Quantum Theory of an Electron in External Fields

Viraht Sahni, Xiao‐Yin Pan, Tao Yang

Local effective potential theory, both stationary-state and time-dependent, constitutes the mapping from a system of electrons in an external field to one of the noninteracting fermions possessing the same basic variable such as the density, thereby enabling the determination of the energy and other properties of the electronic system. This paper is a description via Quantal Density Functional Theory (QDFT) of the electron correlations that must be accounted for in such a mapping. It is proved through QDFT that independent of the form of external field, (a) it is possible to map to a model system possessing all the basic variables; and that (b) with the requirement that the model fermions are subject to the same external fields, the only correlations that must be considered are those due to the Pauli exclusion principle, Coulomb repulsion, and Correlation–Kinetic effects. The cases of both a static and time-dependent electromagnetic field, for which the basic variables are the density and physical current density, are considered. The examples of solely an external electrostatic or time-dependent electric field constitute special cases. An efficacious unification in terms of electron correlations, independent of the type of external field, is thereby achieved. The mapping is explicated for the example of a quantum dot in a magnetostatic field, and for a quantum dot in a magnetostatic and time-dependent electric field.

P. A. M. Dirac

H. Muirhead

Viraht Sahni

A complementary perspective to the Göttingen-Copenhagen interpretation of stationary-state quantum theory of electrons in an electromagnetic field is described. The perspective, derived from Schrödinger-Pauli theory, is that of the individual electron via its equation of motion or 'Quantal Newtonian' First Law. The Law is in terms of 'classical' fields experienced by each electron: the sum of the external and internal fields vanishes. The external field is a sum of the electrostatic and Lorentz fields. The internal field is a sum of fields' representative of Pauli and Coulomb correlations; kinetic effects; electron density; and internal magnetic component. The energy is obtained from these fields. The sources of the fields are expectation values of Hermitian operators. The perspective is elucidated by application to quantum dots in a magnetic field in a ground, excited singlet and triplet states. The relationship of the perspective to Quantal and traditional density functional theories is briefly explained.

Edward G. Harris

Viraht Sahni

Abstract The Schrödinger–Pauli (SP) theory of electrons in an electromagnetic field explicitly accounts for the electron spin moment. The many‐electron theory is complemented via a new descriptive perspective namely, that 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 system wave function. The law states that each electron experiences an external and an internal field, the sum of which vanish. The external field is the sum of the binding electrostatic and a Lorentz field. The internal field is a sum of fields representative respectively of electron correlations due to the Pauli principle and Coulomb repulsion; the electron density; kinetic effects; and an internal magnetic field. The energy can be expressed in integral virial form in terms of these fields. The law is elucidated by application to the 23S state of a quantum dot in a magnetic field. It is proved that the Hamiltonian is an exactly known and universal functional of the wave function. This generalizes the SP equation, and reveals that its eigenfunctions and eigenvalues may be determined self‐consistently. A quantal density functional theory (QDFT) of the SP system is developed whereby additional properties are determined. A physical interpretation of Spin‐DFT based on the QDFT mapping is provided. Further generalizations of the present work to the temporal case, and relativistic Dirac theory, are proposed.