Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Position-operator method for evaluating the shift of a totally reflected electron

David Milton Fradkin, R. J. Kashuba

Published Aug 15, 1974• 9 citations
Publisher

Abstract

A method based on expectation values of a position operator is developed for evaluating the spatial shift of a polarized Dirac electron undergoing total reflection. Longitudinal and transverse components of the shift, as well as the associated time delay, are obtained by considering the differences in the expectation values for the physically reflected wave packet and a purely geometrically reflected wave packet. For a number of position operators, this method is used to calculate the mean shift arising from a single reflection of an electron from a step-function potential. Eigenpolarization states are determined by means of a localization form-invariance argument. For these states, results obtained employing the Newton-Wigner position operator are shown to be identical to those obtained using the phase-shift analysis method. A comparison calculation giving different results is made using the current-flux method. The current-flux results are shown to be inconsistent with conservation of total angular momentum normal to the reflecting boundary.

DOI
10.1103/physrevd.10.1137

Related papers

Spatial displacement of electrons due to multiple total reflections

David Milton Fradkin, R. J. Kashuba

A theoretical investigation is made of the spatial displacements of a Dirac electron which undergoes a series of total internal reflections from finite potential barriers of arbitrary smoothness. These displacements are analogous to the shifts observed for optical total reflection (the Goos-H\"anchen effect). For an electron, the polarization states that are invariant to a single-reflection process are identified and the change in polarization for an arbitrary incident polarized state is determined. A calculation of longitudinal and transverse shifts is made for a double-reflection process from surfaces with a specified orientation. This calculation is based on a phase-shift analysis in which a localization form-invariance argument for the wave packet is used to determine both the eigenpolarization states of the double-reflection process and the relevant phase shifts. An unpolarized beam of electrons will be spatially split into these geometry-dependent eigenpolarization states. In general, it is found that the total spatial displacement of the wave-packet center for each eigenstate consists of two parts, each of which has longitudinal and transverse components. There is a common shift (dependent on the potential) which near critical reflection is of the order of the de Broglie wavelength, and a polarization-dependent shift (independent of the potential) that causes a split that is at most of the order of a Compton wavelength. There is no splitting for reflections between parallel surfaces. In principle, a macroscopically observable common shift can be produced through multiple reflections.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•May 15, 1974•22 citations

Vortex-enhanced Zitterbewegung in relativistic Dirac wave packets

Zhongze Guo, Bei Xu, Qiang Gu

Zitterbewegung (ZBW), arising from the interference between positive- and negative-energy solutions of the Dirac equation, is a characteristic feature of relativistic single-particle dynamics, but its explicit manifestation depends on the choice of position operator. We formulate relativistic vortex electron wave packets as coherent superpositions of Dirac plane-wave states and derive their exact spacetime evolution. For the canonical Dirac position, we show that orbital angular momentum substantially enhances and spatially structures the relative-coordinate contribution conventionally associated with ZBW. We then analyze the operator dependence of this result by separating the canonical position into a Foldy–Wouthuysen/Newton–Wigner (FW/NW) mean position and an oscillatory relative coordinate. For the vortex packets considered here, the FW/NW mean position exhibits only the ordinary centroid drift and no transverse trembling. However, the same vortex-enhanced radial scale survives in the positive-energy sector as a Berry-type geometric dressing of the projected position operator. The present work therefore identifies not a representation-independent trembling of the one-particle centroid, but a vortex-amplified remnant of Dirac interband mixing in an operationally meaningful position observable.

Nuclear Physics B•Jun 2, 2026•0 citations

Scattering of relativistic electrons and analogies with optical phenomena: A study of longitudinal and transverse shifts at step potentials

Yue Ban, Xi Chen

We investigate the behavior of relativistic electrons encountering a potential step through analogies with optical phenomena. By accounting for the conservation of the Dirac current, we elucidate that the Goos-H\"anchen shift can be understood as a combination of two components: one arising from the current entering the transmission region and the other originating from the interference between the incident and reflected beams. This result has been proven to be consistent with findings obtained utilizing the stationary phase method. Moreover, we explore the transverse Imbert-Fedorov shift by applying both current conservation and total angular momentum conservation, revealing intriguing parallel to the spin Hall effect. Beyond enriching our comprehension of fundamental quantum phenomena, our findings have potential applications for designing and characterizing devices using Dirac and topological materials.

