On Ehrenfest's theorem

Abstract

Abstract Ehrenfest's theorem for a system of electrons in a time‐dependent external force is the quantal analogue to Newton's second law of motion for interacting classical particles. The theorem is in terms of the averaged force and position of the electrons. Here we draw the quantal analogue to the equation of motion for the individual classical particle and thereby describe the internal forces experienced by each electron and the evolution of these forces with time. This explanation is based on the description of Schrödinger theory in terms of fields and their quantal sources. The quantum mechanical average of the internal fields and their averaged torques, taken over all the electrons, vanish as they must. However, unlike classical physics, the vanishing of the averaged internal fields cannot be attributed solely to Newton's third law but is additionally a consequence of quantum mechanics. The field perspective thus leads to a new derivation of Ehrenfest's theorem. Furthermore, the theorem can be expressed in terms of the response of the electrons to the external field as described by a field representative of the electronic current density. In addition, the vanishing of the averaged torque of the internal fields leads to a quantal torque–angular momentum equation analogous to that of classical physics. The structure of the internal fields is demonstrated for both a ground and an excited state of the analytically solvable Hooke's atom. The concept of the internal field is extended via quantal density functional theory to the S system of noninteracting Fermions with density equivalent to that of the Schrödinger theory of electrons. By invoking Ehrenfest's theorem and the quantal torque–angular momentum relationship, sum rules for the S system are derived. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004