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Chinese Physics C

Uncertainty principle in larmor clock

Chuan Qiao, Zhongzhou Ren

Published Nov 1, 2011• 3 citations
Publisher

Abstract

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.

DOI
10.1088/1674-1137/35/11/002

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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.
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.
A controversy surrounding the "tunnelling time problem" stems from the seeming inability of quantum mechanics to provide, in the usual way, a definition of the duration a particle is supposed to spend in a given region of space. For this reason, the problem is often approached from an "operational" angle. Typically, one tries to mimic, in a quantum case, an experiment which yields the desired result for a classical particle. One such approach is based on the use of a Larmor clock. We show that the difficulty with applying a non-perturbing Larmor clock in order to "time" a classically forbidden transition arises from the quantum Uncertainty Principle. We also demonstrate that for this reason a Larmor time (in fact, any Larmor time) cannot be interpreted as a physical time interval. We provide a theoretical description of the quantities measured by the clock.
The tunneling-time problem is shown to be analogous to the interpretation of the two-slit interference experiment. A measurement assuming an arbitrarily small interaction between a particle and a clock is shown to contradict the uncertainty principle and leads to complex times. A real non-negative traversal time is obtained in a measurement which selects Feynman paths that spend in the barrier a known amount of time; this, however, strongly perturbs the tunneling. The Larmor clock demonstrates both types of behavior.
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.
We numerically study two methods of measuring tunneling times using a quantum clock. In the conventional method using the Larmor clock, we show that the Larmor tunneling time can be shorter for higher tunneling barriers. In the second method, we study the probability of a spin-flip of a particle when it is transmitted through a potential barrier including a spatially rotating field interacting with its spin. According to the adiabatic theorem, the probability depends on the velocity of the particle inside the barrier. It is numerically observed that the probability increases for higher barriers, which is consistent with the result obtained by the Larmor clock. By comparing outcomes for different initial spin states, we suggest that one of the main causes of the apparent decrease in the tunneling time can be the filtering effect occurring at the end of the barrier.