Energy-Momentum Transfer and the Quantum Potential

M. R. Brown

The composition of the quantum potential and its role in the breakdown of classical symplectic symmetry in quantum mechanics is investigated. General expressions are derived for the quantum potential in both configuration space and momentum space representations. By comparing the configuration space and momentum space representations of the causal interpretation of quantum mechanics, the quantum potential is shown to break the symplectic symmetry that exists between these two representations in classical mechanics. In addition, it is shown that the quantum potential in configuration space may be expressed as the sum of a momentum dispersion energy and a spatial localisation energy; a complementary expression for the quantum potential being found in the momentum representation. The composition and role of the quantum potential in both representations is analysed for a particle in a linear potential and for two eigenstates of the quantum harmonic oscillator.

Tai Tsun Wu

Several problems of nonrelativistic scattering by analytic potentials are studied, where both the energy and the momentum transfer are large. Specifically, we find asymptotically the reflection coefficients from the one-dimensional potentials $V{e}^{x}{(1+{e}^{x})}^{\ensuremath{-}2}$, $V{e}^{\frac{x}{2}}{(1+{e}^{x})}^{\ensuremath{-}1}$, and $V\mathrm{exp}(\ensuremath{-}{x}^{2})$, and the scattering amplitudes for their three-dimensional generalizations $V{e}^{r}{(1+{e}^{r})}^{\ensuremath{-}2}$, $V{e}^{\frac{r}{2}}{(1+{e}^{r})}^{\ensuremath{-}1}$, and $V\mathrm{exp}(\ensuremath{-}{r}^{2})$. The results are compared with these obtained from Born approximation, and it is found that the first Born approximation gives the correct answer asymptotically in the cases $V{e}^{\frac{x}{2}}{(1+{e}^{x})}^{\ensuremath{-}1}$ and $V{e}^{\frac{r}{2}}{(1+{e}^{r})}^{\ensuremath{-}1}$, but not in the other cases. Conjectures about more general cases are also given.

Chen Li, Ryan Requist, E. K. U. Gross

The recently developed exact factorization approach condenses all electronic effects on the nuclear subsystem into scalar and vector potentials that appear in an effective time dependent Schrödinger equation. Starting from this equation, we derive subsystem Ehrenfest identities characterizing the energy, momentum, and angular momentum transfer between electrons and nuclei. An effective electromagnetic force operator induced by the electromagnetic field corresponding to the effective scalar and vector potentials appears in all three identities. The effective magnetic field has two components that can be identified with the Berry curvature calculated with (a) different Cartesian coordinates of the same nucleus and (b) arbitrary Cartesian coordinates of two different nuclei. (a) has a classical interpretation as the induced magnetic field felt by the nucleus, while (b) has no classical analog. Subsystem Ehrenfest identities are ideally suited for quantifying energy transfer in electron-phonon systems. With two explicit examples we demonstrate the usefulness of the new identities.

Robert Carroll