Deterministic Solution of the Discrete Wigner Equation

The Wigner formalism provides a convenient formulation of quantum mechanics in the phase space. Deterministic solutions of the Wigner equation are especially needed for problems where phase space quantities vary over several orders of magnitude and thus can not be resolved by the existing stochastic approaches. However, finite difference schemes have been problematic due to the discretization of the diffusion term in this differential equation. A new approach, which uses an integral formulation of the Wigner equation that avoids the problematic differentiation, is shown here. The results of the deterministic method are compared and validated with solutions of the Schrodinger equation. Furthermore, certain numerical aspects pertaining to the demanded parallel implementation are discussed.