Schrödinger Theory from the ‘Newtonian’ Perspective of ‘Classical’ Fields Derived from Quantal Sources

Abstract

Schrödinger theory of the electronic structure of matter—N electrons in the presence of an external time-dependent field—is described from the perspective of the individual electron. The corresponding equation of motion is expressed via the ‘Quantal Newtonian’ second law, the first law being a description of the stationary state case. This description of Schrödinger theory is ‘Newtonian’ in that it is in terms of ‘classical’ fields which pervade space, and whose sources are quantum-mechanical expectations of Hermitian operators taken with respect to the system wave function. In addition to the external field, each electron experiences an internal field, the components of which are representative of correlations due to the Pauli Exclusion Principle and Coulomb repulsion, the kinetic effects, and the density. The resulting motion of the electron is described by a response field. Ehrenfest’s theorem is derived by showing the internal field vanishes on summing over all the electrons. The ‘Newtonian’ perspective is then explicated for both a ground and excited state of an exactly solvable model. Various facets of quantum mechanics such as the Integral Virial Theorem, the Harmonic Potential Theorem, the quantum-mechanical ‘hydrodynamical’ equations in terms of fields, coalescence constraints, and the asymptotic structure of the wave function and density are derived. The equivalence of the ‘Quantal Newtonian’ second law and the Euler equation of Quantum Fluid Dynamics is proved.