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Quantal statistical phase factor accompanying inter-change of two electrons

Boyan Obreshkov

Published Mar 30, 2009• 0 citations
OpenAlex

Abstract

It is shown that effects of particle identity entail reduction in the number of orbital degrees-of-freedom in non-relativistic 2-particle systems from 6 to 5. This effect of redundancy in description of orbital motion is found to be in correspondence to multiplicative phase factor $(-1)^{2s}$ which accompany interchange of the two particles, where $s$ is the spin of one particle.

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Quantal statistical phase factor accompanying inter-change of two particles

Boyan Obreshkov

It is shown that effects of particle identity entail reduction in the number of orbital degrees-of-freedom in non-relativistic 2-particle systems from 6 to 5. This effect of redundancy in description of orbital motion is found to be in correspondence to multiplicative phase factor $(-1)^{2s}$ which accompany interchange of the two particles, where $s$ is the spin of one particle.

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It is shown that effects of particle identity entail reduction in the number of orbital degrees-of-freedom in non-relativistic 2-particle systems from 6 to 5. This effect of redundancy in description of orbital motion is found to be in correspondence to multiplicative phase factor $(-1)^{2s}$ which accompany interchange of the two particles, where $s$ is the spin of one particle.

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The wave function of a system containing identical particles takes into account the relationship between a particle's intrinsic spin and its statistical property. Specifically, the exchange of two identical particles having odd-half-integer spin results in the wave function changing sign, whereas the exchange of two identical particles having integer spin is accompanied by no such sign change. This is embodied in a term (-1)(2s), which has the value +1 for integer s (bosons), and -1 for odd-half-integer s (fermions), where s is the particle spin. All of this is well-known. In the nonrelativistic limit, a detailed consideration of the exchange of two identical particles shows that exchange is accompanied by a 2pi reorientation that yields the (-1)(2s) term. The same bookkeeping is applicable to the relativistic case described by the proper orthochronous Lorentz group, because any proper orthochronous Lorentz transformation can be expressed as the product of spatial rotations and a boost along the direction of motion.

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Exchanging two identical particles

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The phase factor $(-1)^{2s}$ that features in the exchange symmetry for identical spin-$s$ fermions or bosons contributes to but is not identical to the phase factor one may observe when one actually exchanges two such particles. The observable phase contains {\em single-particle} geometric and dynamical phases as well, induced by both spin and spatial exchange transformations. Extending the analysis to (abelian) anyons by incorporating the Aharonov-Bohm effect shows that the topological phase two anyons pick up under an interchange is at bottom a single-particle effect, in contrast to the fermion/boson exchange phase factor $(-1)^{2s}$, which is a two-particle effect.

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

Boyan Obreshkov

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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Parity as the foundation of the non-relativistic spin-statistics connection

D. V. Averin

It is shown that the symmetry under parity of the wavefunctions of two identical particles with an arbitrary spin $s$ in three spatial dimensions accounts for the appropriate wavefunction exchange statistics under the permutations of particles. The standard properties of the angular momentum in non-relativistic quantum mechanics account for the sign factor $(-1)^{2s}$ that the wavefunctions acquire under the permutation of coordinates of the two particles, without any additional requirements, directly relating spin and the particle exchange statistics in the non-relativistic context.

arXiv:2203.05106•Mar 10, 2022•0 citations
It is shown that effects of particle identity entail reduction in the number of orbital degrees-of-freedom in non-relativistic 2-particle systems from 6 to 5. This effect of redundancy in description of orbital motion is found to be in correspondence to multiplicative phase factor (−1)2s(-1)^{2s}(−1)2s which accompany interchange of the two particles, where sss is the spin of one particle.
It is shown that effects of particle identity entail reduction in the number of orbital degrees-of-freedom in non-relativistic 2-particle systems from 6 to 5. This effect of redundancy in description of orbital motion is found to be in correspondence to multiplicative phase factor (−1)2s(-1)^{2s}(−1)2s which accompany interchange of the two particles, where sss is the spin of one particle.
It is shown that effects of particle identity entail reduction in the number of orbital degrees-of-freedom in non-relativistic 2-particle systems from 6 to 5. This effect of redundancy in description of orbital motion is found to be in correspondence to multiplicative phase factor (−1)2s(-1)^{2s}(−1)2s which accompany interchange of the two particles, where sss is the spin of one particle.
The wave function of a system containing identical particles takes into account the relationship between a particle's intrinsic spin and its statistical property. Specifically, the exchange of two identical particles having odd-half-integer spin results in the wave function changing sign, whereas the exchange of two identical particles having integer spin is accompanied by no such sign change. This is embodied in a term (-1)(2s), which has the value +1 for integer s (bosons), and -1 for odd-half-integer s (fermions), where s is the particle spin. All of this is well-known. In the nonrelativistic limit, a detailed consideration of the exchange of two identical particles shows that exchange is accompanied by a 2pi reorientation that yields the (-1)(2s) term. The same bookkeeping is applicable to the relativistic case described by the proper orthochronous Lorentz group, because any proper orthochronous Lorentz transformation can be expressed as the product of spatial rotations and a boost along the direction of motion.
The phase factor (−1)2s(-1)^{2s}(−1)2s that features in the exchange symmetry for identical spin-sss fermions or bosons contributes to but is not identical to the phase factor one may observe when one actually exchanges two such particles. The observable phase contains {\em single-particle} geometric and dynamical phases as well, induced by both spin and spatial exchange transformations. Extending the analysis to (abelian) anyons by incorporating the Aharonov-Bohm effect shows that the topological phase two anyons pick up under an interchange is at bottom a single-particle effect, in contrast to the fermion/boson exchange phase factor (−1)2s(-1)^{2s}(−1)2s, which is a two-particle effect.
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.
It is shown that the symmetry under parity of the wavefunctions of two identical particles with an arbitrary spin sss in three spatial dimensions accounts for the appropriate wavefunction exchange statistics under the permutations of particles. The standard properties of the angular momentum in non-relativistic quantum mechanics account for the sign factor (−1)2s(-1)^{2s}(−1)2s that the wavefunctions acquire under the permutation of coordinates of the two particles, without any additional requirements, directly relating spin and the particle exchange statistics in the non-relativistic context.