Finite-difference Heisenberg approach: an approximate treatment of many-particle systems, energy level separations and the time-dependent Schrِdinger wavefunction

We extend our recently proposed finite-difference Heisenberg (FDH) approach to the quantum treatment of many-particle systems in the context of a time-dependent self-consistent field approximation. A simple method is then proposed for calculating the energy level separations directly from the matrix elements of the time-evolved operators using a time average and the velocity-dipole identity. Finally, we show that the FDH approach is not limited to the time evolution of operators, but that it also provides us, at no extra effort, with the time evolution of the Schrِdinger wavefunction on a dynamic coordinate system. We illustrate our theory by applications to the multi-dimensional Hénon-Heiles model and 1D anharmonic oscillators with promising results in each case.