New Perspectives on the Schr{\"o}dinger-Pauli Theory of Electrons: Part II: Application to the Triplet State of a Quantum Dot in a Magnetic Field

TL;DR

This paper introduces a new perspective on the Schrödinger-Pauli theory by describing electrons through 'Quantal Newtonian' laws, and applies it to analyze the triplet state of a quantum dot in a magnetic field, revealing its self-consistent nature.

Contribution

It presents a novel 'Quantal Newtonian' framework for Schrödinger-Pauli theory and demonstrates its application to a quantum dot's triplet state, highlighting the theory's intrinsic self-consistency.

Findings

Fields satisfy the 'Quantal Newtonian' first law.

Energy components are derived from these fields.

The approach confirms the self-consistent nature of the Schrödinger-Pauli equation.

Abstract

The Schr\"odinger-Pauli theory of electrons in the presence of a static electromagnetic field can be described from the perspective of the individual electron via its equation of motion or `Quantal Newtonian' first law. The law is in terms of `classical' fields whose sources are quantum-mechanical expectation values of Hermitian operators taken with respect to the wave function. The law states that the sum of the external and internal fields experienced by each electron vanishes. The external field is the sum of the binding electrostatic and Lorentz fields. The internal field is the sum of fields representative of properties of the system: electron correlations due to the Pauli exclusion principle and Coulomb repulsion; the electron density; kinetic effects; the current density. Thus, the internal field is a sum of the electron-interaction, differential density, kinetic, and internal magnetic fields. The energy can be expressed in integral virial form in terms of these fields. Via this perspective, the Schr\"odinger-Pauli equation can be written in a generalized form which then shows it to be intrinsically self-consistent. This new perspective is explicated by application to the triplet $2^{3}S$ state of a 2-D 2-electron quantum dot in a magnetic field. The quantal sources of the density; the paramagnetic, diamagnetic, and magnetization current densities; pair-correlation density; the Fermi-Coulomb hole charge; and the single-particle density matrix are obtained, and from them the corresponding fields determined. The fields are shown to satisfy the `Quantal Newtonian' first law. The components of the energy too are determined from these fields. Finally, the example is employed to demonstrate the intrinsic self-consistent nature of the Schr\"odinger-Pauli equation.