The comparison of two Larmor times

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.