arXiv:2308.05413•Physical Review A•Oct 27, 2023•4 citations

Localized states of Dirac equation

Syed Badiuzzaman Faruque, S.D. Shuvo, Pradip Das

In this paper, we introduce an extension of the Dirac equation, very similar to Dirac oscillator, that gives stationary localized wave packets as eigenstates of the equation. The extension to the Dirac equation is achieved through the replacement of the momentum operator by a PT-symmetric generalized momentum operator. In the 1D case, the solutions represent bound particles carrying spin having continuous energy spectrum, where the envelope parameter defines the width of the packet without affecting the dispersion relation of the original Dirac equation. In the 2D case, the solutions are localized wave packets and are eigenstates of the third component of total angular momentum and involve Bessel functions of integral order. In the 3D case, the solutions are localized spherical wave packets with definite total angular momentum.

arXiv:1902.06232•Feb 17, 2019•0 citations

Consistent Evaluation of Operators Involving the Position Operator in the Bloch Representation: Application to the Orbital Moment

Daehyeon An, Junmo Jeon, Se Kwon Kim

The position operator plays a central role in condensed-matter observables such as velocity, orbital moment, and electric polarization. In solid-state physics, the evaluation of operators incorporating the position operator has not reached a consensus, as observed in the operator-level discrepancy between the local circulation of Wannier functions and the self-rotation of wave packets. Here, to achieve a consistent evaluation of such operators, we propose three rules for evaluating operators involving the position operator in the Bloch representation. The rules are devised to satisfy physical conditions: independence from the choice of unit cell, preservation of Hermitian conjugacy for the product of operators, and recovery of the correct intraband velocity. We further address the gauge dependence of the position operator and introduce a scheme termed gauge filtration, which systematically removes gauge-dependent contributions from the operators containing the position operator. This methodology ensures that the quantities obtained from the operator evaluation correspond to observable physical phenomena. By applying our framework, we reconcile the results concerning the self-rotation of the wave packet and the local circulation of the Wannier function. We expect our proposal to establish a consistent framework for evaluating operators involving the position operator.

cond-mat.mes-hallCross-listedarXiv:2606.11679•Jun 11, 2026

ANGULAR MOMENTUM-PHASE WIGNER OPERATOR FOR DESCRIBING ELECTRONS IN UNIFORM MAGNETIC FIELDS

Hong-Yi Fan

By introducing the angular momentum-phase Wigner operator both in <λ| representation and <l,r| representation, which are respectively the eigenstates of position and angular momentum of an electron in a uniform magnetic field, we establish the angular momentum-phase Wigner function whose marginal distributions are physically meaningful.

