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Dynamical interpretation of the Pauli-exclusion principle for a system of two Coulombically interacting electrons

Boyan Obreshkov

Published Mar 31, 2009• 0 citations
OpenAlex

Abstract

In the framework of recently developed description of the non-relativistic constrained quantum mechanics of systems of $N$-Coulombically interacting electrons [Phys. Rev. {\bf A} 78, 032503 (2008)], we study the relation between spin and statistics in two-electron systems. We find that that the Fermi-Dirac statistics implies lowering of the spatial SO(3) symmetry of the wave-functions of electronic states to Abelian SO(2) rotational subgroup, which is accompanied by reduction of the number of spatial degrees-of-freedom from 6 to 5. We find that the relation between spin and statistics is necessarily related to the anti-symmetry of the relative phase of the spatial part of the two-electron wave-function. We find that it leads to distinct classification of the electronic states according to their relative orbital parity $(-1)^m$ and total spin $S=\{0,1\}$. We also find that the Pauli exclusion principle is realized dynamically due to Coulombic interaction between the particles. We find that the broken rotation symmetry of the wave-functions of the electronic states can be restored dynamically, due to conservation of the Laplace-Runge-Lenz vector. We demonstrate that the proposed theoretical description is internally consistent and predictive, and we find the non-equivalence of the above mention representation to the conventional Schr\{o}dinger picture for the description of the quantum dynamics in Coulombic systems with more than one electron.

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Based on a second-quantized representation of the nonrelativistic Hamiltonian of a system of $N$ electrons with pairwise Coulomb interactions, we demonstrate the exact statistical reduction of the $N$-body problem to a three-dimensional Schr\"odinger equation for the motion of a single active electron with all other $N\ensuremath{-}1$ electrons acting as spectators. As a by-product, three-dimensional Schr\"odinger equations for the ground and excited states of two-electron atoms and ions are derived and the dynamical role of Pauli's exclusion principle is established. The classical limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$ of the quantal all-electron equations is examined, and the Thomas-Fermi equation including the Amaldi correction is obtained.

Physical Review A•Sep 3, 2008•0 citations
In the framework of recently developed description of the non-relativistic
constrained quantum mechanics of systems of NNN-Coulombically interacting
electrons [Phys. Rev. {\bf A} 78, 032503 (2008)], we study the relation between
spin and statistics in two-electron systems. We find that that the Fermi-Dirac
statistics implies lowering of the spatial SO(3) symmetry of the wave-functions
of electronic states to Abelian SO(2) rotational subgroup, which is accompanied
by reduction of the number of spatial degrees-of-freedom from 6 to 5. We find
that the relation between spin and statistics is necessarily related to the
anti-symmetry of the relative phase of the spatial part of the two-electron
wave-function. We find that it leads to distinct classification of the
electronic states according to their relative orbital parity (−1)m(-1)^m(−1)m and total
spin S={0,1}S=\{0,1\}S={0,1}. We also find that the Pauli exclusion principle is realized
dynamically due to Coulombic interaction between the particles. We find that
the broken rotation symmetry of the wave-functions of the electronic states can
be restored dynamically, due to conservation of the Laplace-Runge-Lenz vector.
We demonstrate that the proposed theoretical description is internally
consistent and predictive, and we find the non-equivalence of the above mention
representation to the conventional Schr{o}dinger picture for the description
of the quantum dynamics in Coulombic systems with more than one electron.
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.
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.
The connection between the intrinsic angular momentum (spin) of particles and quantum statistics is established by considering the response of identical particles to a common background radiation field. For this purpose, the Hamiltonian analysis previously performed in stochastic electrodynamics to derive the quantum description of a one-particle system is extended to a system of two identical bound particles subject to the same field. Depending on the relative phase of the response of the particles to a common field mode, two types of particles are distinguished by their symmetry or antisymmetry with respect to particle exchange. While any number of identical particles responding in phase can occupy the same energy state, there can only be two particles responding in antiphase. The calculation of bipartite correlations between the response functions reveals maximum entanglement as a consequence of the parallel response of the particles to the common field. The introduction of an internal rotation parameter leads to a direct link between spin and statistics and to a physical rationale for the Pauli exclusion principle.
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.
The modern state of the Pauli Exclusion Principle (PEP) is discussed. PEP can be considered from two viewpoints. On the one hand, it asserts that particles with half-integer spin (fermions) are described by antisymmetric wave functions, and particles with integer spin (bosons) are described by symmetric wave functions. This is the so-called spin-statistics connection (SSC). As we will discuss, the physical reasons why SSC exists are still unknown. On the other hand, according to PEP, the permutation symmetry of the total wave functions can be only of two types: symmetric or antisymmetric, both belong to one-dimensional representations of the permutation group, all other types of permutation symmetry are forbidden; whereas the solution of the Schrödinger equation may have any permutation symmetry. It is demonstrated that the proof in some widespread textbooks on quantum mechanics that only symmetric and antisymmetric states (one-dimensional representations of the permutation group) can exist is wrong. However, the scenarios, in which an arbitrary permutation symmetry (degenerate permutation states) is permitted lead to contradictions with the concepts of particle identity and their independence. Thus, the existence in our nature particles only in nondegenerate permutation states (symmetric and antisymmetric) is not accidental and so-called symmetrization postulate may not be considered as a postulate, since all other symmetry options for the total wave function may not be realized. From this an important conclusion follows: we may not expect that in future some unknown elementary particles can be discovered that are not fermions or bosons.
Based on a second-quantized representation of the nonrelativistic Hamiltonian of a system of NNN electrons with pairwise Coulomb interactions, we demonstrate the exact statistical reduction of the NNN-body problem to a three-dimensional Schr\"odinger equation for the motion of a single active electron with all other $N\ensuremath{-}1$ electrons acting as spectators. As a by-product, three-dimensional Schr\"odinger equations for the ground and excited states of two-electron atoms and ions are derived and the dynamical role of Pauli's exclusion principle is established. The classical limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$ of the quantal all-electron equations is examined, and the Thomas-Fermi equation including the Amaldi correction is obtained.