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arXiv:2203.09437 · Journal of Physics Communications

Electron wave spin in a quantum well

Ju Gao

Published Jul 29, 2022• 9 citations
arXivHTMLPDF

Abstract

Abstract The particle-wave duality of the electron poses a principle question of whether the spin is a property of the particle or the wave. In this paper, the wave nature of the spin is studied for an electron inside a two-dimensional quantum well. By solving the exact 4 − spinor eigen solution to the Dirac equation, we show that a stable circulating total current density exists inside the well with a donut shaped topography. A spin value is modified by the confining geometry of the well. Our analysis also shows that a free electron Gaussian wavepacket is unstable and experiences quick decoherence.

DOI
10.1088/2399-6528/ac8599

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Abstract The particle-wave duality of the electron poses a principle question of whether the spin is a property of the particle or the wave. In this paper, the wave nature of the spin is studied for an electron inside a two-dimensional quantum well. By solving the exact 4 − spinor eigen solution to the Dirac equation, we show that a stable circulating total current density exists inside the well with a donut shaped topography. A spin value is modified by the confining geometry of the well. Our analysis also shows that a free electron Gaussian wavepacket is unstable and experiences quick decoherence.
Our study shows that an evanescent electron wave exists outside both finite and infinite quantum wells, by solving exact solutions of the Dirac equation in a cylindrical quantum well and maintaining wavefunction continuity at the boundary. Furthermore, we have derived analytical expressions of the current density to demonstrate that the evanescent wave spins concurrently with the wave inside the quantum well. Our findings suggest that it is possible to probe or eavesdrop on quantum spin information through the evanescent wave spin without destroying the entire spin state. The wave spin picture interprets spin as a global and deterministic property of the electron wave that includes both the evanescent and confined wavefunctions. This suggests that a quantum process or device based on the manipulation and probing of the electron wave spin is deterministic in nature rather than probabilistic.
The notion introduced by Ohanian that spin is a wave property is implemented, both in Dirac and in Schrödinger quantum mechanics. We find that half-integer spin is the consequence of azimuthal dependence in two of the four spinor components, relativistically and non-relativistically. In both cases the spinor components are free particle wavepackets; the total wavefunction is an eigenstate of the total angular momentum in the direction of net particle motion. In the non-relativistic case we make use of the Lévy-Leblond result that four coupled non-relativistic wave equations, equivalent to the Pauli-Schrödinger equation, represent particles of half-integer spin, with g-factor 2. An example of an exact Gaussian solution of the non-relativistic equations is illustrated.
This study demonstrates the existence of an evanescent electron wave outside both finite and infinite quantum wells by solving the Dirac equation and ensuring the continuity of the spinor wavefunction at the boundaries. We show that this evanescent wave shares the spin characteristics of the wave confined within the well, as indicated by analytical expressions for the current density across all regions. Our findings suggest that the electron cannot be confined to a mathematical singularity and that quantum information, or quantum entropy, can leak through any quantum confinement. These results emphasize that the electron wave, fully characterized by Lorentz-invariant charge and current densities, should be considered the true and sole entity of the electron.
This study demonstrates the existence of an evanescent electron wave outside both finite and infinite quantum wells by solving the Dirac equation and ensuring the continuity of the spinor wavefunction at the boundaries. We show that this evanescent wave shares the spin characteristics of the wave confined within the well, as indicated by analytical expressions for the current density across all regions. Our findings suggest that the electron cannot be confined to a mathematical singularity and that quantum information, or quantum entropy, can leak through any confinement. These results emphasize that the electron wave, fully characterized by Lorentz-invariant charge and current densities, should be considered the true and sole entity of the electron.
Abstract As a continuation of our earlier investigations into electron wave–spin, we analyze the electron spin and its qubit in a cavity by treating the electron as a physical wave obeying the Dirac equation. In this view, a qubit is a current–density configuration whose orientation is fixed by the relative phase, rather than a particle carrying simultaneous ‘up’ and ‘down’ spin states with assigned probabilities. The resulting magnetic–moment density, derived from the current, displays a richer vector distribution and topology than the fixed axial dipole weighted by probability density in the conventional wave–particle model. Both frameworks yield the same total moment of one Bohr magneton and are indistinguishable in uniform external fields, yet their ontological differences predict distinct couplings to structured fields and spin–spin interactions. These contrasts motivate further exploration of dynamical consequences within the wave–entity framework, including Aharonov–Bohm–like responses that provide testable alternatives to conventional wave–particle duality.
Our study reveals electron spin in a cavity as a stable circulating current density, characterized by a torus topology. This current density circulates concentrically beyond the cavity boundary, illustrating the concept of evanescent wave spin. While the interaction with a uniform magnetic field aligns with established spin-field observations, our analysis of regional contributions deviates from particle-based spin predictions. The integration of charge and spin properties into a single Lorentz covariant entity suggests that the electron wave constitutes the fundamental and deterministic reality of the electron.