New Perspectives on the Schr{"o}dinger-Pauli Theory of Electrons: Part I (original) (raw)
Related papers
arXiv (Cornell University), 2019
The Schrödinger-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 Schrödinger-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 Schrödinger-Pauli equation.
Schrödinger Theory of Electrons in Electromagnetic Fields: New Perspectives
Computation
The Schrödinger theory of electrons in an external electromagnetic field is described from the new perspective of the individual electron. The perspective is arrived at via the time-dependent "Quantal Newtonian" law (or differential virial theorem). (The time-independent law, a special case, provides a similar description of stationary-state theory). 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. It is thus path-independent. The conservative field is the sum of the internal and Lorentz fields. Hence, the potential is inherently related to the properties of the system, and its constituent property-related components known. As the sources of the fields are functionals of the wave function, so are the respective fields, and, therefore, the scalar potential is a known functional of the wave function; (c) as such, the system Hamiltonian is a known functional of the wave function. This reveals the intrinsic self-consistent nature of the Schrödinger equation, thereby providing a path for the determination of the exact wave functions and energies of the system; (d) with the Schrödinger 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 two quantum dots in a magnetostatic field, one in a ground state and the other in an excited state. For the time-dependent case, the evolution of the same states of the quantum dots in both a magnetostatic and a time-dependent electric field is described. In each case, the satisfaction of the corresponding "Quantal Newtonian" law is demonstrated.
Properties of the Schr̈odinger Theory of Electrons in Electromagnetic Fields
arXiv (Cornell University), 2016
The Schrödinger 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.);
Generalization of the Schrödinger Theory of Electrons
Journal of Computational Chemistry, 2017
The Schrödinger theory for a system of electrons in the presence of both a static and timedependent electromagnetic field is generalized so as to exhibit the intrinsic self-consistent nature of the corresponding Schrödinger equations. This is accomplished by proving that the Hamiltonian in the stationary-state and time-dependent cases {Ĥ;Ĥ(t)} are exactly known functionals of the corresponding wave functions {Ψ; Ψ(t)}, i.e.Ĥ =Ĥ[Ψ] andĤ(t) =Ĥ[Ψ(t)]. Thus, the Schrödinger equations may be written asĤ[Ψ]Ψ = E[Ψ]Ψ andĤ[Ψ(t)]Ψ(t) = i∂Ψ(t)/∂t. As a consequence the eiegenfunctions and energy eigenvalues {Ψ, E} of the stationary-state equation, and the wave function Ψ(t) of the temporal equation, can be determined self-consistently. The proofs are based on the 'Quantal Newtonian' first and second laws which are the equations of motion for the individual electron amongst the sea of electrons in the external fields. The generalization of the Schrödinger equation in this manner leads to additional new physics. The traditional description of the Schrödinger theory of electrons with the Hamiltonians {Ĥ;Ĥ(t)} known constitutes a special case.
The New Relativistic Quantum Theory of the Electron
In this paper we introduce a framework to unify quantum and special relativity theories conforming the principle of causality through a new concept of fundamental particle mass based on new models of both stochastic process and elementary particles as concentrated energy localized on the surface of 3-dimensional sphere-form (2-manifold without boundary). The natural picture of fundamental connection between quantum and special relativistic aspects of particles is described by the existence of the intrinsic random vibrating motion of an elementary particle in a quantum-sized volume (Planck scale) directly connected with a spin phenomenon, which is playing fundamental role as internal time. The results show that fir st, relativistic effects fundamentally relate to dynamic aspects of a particle. Second, new equations indicate antiparticle (antimatter) must have positive energy. Third, these are different from the Dirac's equation exhibiting an electric moment in a pure imaginary. Our equation presents a real electric moment. We also show that the antiparticles only present in strong potential causing the non-symmetry reality between matter and antimatter in the universe.
Schrödinger Theory of Electrons: A Complementary Perspective
Springer eBooks, 2022
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
A classical model for the electron
The construction of classical and semi-classical models for the electron has had a long and distinguished history. Such models are useful more for what they teach us about field theory than what they teach us about the electron. In this Letter I exhibit a classical model of the electron consisting of ordinary electromagnetism coupled with a self-interacting version of Newtonian gravity. The gravitational binding energy of the system balances the electrostatic energy in such a manner that the total rest mass ofthe electron is finite.
Schrödinger Theory of Electrons: Complementary Perspectives
Springer eBooks, 2022
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
Consistency in the formulation of the Dirac, Pauli, and Schrödinger theories
Journal of Mathematical Physics, 1975
Properties of observables in the Pauli and Schrödinger theories and first order relativistic approximations to them are derived from the Dirac theory. They are found to be inconsistent with customary interpretations in many respects. For example, failure to identify the "Darwin term" as the s-state spin-orbit energy in conventional treatments of the hydrogen atom is traced to a failure to distinguish between charge and momentum flow in the theory. Consistency with the Dirac theory is shown to imply that the Schrödinger equation describes not a spinless particle as universally assumed, but a particle in a spin eigenstate. The bearing of spin on the interpretation of the Schrödinger theory discussed. Conservation laws of the Dirac theory are formulated in terms of relative variables, and used to derive virial theorems and the corresponding conservation laws in the Pauli-Schrödinger theory.
A Proposal on the Structure and Properties of an Electron
A model of the electron is proposed in which it is composed of a known and detectable particle. This model clearly defines what is mass, what is energy and why E = mc 2 . It shows that the special relativity corrections of mass, length and time with velocity are automatically a function of the structure of an electron as it moves. It also proposes the origin of electric charge and uses that origin to derive the equation for the Bohr magnetron. It shows that the electron's spin of ½ħ is simply angular momentum, further explaining why the electron has only two states of spin. This model gives an expression for the radius of an electron, at the same time pointing out why it has been detected as a point particle, yet still has angular momentum of ½ħ. It explains why the mass of an electron increases with velocity while its spin and charge remain the same and its magnetic moment decreases, as well as why its charge spirals when it travels through space. This model gives a physical reason for the existence of the de Broglie wavelength and derives the expression for it. As well as matching the known properties of an electron it also makes predictions of previously unknown properties, pointing out that they may have been detected but not recognized as such.