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

Spatial displacement of electrons due to multiple total reflections

David Milton Fradkin, R. J. Kashuba

Published May 15, 1974• 22 citations
Publisher

Abstract

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.

DOI
10.1103/physrevd.9.2775

Related papers

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

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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.

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In conventional optics for light waves, the pioneering work by Goos-Hänchen in 1947 on the lateral shift (or displacement) of a light beam along the surface of a dielectric boundary under the condition of total reflection has stimulated a large number of studies. The key physics behind the Goos-Hänchen shift is the nature of wave interference. From the perspective of wave optics, the incident beam of a finite transverse width can be viewed as composed of plane wave components, each of which has a slightly different transverse wavevector. Each wave component, after the total internal reflection, undergoes a different phase shift, and the superposition of all the reflected wave components gives rise to the lateral shift of the intensity peak in the reflected beam.

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Quantum mechanical streamlines. I. Square potential barrier

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Photon Mass and New Experimental Results on Longitudinal Displacements of Laser Beams near Total Reflection

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The suggestion of de Broglie and Vigier that an "intermediate shift" might be expected to be obtained in the Goos-Hanchen effect, for plane polarized beams having components both parallel and perpendicular to the plane of incidence of the totally reflective surface, is shown to be untenable. Their quoted experimental results are adequately explained classically.

Physical Review Letters•Jun 5, 1972•12 citations
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.
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.
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 conventional optics for light waves, the pioneering work by Goos-Hänchen in 1947 on the lateral shift (or displacement) of a light beam along the surface of a dielectric boundary under the condition of total reflection has stimulated a large number of studies. The key physics behind the Goos-Hänchen shift is the nature of wave interference. From the perspective of wave optics, the incident beam of a finite transverse width can be viewed as composed of plane wave components, each of which has a slightly different transverse wavevector. Each wave component, after the total internal reflection, undergoes a different phase shift, and the superposition of all the reflected wave components gives rise to the lateral shift of the intensity peak in the reflected beam.
The Goos-H\"anchen (GH) shift is a specifical optical phenomenon that describes a shift parallel to the reflected light inside the plane of incidence, when a finite-width light undergoes total internal reflection at the interface of medium. Although the GH shift in optics has been widely observed experimentally, its generalization remains uncovered completely in relativistic quantum mechanics for the spinor structure and the possible existence of Klein's paradox. Through approximation and angular spectrum analysis, we calculate the GH shift for Dirac fermions under relativistic conditions when they are incident obliquely on a three-dimensional infinite potential barrier. Furthermore, we find that the relativistic quantum GH shift will degenerate into the result based on Schr\"odinger equation under nonrelativistic limit, while the GH shift can be negative in Dirac's framework, which is different from the nonrelativistic case.
Exact numerical calculations are made for the scattering of quantum mechanical particles from a square two-dimensional potential barrier. This treatment is an exact analog of both frustrated total reflection of perpendicularly polarized light and the longitudinal Goos-Hänchen shift. Quantum mechanical streamlines (which are analogous to either classical mechanical trajectories or optical rays) are plotted. These streamlines are smooth, continuous, and have continuous first derivatives even through the classically forbidden region. The streamlines form quantized vortices surrounding each of the nodal points (which result from interference between the incident and reflected waves). Similar vortices occur in reactive collisions of H + H2 (McCullough and Wyatt; Kuppermann, Adams, and Truhlar) and undoubtedly play an important role in molecular collision dynamics. The theory for these vortices is given in a companion paper. A comparison is given between our numerical calculations and the stationary phase approximation (SPA). Although our incident wave packet has a half-width of more than one de Broglie wavelength in contrast to the SPA which replaces the diffuse beams by rays with delta function profiles, the agreement was surprisingly good for both the Goos-Hänchen shifts and for the reflection coefficients. However, we found that the Goos-Hänchen shift for the transmitted beam is significantly smaller than for the reflected beam, although in the stationary phase approximation these two shifts are equal. Furthermore, we found that the scattering from the potential barrier has very little effect on the shape of the wave packets. The power series expansion of the incident Debye-Picht wave packet ψI has an extremely small radius of convergence, whereas the power series for ψI*ψI has a radius of convergence of more than two de Broglie wavelengths. The imaginary velocity viis introduced into the Madelung-Landau-London hydrodynamical formulation of quantum mechanics. The corresponding imaginary streamlines will be considered in a forthcoming paper. The time-independent Schrödinger equation for real wavefunctions is reduced to solving the nonlinear first order partial differential equation: ℏ▿·vi = 2(E − V) + vi2. Here, vi is irrotational. This equation may lead to interesting new methods of solving the Schrödinger equation. It does lead to a generalization of the Prager-Hirschfelder perturbation scheme which invokes an electrostatic analogy. The philosophical implications of the hydrodynamical formulation of quantum mechanics are discussed. Penetration of a one-dimensional square potential barrier is used to demonstrate exactly how tunneling occurs by particles ``riding over the barrier'' à la Bohm. Cases are cited where quantum and classical mechanical motions are identical.