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Animation of Quantum Monte Carlo: Quantum mechanics by counting

german (in german)

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Movie: QMC Simulation (GaAs) [1,4MB mpeg movie for e.g. mpeg_play]

Explanation:

In the upper part, illustrating calculations for bulk gallium arsenide, the course of a typical simulation is shown.

The cube represents the supercell, which is periodically repeated in all three directions of space to construct the infinite bulk system. In the chosen example 32 fcc unit cells with 64 ions are contained. Think of the green resp. red spots as ion sites of gallium resp. arsenic. One tetrahedral bond typical of the zincblende structure is shown. The tiny blue points, filling the simulation cell denser and denser, mark where the electrons move during the random walk. Altogether 1000 for each of the 256 electrons, they are distributed according to the square of the many-particle wave function (Metropolis, importance sampling). At these points in the high dimensional (3*256) configuration space, the Hamiltonian is actually evaluated.

In the lower part the development of energy statistics with growing number of integration points is shown. One blue spot denotes the total energy when each of the 256 electrons is moved once. By taking the mean of these values one gets a proper estimator for the total energy (thick red line). With growing number of integration points (realizations) the statistical fluctuations get smaller and smaller as the central limit theorem demands. The interval, in which the measured total energies fall with a probability of 68 percent, is limited by the thin red lines. So their separation gives a value for the proper estimator for the error in total energy.

In summary, one gets the exact total energy of a interacting many-body system with known, purely statistical error!
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CAU Kiel
Physics
Theory
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6 Sept 2000
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