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Physical Review A

Time-dependent method to calculate the relativistic dwell time and Larmor time

Jing Zhao, Chao‐Yun Long

Published Jul 10, 2007• 6 citations
Publisher

Abstract

A general relation between the relativistic dwell time and Larmor time is derived by a time-dependent method. It is explicitly shown that the relativistic dwell time is equal to the Larmor time in an infinitesimal magnetic field.
DOI
10.1103/physreva.76.012105

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Journal of Physics A Mathematical and General•Jun 21, 1992•13 citations
The Larmor precession of a relativistic neutral spin-½ particle in a uniform constant magnetic field confined to the region of a one-dimensional rectangular potential well is investigated. The spin precession serves as a clock to measure the time spent by a quantum particle dwelling at a potential well. With the help of a general spin coherent state it is explicitly shown that the spin precession time is equal to the dwell time in the first-order approximation of the infinitesimal field limit. The comparison of the time in a potential well with that in free space shows apparent superluminality.
The Larmor clock which was proposed to determine tunneling times of particles with spin through barriers is studied. Tunneling of wave packets is numerically derived from the time-dependent Schr\"odinger-Pauli equation. The Larmor clock often does not proceed unidirectionally in time. The asymptotic readings are well defined. For wide wave packets the asymptotic Larmor times agree with the stationary-state predictions; deviations are seen for narrow wave packets. B\"uttiker's dwell time is investigated and its connection with the Larmor times is confirmed in the case of wave packets. \textcopyright{} 1996 The American Physical Society.
For the relativistic tunneling effect described using Dirac's equation, Winful et al. [Phys. Rev. A 70, 052112 (2004)] presented the deduction of a general result that allows for the determination of the phase time (group delay) as the sum of the particle dwell time inside a potential barrier and of the self-interference delay associated with the incident and reflected wave functions interaction. In this paper, a mathematical model is derived through a construction analogous to the proposal mentioned above, but based on an alternative representation for Dirac's equation. This representation is similar to the one introduced by Ajaib [Found. Phys. 45, 1586 (2015)]. Thus, from the application of this model in the study of the tunneling effect in the absence of an external magnetic field, the influence of spin on the tunneling times is described. More specifically, the tunneling time is obtained as the sum of the dwell times inside the potential barrier for particles with spin up and spin down and the self-interaction time associated with the incident and reflected wave functions for particles with spin up.
Using the time-dependent scattering theory the authors prove that, in any spatial dimension and for arbitrary spin, the reading of the Larmor clock agrees with the global Eisenbud-Wigner time delay in the limit of an infinitesimal magnetic field. They show that convergence is also achieved at fixed energy (without oscillating terms) in the limit where the spatial switching on the field occurs on a much larger scale than the de Broglie wavelength of the particle. Finally, they investigate the functioning of the spin clock beyond the linear response regime.