High Quality Content by WIKIPEDIA articles! In mathematical physics, variational Monte Carlo (VMC) is a quantum Monte Carlo method that applies the variational method to approximate the ground state of the system. The expectation value necessary can be written in the x representation as frac{langle Psi(a) | H | Psi(a) rangle} {langle Psi(a) | Psi(a) rangle } = frac{int | Psi(X,a) | ^2 frac{HPsi(X,a)}{Psi(X,a)} , dX} { int | Psi(X,a)|^2 , dX}. Following the Monte Carlo method for evaluating integrals, we can interpret frac{ | Psi(X,a) | ^2 } { int | Psi(X,a) | ^2 , dX } as a probability distribution function, sample it, and evaluate the energy expectation value E(a) as the average of the local function frac{HPsi(X,a)}{Psi(X,a)} , and minimize E(a). VMC is no different from any other variational method, except that since the many-dimensional integrals are evaluated numerically, we only need to calculate the value of the possibly very complicated wave function, which gives a large amount of flexibility to the method.

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