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Monte Carlo techniques have been widely employed in statistical physics as well as in quantum theory in the Lagrangian formulation. However, in some areas of application to quantum theories computational progress has been slow. Here we present a recently developed approach: the Monte Carlo Hamiltonian method, designed to overcome the difficulties of the conventional approach.

We further study the validity of the Monte Carlo Hamiltonian method. The advantage of the method, in comparison with the standard Monte Carlo Lagrangian approach, is its capability to study the excited states. We consider two quantum mechanical models: a symmetric one V (x) = jxj=2 and an asymmetric one V (x) = 1, for x < 0 and V (x) = x, for x 0. The results for the spectrum, wave functions and thermodynamical observables are in agreement with the analytical or Runge{Kutta calculations.

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We suggest how to construct an effective low energy Hamiltonian via Monte Carlo starting from a given action. We test it by computing thermodynamical observables like average energy and specific heat for simple quantum systems.Comment: Contribution to Lattice'99 (Theoretical developments) Text (LaTeX file) + 2 figures (ps files