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Schrödinger equation

The wavefunction of the system satisfies the Schrödinger equation
$\displaystyle i\hbar \frac{\partial}{\partial \tau} \varphi ({\bf R},\tau) =
\hat H \varphi ({\bf R},\tau),$     (3.1)

where ${\bf R} = (\vec r_1, \vec r_2, ...)$ denotes the particle coordinates. This equation can be rewritten in imaginary time $t =
i\tau / \hbar$.
$\displaystyle -\frac{\partial}{\partial t} \varphi ({\bf R},t) =
(\hat H -E) \varphi ({\bf R},t),$     (3.2)

where $E$ is an energy shift whose meaning will become clearer later. The formal solution of this equation is
$\displaystyle \psi ({\bf R},t) = e^{-(\hat H - E)t} \psi ({\bf R},0)$     (3.3)

This solution can be expanded in eigenstate functions of the Hamiltonian $\hat H\phi_n~=~E_n\phi_n$, $E_0 < E_1 < ...$
$\displaystyle \psi ({\bf R},t) =
\sum\limits_n c_n \phi_n({\bf R},t) =
\sum\limits_n c_n \phi_n({\bf R},0) e^{-(E_n - E)t}$     (3.4)

The amplitudes of the components change with time, either increasing or decreasing depending on the sign of $(E_n-E)$. 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
$\displaystyle \psi ({\bf R},t) \to c_0 \phi_0({\bf R},0) e^{-(E_0 - E)t}
\qquad \mbox{if } t \to \infty$     (3.5)

In the long time limit the wavefunction remains finite only when $E$ is equal to $E_0$. This provides a method to obtain the ground state energy by adjusting the parameter $E$ in a way that the norm of $\psi({\bf R},t)$ is constant. Let us consider system of $N$ particles, introducing the Hamiltonian through a pair-wise potential
$\displaystyle \hat H =
-\frac{\hbar^2}{2m} \sum\limits_{i=1} \Delta_i
+ \sum\limits_{i<j}^N V(\vert\vec r_i -\vec r_j\vert),$     (3.6)

and the Schrödinger equation reads
$\displaystyle -\frac{\partial}{\partial t} \psi ({\bf R},t) =
-D \sum\limits_{i...
...N \Delta_i \psi ({\bf R},t)
+ V({\bf R}) \psi ({\bf R},t) - E \psi ({\bf R},t),$     (3.7)

where the following notation is used: $D = \hbar^2 /2m$ and $V({\bf
R}) = \sum\limits_{i<j} V(\vert\vec r_i - \vec r_j\vert)$. 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 $V({\bf R})$ 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
$\displaystyle f({\bf R},t) = \psi_T ({\bf R},t) \psi ({\bf R},t)$     (3.8)

Here $\psi_T({\bf R},t)$ is the trial wavefunction which approximates the true wavefunction $\psi({\bf R},t)$ of the system. The distribution function $f$ satisfies the following equation
$\displaystyle -\frac{\partial}{\partial t} f ({\bf R},t) =
-D \sum\limits_{i=1}...
... R},t)
+ D \vec \nabla (\vec F f({\bf R},t))
+ (E_L({\bf R}) - E) f({\bf R},t),$     (3.9)

here $E_L$ denotes the local energy which is the average of the Hamiltonian with respect to trial wavefunction
$\displaystyle E_L({\bf R}) =
\frac{\psi_T^*({\bf R}) \hat H \psi_T({\bf R})}{\psi_T^*({\bf R})\psi_T({\bf R})}$     (3.10)

and $\vec F$ is the drift force which is proportional to the gradient of the trial wavefunction and consequently always points in the direction where $\psi_T$ increases
$\displaystyle \vec F = \frac{2}{\psi_T({\bf R})} \vec \nabla \psi_T({\bf R})$     (3.11)


next up previous contents
Next: Green's function Up: Diffusion Monte Carlo Previous: Introduction   Contents