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. This can be different, if one applies approximations to the wave function. For illustration, we treat one- and two-dimensional dynamics in coupled electron-nuclear systems. Using the adiabatic expansion of the total wave function, the expectation values are decomposed into different contributions. This allows us to discuss the validity of the Born-Oppenheimer (BO) approximation applied to the calculation of the expectation values from probability density- and flux density- integrals. Choosing force- and torque operators as examples, we illustrate the different spatiotemporal characteristics of the various integrands.