Properties of the Schr\"odinger Theory of Electrons in Electromagnetic Fields

TL;DR

This paper explores the properties of Schr"odinger theory for electrons in electromagnetic fields, introducing a self-consistent, property-based framework that enhances understanding of electron behavior in static and dynamic fields.

Contribution

It presents a novel perspective using quantal Newtonian laws, revealing the self-consistent, property-dependent nature of the Schr"odinger equation and extending it to include external magnetic fields.

Findings

Reveals internal and external field effects on electrons.

Shows the Hamiltonian as a wave function functional.

Applies theory to quantum dots in magnetic and electric fields.

Abstract

The Schr\"odinger theory of electrons in an external electromagnetic field can be described from the perspective of the individual electron via the `Quantal Newtonian' laws (or differential virial theorems). These laws are in terms of `classical' fields whose sources are quantal expectations of Hermitian operators taken with respect to the wave function. The laws reveal the following physics: (a) In addition to the external field, each electron experiences an internal field whose components are representative of a specific property of the system such as the correlations due to the Pauli exclusion principle and Coulomb repulsion, the electron density, kinetic effects, and an internal magnetic field component. (The response of the electron is described by the current density field.); (b) The scalar potential energy of an electron is the work done in a conservative field which is the sum of the internal and Lorentz fields. It is thus inherently related to the properties of the system. Its constituent property-related components are hence known. It is a known functional of the wave function; (c) As such the Hamiltonian is a functional of the wave function, thereby revealing the intrinsic self-consistent nature of the Schr\"odinger equation. This then provides a path for the determination of the exact wave function. (d) With the Schr\"odinger equation written in self-consistent form, the Hamiltonian now admits via the Lorentz field a new term that explicitly involves the external magnetic field. The new understandings are explicated for the stationary state case by application to a quantum dot in a magnetostatic field in both a ground and excited state. For the time-dependent case, the same states of the quantum dot in both a magnetostatic and a time-dependent electric field are considered.