔτ, where T and V are the kinetic and potential energy operators. The matrix elements of these operators in coordinate space are

Random-walk simulation of the propagation process. The wave-function is represented as density distribution of "walkers". Then the e^-iTΔτ operator acts on this distribution as: take a random number Δx from a gaussian distribution with our σ and move the walker this distance. The e^-iVΔτ operator acts as: if V(x)>E_o kill the walker with probability 1 - e^-(V-E_o)Δτ; if V(x)<E_o duplicate the walker with probability e^-(V-E_o)Δτ - 1. Here E_o is some parameter which governs the number of walkers.

Energy estimate. Adjust the parameter E_o such that the number of walkers is constant. When you reach equilibrium this will be the ground state energy.

Importance sampling. We probably won't reach that far in one lecture anyway. Read [F.Mentch, J.B.Anderson, J.Chem.Phys. 74 (1981) 6307] and references therein.

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