Variational Monte Carlo
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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, samp...
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 flexibilityto the method.
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