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International Journal of Quantum Chemistry

Quantum‐Mechanical Definition of the Classical Scalar Potential in Schrödinger‐Pauli and Schrödinger Theory

Viraht Sahni

Published Aug 18, 2025• 0 citations
Publisher

Abstract

ABSTRACT According to the Bohr correspondence principle, the external temporal scalar potential in the classical equation of motion is replicated in quantum theory as a multiplicative operator. An equivalent quantum‐mechanical definition of the scalar potential in Schrödinger‐Pauli/Schrödinger theory is provided. The potential is a known universal functional of the wave function. At each instant of time, it is the work done in a conservative “classical” field representative of internal properties of the system: Pauli and Coulomb correlations, kinetic effects, the density, the Lorentz force, an internal magnetic component, and the current density response. The Hamiltonians are thus rewritten in a new physical manner indicative of these properties. An application to the triplet state of a 2‐electron semiconductor quantum dot in a magnetic field is provided. The significance of the definition to the Hohenberg‐Kohn theorem, and its relationship to Quantum and Kohn‐Sham density functional theory, is discussed.

DOI
10.1002/qua.70096

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ABSTRACT According to the Bohr correspondence principle, the external temporal scalar potential in the classical equation of motion is replicated in quantum theory as a multiplicative operator. An equivalent quantum‐mechanical definition of the scalar potential in Schrödinger‐Pauli/Schrödinger theory is provided. The potential is a known universal functional of the wave function. At each instant of time, it is the work done in a conservative “classical” field representative of internal properties of the system: Pauli and Coulomb correlations, kinetic effects, the density, the Lorentz force, an internal magnetic component, and the current density response. The Hamiltonians are thus rewritten in a new physical manner indicative of these properties. An application to the triplet state of a 2‐electron semiconductor quantum dot in a magnetic field is provided. The significance of the definition to the Hohenberg‐Kohn theorem, and its relationship to Quantum and Kohn‐Sham density functional theory, is discussed.
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.
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 viz. 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.
Abstract The quantal density functional theory (Q‐DFT) of excited states is the description of the physics of the mapping from any bound nondegenerate excited state of Schrödinger theory to that of the s‐system of noninteracting Fermions with equivalent density ρk(r), energy Ek, and ionization potential Ik. The s‐system may either be in an excited state with the same configuration as in Schrödinger theory or in a ground state with a consequently different configuration. The Q‐DFT description of the s‐system is in terms of a conservative field ℱk(r), whose electron‐interaction ℰee(r) and correlation‐kinetic 𝒵 (r) components are separately representative of electron correlations due to the Pauli exclusion principle and Coulomb repulsion, and correlation‐kinetic effects, respectively. The sources of these fields are expectations of Hermitian operators taken with respect to the system wavefunction. The local electron‐interaction potential vee(r) of the s‐system, representative of all the many‐body correlations, is the work done to move an electron in the force of the field ℱk(r). The electron interaction Eee and correlation‐kinetic Tc components of the total energy Ek may be expressed in integral virial form in terms of their respective fields. The difference between the s‐system in its ground or excited state representation is due entirely to correlation‐kinetic effects. The highest occupied eigenvalue of the s‐system differential equation in either case is minus the ionization potential Ik. In this work we demonstrate the transformation of an excited state of Schrödinger theory, as represented by the first excited singlet state of the exactly solvable Hooke's atom, to that of noninteracting Fermions in their ground state with equivalent excited state density, energy, and ionization potential. To further prove the Fermions are in a ground state, we solve the corresponding singlet s‐system differential equation numerically for the vee(r) determined, and obtain the excited state density from the zero node orbitals generated. The resulting total energy is also the same. In addition, the single eigenvalue determined corresponds to minus the ionization potential of the excited state. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001
A dynamical treatment of a quantum in a potential is proposed by using the formulation of quantum mechanics with classical pictures and defining the quantum mechanical (abbreviated to q. m.) principal function. It is shown in the case of Coulomb potential that the q. m. momentum with some parameters can be uniquely determined and the q. m. principal function can be obtained in principle. On this occasion a new dynamical postulate is introduced. In addition to the above, it is qualitatively demonstrated that the q. m. quantities in the classical region are approximately equal to the classical ones and are larger than those in the non-classical region. Accordingly we may consider the effect of the so-called quantum potential to be small in comparison with that of the usual potential.
Schr{ö}dinger-Pauli (SP) theory is a description of electrons in the presence of a static electromagnetic field in which the interaction of the magnetic field with both the orbital and spin moments is explicitly considered. The theory is described from a new perspective, viz. that of the individual electron via its equation of motion or `Quantal Newtonian' first law. The law leads to new physical and mathematical insights into the theory. 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 electrostatic and a Lorentz field. The internal field is a sum of fields: the electron-interaction, differential density, kinetic, and internal magnetic fields. These fields are respectively representative of a property of the system: electron correlations due to the Pauli exclusion principle and Coulomb repulsion, the electron density, kinetic effects, and the physical current density. The energy components can be expressed in integral virial form in terms of these fields. The law leads to the further understanding that the Hamiltonian is an exactly known and universal functional of the wave function. This allows for the generalization of the SP equation, which then proves it to be intrinsically self-consistent. A Quantal density functional (local effective potential) theory of the SP system is developed. Further generalizations of the present work to the temporal case, and relativistic Dirac theory are proposed.
We show that the quantum wavefunction, interpreted as the probability density of finding a single non-localized quantum particle, which evolves according to classical laws of motion, is an intermediate description of a material quantum particle between the quantum and classical realms. Accordingly, classical and quantum mechanics should not be treated separately, a unified description in terms of the Wigner distribution function being possible. Although defined on classical phase space coordinates, the Wigner distribution function accommodates the nonlocalization property of quantum systems, and leads to both the Schrodinger equation for the quantum wavefunction and to the definition of position and momentum operators.