The drift vector should be of the form \( \mathbf{F} = g(\mathbf{x}) \frac{\partial P}{\partial \mathbf{x}} \). Then,

$$ \frac{\partial^2 P}{\partial {\mathbf{x_i}^2}} = P\frac{\partial g}{\partial P}\left( \frac{\partial P}{\partial {\mathbf{x}_i}} \right)^2 + P g \frac{\partial ^2 P}{\partial {\mathbf{x}_i^2}} + g \left( \frac{\partial P}{\partial {\mathbf{x}_i}} \right)^2. $$

The condition of stationary density means that the left hand side equals zero. In other words, the terms containing first and second derivatives have to cancel each other. It is possible only if \( g = \frac{1}{P} \), which yields

$$ \mathbf{F} = 2\frac{1}{\Psi_T}\nabla\Psi_T, $$

which is known as the so-called quantum force. This term is responsible for pushing the walker towards regions of configuration space where the trial wave function is large, increasing the efficiency of the simulation in contrast to the Metropolis algorithm where the walker has the same probability of moving in every direction.