Modern Physics Letters B•Jun 10, 2001•2 citations
A method based on expectation values of a position operator is developed for evaluating the spatial shift of a polarized Dirac electron undergoing total reflection. Longitudinal and transverse components of the shift, as well as the associated time delay, are obtained by considering the differences in the expectation values for the physically reflected wave packet and a purely geometrically reflected wave packet. For a number of position operators, this method is used to calculate the mean shift arising from a single reflection of an electron from a step-function potential. Eigenpolarization states are determined by means of a localization form-invariance argument. For these states, results obtained employing the Newton-Wigner position operator are shown to be identical to those obtained using the phase-shift analysis method. A comparison calculation giving different results is made using the current-flux method. The current-flux results are shown to be inconsistent with conservation of total angular momentum normal to the reflecting boundary.
A theoretical investigation is made of the spatial displacements of a Dirac electron which undergoes a series of total internal reflections from finite potential barriers of arbitrary smoothness. These displacements are analogous to the shifts observed for optical total reflection (the Goos-H\"anchen effect). For an electron, the polarization states that are invariant to a single-reflection process are identified and the change in polarization for an arbitrary incident polarized state is determined. A calculation of longitudinal and transverse shifts is made for a double-reflection process from surfaces with a specified orientation. This calculation is based on a phase-shift analysis in which a localization form-invariance argument for the wave packet is used to determine both the eigenpolarization states of the double-reflection process and the relevant phase shifts. An unpolarized beam of electrons will be spatially split into these geometry-dependent eigenpolarization states. In general, it is found that the total spatial displacement of the wave-packet center for each eigenstate consists of two parts, each of which has longitudinal and transverse components. There is a common shift (dependent on the potential) which near critical reflection is of the order of the de Broglie wavelength, and a polarization-dependent shift (independent of the potential) that causes a split that is at most of the order of a Compton wavelength. There is no splitting for reflections between parallel surfaces. In principle, a macroscopically observable common shift can be produced through multiple reflections.
Zitterbewegung (ZBW), arising from the interference between positive- and negative-energy solutions of the Dirac equation, is a characteristic feature of relativistic single-particle dynamics, but its explicit manifestation depends on the choice of position operator. We formulate relativistic vortex electron wave packets as coherent superpositions of Dirac plane-wave states and derive their exact spacetime evolution. For the canonical Dirac position, we show that orbital angular momentum substantially enhances and spatially structures the relative-coordinate contribution conventionally associated with ZBW. We then analyze the operator dependence of this result by separating the canonical position into a Foldy–Wouthuysen/Newton–Wigner (FW/NW) mean position and an oscillatory relative coordinate. For the vortex packets considered here, the FW/NW mean position exhibits only the ordinary centroid drift and no transverse trembling. However, the same vortex-enhanced radial scale survives in the positive-energy sector as a Berry-type geometric dressing of the projected position operator. The present work therefore identifies not a representation-independent trembling of the one-particle centroid, but a vortex-amplified remnant of Dirac interband mixing in an operationally meaningful position observable.
We investigate the behavior of relativistic electrons encountering a potential step through analogies with optical phenomena. By accounting for the conservation of the Dirac current, we elucidate that the Goos-H\"anchen shift can be understood as a combination of two components: one arising from the current entering the transmission region and the other originating from the interference between the incident and reflected beams. This result has been proven to be consistent with findings obtained utilizing the stationary phase method. Moreover, we explore the transverse Imbert-Fedorov shift by applying both current conservation and total angular momentum conservation, revealing intriguing parallel to the spin Hall effect. Beyond enriching our comprehension of fundamental quantum phenomena, our findings have potential applications for designing and characterizing devices using Dirac and topological materials.
In this paper, we introduce an extension of the Dirac equation, very similar to Dirac oscillator, that gives stationary localized wave packets as eigenstates of the equation. The extension to the Dirac equation is achieved through the replacement of the momentum operator by a PT-symmetric generalized momentum operator. In the 1D case, the solutions represent bound particles carrying spin having continuous energy spectrum, where the envelope parameter defines the width of the packet without affecting the dispersion relation of the original Dirac equation. In the 2D case, the solutions are localized wave packets and are eigenstates of the third component of total angular momentum and involve Bessel functions of integral order. In the 3D case, the solutions are localized spherical wave packets with definite total angular momentum.
The position operator plays a central role in condensed-matter observables such as velocity, orbital moment, and electric polarization. In solid-state physics, the evaluation of operators incorporating the position operator has not reached a consensus, as observed in the operator-level discrepancy between the local circulation of Wannier functions and the self-rotation of wave packets. Here, to achieve a consistent evaluation of such operators, we propose three rules for evaluating operators involving the position operator in the Bloch representation. The rules are devised to satisfy physical conditions: independence from the choice of unit cell, preservation of Hermitian conjugacy for the product of operators, and recovery of the correct intraband velocity. We further address the gauge dependence of the position operator and introduce a scheme termed gauge filtration, which systematically removes gauge-dependent contributions from the operators containing the position operator. This methodology ensures that the quantities obtained from the operator evaluation correspond to observable physical phenomena. By applying our framework, we reconcile the results concerning the self-rotation of the wave packet and the local circulation of the Wannier function. We expect our proposal to establish a consistent framework for evaluating operators involving the position operator.