Green's function

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

$$\langle{\bf R}\vert f(t)\rangle = \sum\limits_{{\bf R'}} \langle{... ...R}\vert e^{-(\hat H-E)t}\vert {\bf R'}\rangle \langle{\bf R'}\vert f(0)\rangle,$$ (3.12)

or, expressed in terms of the Green's function $G({\bf R},{\bf R'},t) = \langle{\bf R}\vert e^{-(\hat H-E)t}\vert{\bf R'}\rangle$, the above equation reads

$$f({\bf R},t) = \int G({\bf R},{\bf R'},t) f({\bf R'},0)\,{\bf dR'}$$ (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 $G({\bf R'},{\bf R},t)$ is not known, it can be approximated for small values of the argument $t$, and then equation (3.13) can be solved step by step

$$f({\bf R},t + \triangle t) = \int G({\bf R},{\bf R'},\triangle t) f({\bf R'},t)\,{\bf dR'}$$ (3.14)

For further convenience let us split the Hamiltonian into three operators

$$\hat H = \hat H_1 + \hat H_2 + \hat H_3,$$ (3.15)

where

$$\begin{array}{lcl} \hat H_1&=&-D \Delta,\\ \hat H_2&=&D((\vec \nabla \vec F) + \vec F \vec \nabla)),\\ \hat H_3&=&E_L({\bf R}) - E \end{array}$$ (3.16)

and let us introduce the corresponding Green's functions

$$G_i({\bf R},{\bf R'},t) = \langle{\bf R}\vert e^{-\hat H_i t}\vert{\bf R'}\rangle, \qquad i = 1,2,3$$ (3.17)

The exponential operator can be approximated as (the error comes from the noncommutativity of the $\hat H_i$'s, $i = 1,2,3$)

$$e^{-\hat H t} = e^{-\hat H_1 t}e^{-\hat H_2 t}e^{-\hat H_3 t}+O(t^2)$$ (3.18)

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

$$G({\bf R},{\bf R'},t) = \int\int G_1({\bf R},{\bf R_1},t) G_2({\bf R_1},{\bf R_2},t) G_3({\bf R_2},{\bf R'},t)\, {\bf dR}_1 {\bf dR}_2$$

To obtain the three Green's functions one must solve the differential equations

$$\left\{ {\begin{array}{rcll} \displaystyle -\frac{\partial}{\part... ...},{\bf R'},0)& =&\displaystyle\delta({\bf R}-{\bf R'})&\\ \end{array}} \right.$$ (3.19)

The equation for the kinetic term has the form

$$-\frac{\partial G_1({\bf R},{\bf R'},t)}{\partial t} = -D\triangle G_1({\bf R},{\bf R'},t)$$ (3.20)

This is the diffusion equation with diffusion constant $D = \hbar^2 /2m$ and its solution is a Gaussian

$$G_1({\bf R},{\bf R'},t) = (4 \pi D t)^{3N/2} \exp\left(-\frac{({\bf R}-{\bf R'})^2}{4Dt}\right)$$ (3.21)

The equation for the drift force term is

$$-\frac{\partial G_2({\bf R},{\bf R'},t)}{\partial t} = -D \vec \nabla (\vec F G_2({\bf R},{\bf R'},t))$$ (3.22)

and its solution is

$$G_2({\bf R},{\bf R'},t) = \delta({\bf R}-{\bf R}(t)),$$ (3.23)

here ${\bf R}(t)$ is the solution of the classical equation of motion

$$\left\{ {\begin{array}{rcl} \displaystyle\frac{{\bf dR}(t)}{dt}&=... ...R}(t)),\\ \displaystyle{\bf R}(0)&=&\displaystyle{\bf R'} \end{array}} \right.$$ (3.24)

The last equation from (3.19) has trivial solution, which describes the rate term

$$G_3({\bf R},{\bf R'},t) = \exp((E-E_L({\bf R}))t) \delta({\bf R}-{\bf R'})$$ (3.25)