Semi-classical Schr\"odinger numerics in the residual representation

Abstract

The numerical treatment of quantum mechanics in the semi-classical regime is known to be computationally demanding, due to the highly oscillatory behaviour of the wave function and its large spatial extension. A recently proposed representation of quantum mechanics as a residual theory on top of classical Hamiltonian mechanics transforms a semi-classical wave function into a slowly-fluctuating, spatially confined residual wave function. This representation is therefore well-suited for the numerical solution of semi-classical quantum problems. In this note I outline the formulation of the theory and demonstrate its applicability to a set of semi-classical scenarios, including a discussion of limitations. I work out the connection to established numerical approaches, such as the Gaussian beam approximation and the Gaussian wave packet transform by Russo and Smereka. A prototypical implementation of the method has been published as open-source software.