The formal solution of the Schrödinger equation written in
coordinate space is given by
(3.12)
or, expressed in terms of the Green's function
,
the above equation reads
(3.13)
In other words, the differential Schrödinger equation
(3.3) corresponds to the integral equation
(3.13), which can be integrated with help of Monte
Carlo methods. Although the Green's function
is not
known, it can be approximated for small values of the argument ,
and then equation (3.13) can be solved step by step
(3.14)
For further convenience let us split the Hamiltonian into three
operators
(3.15)
where
(3.16)
and let us introduce the corresponding Green's functions
(3.17)
The exponential operator can be approximated as (the error comes
from the noncommutativity of the 's, )
(3.18)
This formula, rewritten in coordinate representation, gives approximation
for the Green's function
To obtain the three Green's functions one must solve the
differential equations
(3.19)
The equation for the kinetic term has the form
(3.20)
This is the diffusion equation with diffusion constant
and its solution is a Gaussian
(3.21)
The equation for the drift force term is
(3.22)
and its solution is
(3.23)
here is the solution of the classical equation of motion
(3.24)
The last equation from (3.19) has trivial solution, which
describes the rate term