Quantum Monte-Carlo

The term Quantum Monte-Carlo does not refer to a particular method but rather to any method using Monte-Carlo type methods to solve quantum (usually many-body) problems. As an example consider the evaluation of the energy of a trial wave function $\Psi(\bi{r})$ where $\bi{r}$ is a 3N dimensional position coordinate of N particles,

$$E = {\int\d\bi{r} \Psi^*(\bi{r})\hat\bss{H}\Psi(\bi{r})\over \int\d\bi{r} \Psi^*(\bi{r})\Psi(\bi{r})}$$ (4.17)

where $\hat\bss{H}$ is the Hamiltonian operator. This can be turned into an appropriate form for Monte-Carlo integration by rewriting as

$$E = \int\d\bi{r} \left\{\left\vert\Psi(\bi{r})\right\vert^2\... ...right\} \left[\Psi(\bi{r})^{-1}\hat\bss{H}\Psi(\bi{r})\right]$$ (4.18)

such that the quantity in braces ($\{\}$) has the form of a probability distribution. This integral can now easily be evaluated by Monte-Carlo integration. Typically a sequence of $\bi{r}$'s is generated using the Metropolis algorithm, so that it is not even necessary to normalise the trial wave function, and the quantity in square brackets $[]$ is averaged over the $\bi{r}$'s.

This method, variational quantum Monte-Carlo, presupposes we have a good guess for the wave function and want to evaluate an integral over it. It is only one of several different techniques which are referred to as quantum Monte Carlo. Others include, Diffusion Monte-Carlo, Green's function Monte-Carlo and World Line Monte-Carlo.