Time-dependent solution of the Liouville-von Neumann equation: non-dissipative evolution

Abstract A method for solving the Liouville-von Neumann equation is presented. The action of operators is calculated locally in coordinate and/or momentum representation. The Fast Fourier Transform (FFT) is used to pass back and forth between coordinate and momentum representations, this transformation preserving all exact commutation relations. The time propagation is calculated by a Chebychev expansion of the time evolution operator. The accuracy and convergence properties of the method are investigated and compared with an exactly solvable model problem. Accurate converged results are obtained using a phase space boundary slightly exceeding the minimum theoretical value. Efficiency is attained by the natural vectorization options provided by the algorithm. A typical non-trivial application is presented, namely, the splitting up of the probability density of a non-pure state into transmitted and reflected branches due to scattering off a potential barrier.