Computational Physics

390 Quantum Monte Carlo methods

0 0.5 1 1.5 2 2.5 3 3.5 4

r

Figure 12.1. Ground state wave function (timesr^2 ) for the three-dimensional

harmonic oscillator as resulting from the DMC calculation (dots) compared with

the exact form, scaled to match the numerical solution best.

the transition probability should be positive. This is called thefermion problem.We
shall come back to this later. Another problem arises when the interaction potential
assumes strongly negative values. This will be discussed in some detail in the next
section and then we shall consider a refinement of the DMC which is not susceptible
to this problem.

12.3.2 Applications

We apply the DMC procedure first to the three-dimensional harmonic oscillator.
The exact ground state wave function is given by

ψ(r)=

1

( 2 π)^3 /^2

e−r

(^2) / 2
; (12.56)
the energy is 3/2 (in dimensionless units). It should be noted that the probability
distribution for finding a walker at a distancerfrom the origin is given by the
wave function timesr^2 , because the volume of a spherical shell of thickness dr
is 2πr^2 dr. For an average population of 300 walkers executing 4000 steps and
a time stepτ =0.05, we findEG =1.506±0.015, to be compared with the
exact value 1^12. The distribution histogram is shown inFigure 12.1, together with
the exact wave function, multiplied byr^2 and scaled in amplitude to fit the DMC
results best. Ground state energy and wave function are calculated with reasonable
accuracy. Note that these results are obtained without using any knowledge of the
exact solution: the diffusion process ‘finds’ the ground state by itself.