Instantaneous velocity during quantum tunnelling

TL;DR

This paper investigates the dynamics of particle velocity during quantum tunnelling, revealing how velocity relaxes within the barrier and approaches zero for wide barriers, clarifying longstanding debates about tunnelling speed.

Contribution

It provides a detailed analysis of the temporal evolution of tunnelling particles, deriving explicit relations between velocity and barrier width, and clarifies misconceptions about stationary tunnelling speed.

Findings

Particle velocity relaxes from high initial values to near zero inside the barrier.

Probability density builds up gradually, contrasting previous assumptions.

Velocity approaches zero as barrier width increases, resolving paradoxes.

Abstract

Quantum tunnelling, a hallmark phenomenon of quantum mechanics, allows particles to pass through the classically forbidden region. It underpins fundamental processes ranging from nuclear fusion and photosynthesis to the operation of superconducting qubits. Yet the underlying dynamics of particle motion during tunnelling remain subtle and are still the subject of active debate. Here, by analyzing the temporal evolution of the tunnelling process, we show that the particle velocity inside the barrier continuously relaxes from a large initial value toward a smaller one, and may even approach zero in the evanescent regime. Meanwhile, the probability density within the barrier gradually builds up before reaching its stationary profile, in contrast to existing inherently. In addition, starting from the steady-state equations, we derive an explicit relation between the particle velocity and the barrier width, and show that the velocity in evanescent states approaches zero when the barrier is sufficiently wide. These findings resolve the apparent paradox of a vanishing steady-state velocity coexisting with a finite particle density. We point out that defining an effective speed from the probability density, rather than from the probability current, can lead to spuriously nonzero "stationary speed," as appears to be the case in Ref. [Nature 643, 67 (2025)]. Our work establishes a clear dynamical picture for the formation of tunnelling flow and provides a theoretical foundation for testing time-resolved tunnelling phenomena.