Monte Carlo is an extremely useful technique for solving many-bodied quantum systems.

The basis for quantum mechanics is the Schroedinger equation which is a second-order differential equation that relates the wavefunction of a system to its total energy: [1]

Among the problems in solving this equation is the fact that upon adding multiple particles to the system (say, an additional electron or nucleus), the potential energy component becomes increasingly complicated. To put this in perspective, a two-bodied system (eg. the hydrogen atom) involves one potential function, a three-bodied system (eg. helium) involves three potential interactions (one between electrons and two between each electron and the nucleus), and a 4-bodied system requires 6. It is noteworthy that no systems involving more than 3 bodies have been solved analytically [2].

To solve such higher ordered systems, Quantum Monte Carlo methods are employed. QMC is most generally used to determine the ground state wavefunction of a system, as this describes the status of the system at normal temperatures [3].

One of the most common QMC methods is Variational Monte Carlo. The total energy in the system can be determined by applying the Hamiltonian operator to the trial wavefunction:

The trial wavefunction is an estimated representation of the system and as the expectation value approaches its minimum value (the energy of the ground state), the trial wavefunction becomes more accurate in describing the system.

In order to computer this using a Monte Carlo method, the expectation value can be rearranged as

Therefore, one can simply define a new function, the local energy as

and from there, take an average sampling over the system to find the expectation value. Note that computing the local energy can be extremely expensive computationally as it requires calculating the second derivative of the wavefunction and potentials between all electrostatic particles (via the Hamiltonian) and therefore is often subject to certain optimizations.

Choosing these random sites for computing local energies is critical. The sites that are chosen are weighted upon the probability that a particle will be found in that location; this probability is given by computing the wavefunction times its complex conjugate. As a result, more samplings are taken from areas where particles are more likely to be found. In essence, this is accomplished by picking an initial location, proposing a a new position by adding a random vector to it, computing the probability distribution at the new location using the wavefunction, and accepting the new position if the probability is greater, else rejecting it.

In application, Monte Carlo methods are extremely useful for computing the ground state status of a system with many bodies. It provides the greatest accuracy possible for a computer simulation by taking into account all electrons in the system, whereas other methods use only one (Density Functional Theory) or none at all (Molecular Dynamics). At the same time, it is extremely expensive computationally and therefore cannot be applied to big systems (generally, QMC becomes impractical after surpassing 1000 electrons on standard computers). At the same time, QMC has been successfully applied to describe a diverse field of molecular systems and as computers become faster and more powerful, greater successes in the field of QMC will surely prevail.

Additional sources:
http://farside.ph.utexas.edu/teaching/qmech/lectures/node30.html
http://www.physics.uc.edu/~pkent/thesis/pkthnode17.html