Time-Dependent Expectation Values from Integral Equations for Quantum Flux and Probability Densities.

We compare the calculation of time-dependent quantum expectation values performed in different ways. In one case, they are obtained from an integral over a function of the probability density, and in the other case, the integral is over a function of the probability flux density. The two kinds of coordinate-dependent integrands are very different in their appearance, but integration yields identical results, if the exact wave function enters into the computation.

Quantum flux densities for electronic-nuclear motion: exact versus Born-Oppenheimer dynamics.

We study the coupled electronic-nuclear dynamics in a model system to compare numerically exact calculations of electronic and nuclear flux densities with those obtained from the Born-Oppenheimer (BO) approximation. Within the adiabatic expansion of the total wave function, we identify the terms which contribute to the flux densities. It is found that only off-diagonal elements that involve the interaction between different electronic states contribute to the electronic flux whereas in the nuclear case the major contribution belongs to the BO electronic state.

Correlated three-dimensional electron-nuclear motion: Adiabatic dynamics vs passage of conical intersections.

We study the three-dimensional correlated motion of an electron and a proton. In one situation, the dynamics is restricted to the electronic ground state and is, thus, well described within the Born-Oppenheimer (BO) approximation. The probability and flux densities yield information about the coupled dynamics.

Wave packet dynamics in an harmonic potential disturbed by disorder: Entropy, uncertainty, and vibrational revivals.

We investigate the quantum and classical wave packet dynamics in an harmonic oscillator that is perturbed by a disorder potential. This perturbation causes the dispersion of a Gaussian wave packet, which is reflected in the coordinate-space and the momentum-space Shannon entropies, the latter being a measure for the amount of information available on a system. Regarding the sum of the two quantities, one arrives at an entropy that is related to the coordinate-momentum uncertainty.

Time-dependent momentum expectation values from different quantum probability and flux densities.

Based on the Ehrenfest theorem, the time-dependent expectation value of a momentum operator can be evaluated equivalently in two ways. The integrals appearing in the expressions are taken over two different functions. In one case, the integrand is the quantum mechanical flux density j̲, and in the other, a different quantity j̲̃ appears, which also has the units of a flux density.