The wavefunction of the system satisfies the Schrödinger equation
(3.1)
where
denotes the particle
coordinates. This equation can be rewritten in imaginary time
.
(3.2)
where is an energy shift whose meaning will become clearer later.
The formal solution of this equation is
(3.3)
This solution can be expanded in eigenstate functions of the
Hamiltonian
,
(3.4)
The amplitudes of the components change with time, either
increasing or decreasing depending on the sign of . At
large times the term that corresponds to the projection on the
ground state dominates the sum. In other words all excited states
decay exponentially fast and only contribution from ground state
survives
(3.5)
In the long time limit the wavefunction remains finite only when
is equal to . This provides a method to obtain the ground
state energy by adjusting the parameter in a way that the norm
of
is constant.
Let us consider system of particles, introducing the
Hamiltonian through a pair-wise potential
(3.6)
and the Schrödinger equation reads
(3.7)
where the following notation is used:
and
. In principle,
any external field which is independent of the particle momenta and
is a function only of the particle coordinates can be included into
without any harm to the reasoning.
Better efficiency is achieved if the importance sampling is used.
In the DMC method this means that one has to solve the
Schrödinger equation for the modified wavefunction
3.1
(3.8)
Here
is the trial wavefunction which
approximates the true wavefunction
of the system.
The distribution function satisfies the following equation
(3.9)
here denotes the local energy which is the average of
the Hamiltonian with respect to trial wavefunction
(3.10)
and is the drift force which is proportional to the
gradient of the trial wavefunction and consequently always points
in the direction where increases