Larmor time and proper time

Juan Antonio López-Villanueva, V. Gasparian

J. P. Falck, E. H. Hauge

The Larmor clock, a thought experiment designed by Baz' to measure scattering times, is reexamined in the context of one-dimensional tunneling by wave packets, narrow in wave-number space, through arbitrary static barriers. It is shown that the Larmor clock is, in general, inaccurate when used on a single Fourier component, i.e., on a stationary state. A reliable Larmor clock depends for its operation on properly designed wave packets. Precise conditions for its reliability are given. Thus, the apparent conflict between previous Larmor clock results for tunneling times, and results arrived at by other methods, is removed. When properly set, the Larmor clock shows the classic phase times. Stationary-state calculations of Larmor and dwell ``times'' are shown to represent coherent superpositions of, in general, widely different collision events. The relative merits of the Larmor clock and the center-of-gravity clock are discussed. Finally, it is shown that, for an arbitrary symmetric potential, the reflection and transmission times coincide.

G. Stranathan

Chuan Qiao, Zhongzhou Ren

It is well known that the spin operators of a quantum particle must obey uncertainty relations. We use the uncertainty principle to study the Larmor clock. To avoid breaking the uncertainty principle, Larmor time can be defined as the ratio of the phase difference between a spin-up particle and a spin-down particle to the corresponding Larmor frequency. The connection between the dwell time and the Larmor time has also been confirmed. Moreover, the results show that the behavior of the Larmor time depends on the height and width of the barrier.

Evert Jan Post

Tong-Jia Li, Zhi Xiao

This paper explores the Larmor time and compares it for various shapes of barriers such as rectangular barrier and triangular barrier, and we arrive at consistent results between the traditionally defined Larmor time and a newly defined Larmor time. Interestingly, we have found some regular sharp jumps (discontinuities) in reflection Larmor times for a given rectangular barrier. With further investigation, it was found that the wave numbers where the Larmor time jumps correlate with those where the reflection probability reaches minimal values. Based on known results for triangular barriers, this paper also presents the Larmor times for positive and negative slope trapezoidal barriers, whose treatments are similar to the positive and negative triangular barriers, respectively, though the trapezoidal barriers exhibit more oscillatory behaviors for transmission times in comparison to their triangular counterparts. Finally, we demonstrate that the transmission and reflection Larmor times for M-shaped barriers exhibit more pronounced peaks, as expected. Also interestingly, some of the sharp jumps in reflection time are associated with sharp variations in the corresponding reflection probabilities, which is similar to the correlation mentioned earlier between the sharp jumps in reflection time and reflection probabilities in wave numbers for rectangular barriers.