Computational Physics

12.2 The variational Monte Carlo method 383

A particular diffusion equation which we shall encounter later in this chapter is

∂ρ

∂τ

=

1

2

∂^2 ρ(x,τ)

∂x^2

−V(x)ρ(x,τ). (12.33)

This looks very much like the one-dimensional time-dependent Schrödinger
equation for a zero-mass particle; in fact, this equation is recovered when we con-
tinue the time analytically into imaginary timeτ =it(we useτfor imaginary
time). Using(12.31), we can write the solution as

ρ(x,τ)=eτ(−K−V)ρ(x,0) (12.34)

whereKis the kinetic energy operatorK =p^2 / 2 =− 1 / 2 (∂^2 /∂x^2 )(pis the
momentum operatorp=−i(∂/∂x)of quantum mechanics). The exponent cannot
be evaluated because the operatorsKandVdo not commute. However, we might
neglect Campbell–Baker–Hausdorff (CBH) commutators – this is only justified
whenτis small. To emphasise that the following is only valid for smallτ, we shall
use the notation
τinstead ofτ.Wehave

e−

τ (K+V)=e−

τKe−

τV+O(

τ^2 ) (12.35)

where the order
τ^2 error term results from the neglect of CBH commutators.
To find the Green’s function explicitly, we must find the matrix element of the
exponential operator on the right hand side. The term involving the potential is
not a problem as this is simply a function ofx. It remains then to find the matrix
elements of the kinetic operator:

GKin(x,y;

τ )=〈x|e−

τˆp

(^2) / 2
|y〉 (12.36)
wherepˆis the momentum operator – we have used the caret ˆ to distinguish the
operator from its eigenvalue.
The Green’s function can be evaluated explicitly by inserting two resolutions of
the unit operator of the form

∫

dp|p〉〈p|and using the fact that

〈x|p〉=

1

√

2 π

eipx (
≡ 1 ). (12.37)

As the kinetic operator is diagonal in thep-representation, the matrix element is
then found simply by performing a Gaussian integral. The result is

GKin(x,y;

τ )=

1

√

2 π

τ

e−(x−x

′) (^2) /( 2
τ )

. (12.38)

This form is recognised as the Green’s function of the simple diffusion equation;
indeed our imaginary-time Schrödinger equation reduces to this equation forV≡0,
and therefore the kinetic part of our Green’s function should precisely be equal to
the Green’s function of the simple diffusion equation. We have derived this form