The formal solution of the Schrödinger equation written in coordinate space is
given by
(2.16) |
In other words, the differential Schrödinger equation (2.6)
corresponds to the integral equation (2.17), which can be integrated with
help of Monte Carlo methods. Although the Green's function
is not
known, it can be approximated at small times , and then equation (2.17)
can be solved step by step
The asymptotic solution for large times can be obtained by propagating
for a large number
of time steps .
For further convenience let us split the Hamiltonian into three pieces
(2.21) |
Let us introduce the corresponding Green's functions:
The exponent of a sum of two operators (on the contrary to an exponent of
-numbers) in general can not be written as a product of two exponents. The exact
relation takes into account the non-commutativity
. The primitive
approximation consists in neglecting the noncommutativity
This formula, rewritten in the coordinate representation, gives the expression for
the Green's function
From Eq. 2.22 we find that the Green's function should satisfy Bloch
differential equation:
The equation for the kinetic term has the form
(2.25) |
This is the diffusion equation with diffusion constant
. It can be
conveniently solved in a momentum representation where the kinetic energy operator
is diagonal. Going back to the coordinate representation one finds that the solution
is a Gaussian
The equation for the drift force term is
(2.27) |
The last equation from (2.24) has a trivial solution, which describes the branching term