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Green's function

The formal solution of the Schrödinger equation written in coordinate space is given by

$\displaystyle \langle{\bf R}\vert f(\tau)\rangle =
\sum\limits_{{\bf R}'} \lang...
...vert e^{-(\hat H-E)\tau}\vert {\bf R}'\rangle
\langle{\bf R}'\vert f(0)\rangle,$     (2.16)

or, expressed in terms of the Green's function $G({\vec r},{\bf R}',\tau) =
\langle{\bf R}\vert e^{-(\hat H-E)t}\vert{\bf R}'\rangle$, the above equation reads as
$\displaystyle f({\vec r},\tau) = \int G({\vec r},{\bf R}',\tau) f({\bf R}',0)\,{\bf dR'}$     (2.17)

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 $G({\bf R}',{\vec r},\tau)$ is not known, it can be approximated at small times $\tau$, and then equation (2.17) can be solved step by step

$\displaystyle f({\vec r},t + \Delta\tau) = \int G({\vec r},{\bf R}',\Delta\tau) f({\bf R}',\tau)\,{\bf dR'}$     (2.18)

The asymptotic solution for large times can be obtained by propagating $f({\vec r},\tau)$ for a large number of time steps $\Delta \tau$.

$\displaystyle f({\vec r},\tau)\to\psi_T({\bf R})\phi_0({\bf R}),\quad\tau\to\infty$     (2.19)

For further convenience let us split the Hamiltonian into three pieces

$\displaystyle \hat H = \hat H_1 + \hat H_2 + \hat H_3,$     (2.20)

where
$\displaystyle \begin{array}{lcl}
\hat H_1&=&-D \Delta,\\
\hat H_2&=&D((\nabla_...
...f F}) + {\bf F}\nabla_{\bf R})),\\
\hat H_3&=&E^{loc}({\bf R}) - E
\end{array}$     (2.21)

Let us introduce the corresponding Green's functions:

$\displaystyle G_i({\vec r},{\bf R}',\tau) = \langle{\bf R}\vert e^{-\hat H_i\tau}\vert{\bf R}'\rangle,\qquad i = 1,2,3$     (2.22)

The exponent of a sum of two operators (on the contrary to an exponent of $c$-numbers) in general can not be written as a product of two exponents. The exact relation takes into account the non-commutativity $\exp\{-(\hat A+\hat B)\tau+[\hat
A,\hat B]\tau^2/2\}=\exp\{-\hat A\tau\}\exp\{-\hat B\tau\}$. The primitive approximation consists in neglecting the noncommutativity

$\displaystyle e^{-\hat H\tau} = e^{-\hat H_1\tau}e^{-\hat H_2\tau}e^{-\hat H_3\tau}+O(\tau^2)$     (2.23)

This formula, rewritten in the coordinate representation, gives the expression for the Green's function

$\displaystyle G({\vec r},{\bf R}',\tau) = \int\!\!\!\!\int
G_1({\vec r},{\bf R_...
...({\bf R_1},{\bf R_2},\tau) G_3({\bf R_2},{\bf R}',\tau)\,
{\bf dR}_1 {\bf dR}_2$      

From Eq. 2.22 we find that the Green's function should satisfy Bloch differential equation:

$\displaystyle \left\{
{\begin{array}{rcll}
\displaystyle -\frac{\partial}{\part...
...},{\bf R}',0)& =&\displaystyle\delta({\bf R}-{\bf R}')&\\
\end{array}}
\right.$     (2.24)

The equation for the kinetic term has the form

$\displaystyle -\frac{\partial G_1({\vec r},{\bf R}',\tau)}{\partial \tau} =-D\Delta G_1({\vec r},{\bf R}',\tau)$     (2.25)

This is the diffusion equation with diffusion constant $D = \hbar^2/2m$. 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

$\displaystyle G_1({\vec r},{\bf R}',\tau) = (4 \pi D\tau)^{-3N/2}\exp\left\{-\frac{({\bf R}-{\bf R}')^2}{4D\tau}\right\}$     (2.26)

The equation for the drift force term is

$\displaystyle -\frac{\partial G_2({\vec r},{\bf R}',\tau)}{\partial \tau} =-D \nabla_{\bf R}({\bf F}G_2({\vec r},{\bf R}',\tau))$     (2.27)

and its solution is
$\displaystyle G_2({\vec r},{\bf R}',\tau) = \delta({\bf R}-{\bf R}(\tau)),$     (2.28)

here ${\bf R}(\tau)$ is the solution of the classical equation of motion
$\displaystyle \left\{
{\begin{array}{rcl}
\displaystyle\frac{{\bf dR}(\tau)}{d\...
...\tau)),\\
\displaystyle{\bf R}(0)&=&\displaystyle{\bf R}'
\end{array}}
\right.$     (2.29)

The last equation from (2.24) has a trivial solution, which describes the branching term

$\displaystyle G_3({\vec r},{\bf R}',\tau) = \exp\{(E-E^{loc}({\bf R}))\,\tau\}~\delta({\bf R}-{\bf R}')$     (2.30)


next up previous contents
Next: Primitive algorithm Up: Diffusion Monte Carlo Previous: Schrödinger equation   Contents
G.E. Astrakharchik 15-th of December 2004