My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 3.
QUANTUM MONTE CARLO METHODS
a single particle; the diffusion takes place in Nd dimensions, where N is the number
of particles and d is the dimension of the physical space.
The challenge is to incorporate the potential into the simulation. This can be
achieved by assigning a weight w i to each walker; the weights are allowed to vary
as the walker explores configuration space, and ultimately the density of the walker
weights (rather than the walkers themselves) is used to represent Ψ. Neglecting the
diffusion term in equation (3.28) leaves the following rate equation:
which has the solution
∂Ψ(R, τ)
∂τ
= −V (R)Ψ(R, τ) (3.29)
Ψ(R, τ) = Ψ(R, 0)e −τV (R) . (3.30)
The potential has the effect of increasing the wave function where V is negative,
and decreasing it where V is positive. When a walker arrives at the position R, its
weight should be multiplied by e −τV (R) , in accordance with equation (3.30). After
n steps of size ∆τ, the walker’s weight becomes
[
n∑
w i (n∆τ) = exp −∆τ V ( R i (m∆τ) )] . (3.31)
m=1
In fact, the weight of a walker should reflect the whole path in configuration
space along which it has travelled; however, in a simulation, the complete path of
the walker is not specified because walkers must move in discrete steps.
This is
a very important point: the procedure described here is valid only for small time
steps, when the potential V can be assumed to remain constant over the course of
a move. In the limit of infinitely small time steps, the walker weight becomes
[ ∫ τ
−
as it should.
lim w i (τ) = exp
∆τ→0
n∆τ=τ
0
V ( R i (τ ′ ) ) dτ ′ ]
, (3.32)
The difference between the simple diffusion equation and the imaginary-time
Schrödinger equation is that the Green’s function for the latter is not known precisely.
The simulation technique outlined here is equivalent to using an approximation
to the Green’s function which is valid for short time steps.
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