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I understood the formal derivation of various QMC methods like Path Integral Monte Carlo. However, at the end of the day I still have a doubt on how to effectively use these techniques.

Given that we can interpret [itex]\beta[/itex] in the quantum operator [itex]e^{-\beta\hat{H}}[/itex] both as an inverse temperature and an imaginary time, the aim of these algorithms should be to compute an approximation of this operator. Indeed, if we would directly measure quantities from the various configurations sampled along a simulation, in the "inverse temperature" case we would have samples respecting a probability density based on [itex]\beta/M[/itex], where [itex]M[/itex] is the discretization introduced in the Trotter decomposition. Instead, in the "imaginary time" case we would obtain samples at various discrete time-steps, thus getting averages along the time as well (and we wouldn't obtain quantities such as [itex]<\psi_t|\hat{A}|\psi_t>[/itex] at a given time [itex]t[/itex], with [itex]\hat{A}[/itex] being a certain physical quantity). Am I wrong?

Please, help! :)