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Electron exchange-correlation in quantum mechanics

Burke Ritchie

Published Jan 30, 2009• 0 citations
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Abstract

It is shown that Fermi-Dirac statistics is guaranteed by the Dirac current, from which spin-dependent quantum velocity fields and spin-dependent quantum trajectories can be inferred. Pauli's exclusion principle is demonstrated using the spin-dependent quantum trajectories. The Dirac current, unlike the Schroedinger current, is nonzero for stationary bound states due to the permanent magnetic moment of the electron. It is of order c{sup 0} in agreement with observation that Fermi-Dirac statistics is independent of electronic velocity. In summary the physical basis for exchange-correlation is found in Dirac's equation, although Schroedinger's equation may be used to evaluate the Dirac current in the nonrelativistic regime of electronic velocity.

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It is shown that Fermi-Dirac statistics is guaranteed by the Dirac current, from which spin-dependent quantum velocity fields and spin-dependent quantum trajectories can be inferred. Pauli's exclusion principle is demonstrated using the spin-dependent quantum trajectories. The Dirac current, unlike the Schroedinger current, is nonzero for stationary bound states due to the permanent magnetic moment of the electron. It is of order c{sup 0} in agreement with observation that Fermi-Dirac statistics is independent of electronic velocity. In summary the physical basis for exchange-correlation is found in Dirac's equation, although Schroedinger's equation may be used to evaluate the Dirac current in the nonrelativistic regime of electronic velocity.
The relationship between the spin of an individual electron and Fermi‐Dirac statistics (FDS), which is obeyed by electrons in the aggregate, is elucidated. The relationship depends on the use of spin‐dependent quantum trajectories (SDQT) to evaluate Coulomb’s law between any two electrons as an instantaneous interaction in space and time rather than as a quantum‐mean interaction in the form of screening and exchange potentials. Hence FDS depends in an ab initio sense on the inference of SDQT from Dirac’s equation, which provides for relativistic Lorentz invariance and a permanent magnetic moment (or spin) in the electron’s equation of motion. Schroedinger’s time‐dependent equation can be used to evaluate the SDQT in the nonrelativistic regime of electron velocity. Remarkably FDS is a relativistic property of an ensemble of electron, even though it is of order c 0 in the nonrelativistic limit, in agreement with experimental observation. Finally it is shown that covalent versus separated‐atoms limits can be characterized by the SDQT. As an example of the use of SDQT in a canonical structure problem, the energies of the 1 Σ g and 3 Σ u states of H 2 are calculated and compared with the accurate variational energies of Kolos and Wolniewitz.
Shortly after their introduction, the electron's Zweideutigkeit and Pauli's Ausschliessungsregel were embedded into a growing theoretical framework: from Fermi–Dirac statistics in 1926 (Section 4.2), to the non-relativistic quantum mechanics of the magnetic electron with Pauli's spin matrices in 1927 (Section 4.3); from Wigner and von Neumann's group theoretical derivation of the spin matrices in 1927 (Section 4.4), to Jordan's reinterpretation of Pauli's exclusion principle in terms of anticommutation relations for particle creation and annihilation operators in 1928 (Section 4.5), which paved the way for quantum field theory. The most important step in building up this theoretical framework was the transition from non-relativistic to relativistic quantum mechanics, with Dirac's equation for the electron in 1928 (Section 4.6), which finally allowed the derivation of the electron's spin with its anomalous magnetic moment and hence clarified in a conclusive way the origin of spectroscopic anomalies. The negative energy solutions of the Dirac equation, and Dirac's attempt to accommodate them via the hole theory in 1930, anticipated the experimental discovery of the antiparticle of the electron, the positron. Sections 4.7-4.9 reconstruct Pauli's sustained criticism of Dirac's hole theory. The history of this debate is intertwined with Pauli's search for a spin–statistics connection, which finally culminated in the spin–statistics theorem in 1940. With this theorem, the nomological shift from the status of a phenomenological rule to that of a fundamental scientific principle is finally completed. Introduction The year 1925 was the annus mirabilis for quantum mechanics.
Abstract Electron or ion dynamics are treated using spin‐dependent quantum trajectories. These trajectories are inferred from the Dirac current, which contributes Schroedinger's current and additional spin‐dependent terms, all of which are of order c0 in the nonrelativistic regime of particle velocity, where c is the speed of light. The many‐body problem is treated precisely as in classical dynamics. Each electron or ion has its own equation of motion, which is the time‐dependent Dirac or the time‐dependent Schroedinger equation in the relativistic or nonrelativistic regime of particle velocity, respectively. As an example the theory is applied to the electronic structure of the helium atom, in which two electrons with opposite spin states are shown to correlate such that their quantum trajectories keep them on average on opposite sides of the nucleus. As the theory is time dependent, excited states are also generated. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011
Despite the success of the Dirac equation, the incorporation of the special theory of relativity into quantum mechanics predicts some paradoxes, like that the spin prediction from Dirac equation can be only identified with non- relativistic approximations (Pauli and Foldy-Wouthysen), as well as the spin predication is a relativistic quantum phenomena because the spin prediction is a necessary requirement of the relativistic quantum mechanics only. In this paper we show that the derivation of the spin and its magnetic moment can be done with a pure classical treatment. Since we start from the classical physical laws and the classical relativity principle to get the linear Schrödinger equation as a result the derivation of the spin and its magnetic moment can be done with a pure classical treatment. This approach result a Schrödinger equation, in which we show that the spin of the electron is a non relativistic quantum phenomenon too.
Dirac sought a relativistic quantum free-particle Hamiltonian that imposes space-time symmetry on the Schroedinger equation in configuration representation; he ignored the Lorentz covariance of energy-momentum. Dirac free-particle velocity therefore is momentum-independent, breaching relativity basics. Dirac also made solutions of his equation satisfy the Klein-Gordon equation via requirements imposed on its operators. Dirac particle speed is thereby fixed to the unphysical value of c times the square root of three, and anticommutation requirements prevent four observables, including the components of velocity, from commuting when Planck's constant vanishes, a correspondence-principle breach responsible for Dirac free-particle spontaneous acceleration (zitterbewegung) that diverges in the classical limit. Nonrelativistic Pauli theory contrariwise is physically sensible, and its particle rest-frame action can be extended to become Lorentz invariant. The consequent Lagrangian yields the corresponding closed-form relativistic Hamiltonian when magnetic field is absent, otherwise a successive-approximation regime applies.
The mean arrival time of free particles is computed using the quantum-mechanical probability current. This is uniquely determined in the nonrelativistic limit of Dirac equation, although the Schr\"odinger probability current has an inherent nonuniqueness. Since the Dirac probability current contains a spin-dependent term, an arrival time distribution based on the probability current shows an observable spin-dependent effect, even for free particles. This arises essentially from relativistic quantum dynamics, but persists even in the nonrelativistic regime.