Hartman effect, time-delays, and the non-spatial nature of quantum particles

TL;DR

The paper investigates the Hartman effect in quantum tunneling, showing that the transmission time-delay $\tau_{\rm tr}(E)$ can exhibit superluminal-like behavior with $\tau_{\rm tr}(E) = -\frac{2a}{v}[1 + O(1/a)]$, while the effective velocity inside the barrier, $v_{\rm eff}=\frac{2(a+b)}{T_{\rm tr}}$, diverges as the barrier width grows. By defining conditional time-delays via the scattering matrix and projection operators, the authors derive exact relations such as $\tau_{\rm tr}(E)=\tfrac{1}{2}(\tau_{\rm re}^{\rm left}(E)+\tau_{\rm re}^{\rm right}(E))$, which for symmetric barriers reduces to $\tau_{\rm tr} = \tau_{\rm re}$. The conundrum that a tunneling particle seemingly traverses a forbidden region is resolved by interpreting the tunneling entity as non-spatial, actualizing in space only upon detection; this view is framed within the Conceptuality Interpretation, where non-spatiality corresponds to abstractness and spatial concreteness arises at measurement. The work thus argues that the tunneling process provides strong evidence for a non-spatial, foundationally conceptual quantum reality, with broader implications for quantum foundations and cognitive-analogy interpretations.

Abstract

The phenomenon of quantum tunneling remains a fascinating and enigmatic one, defying classical notions of particle behavior. This paper presents a novel theoretical investigation of the tunneling phenomenon, from the viewpoint of Hartman effect, showing that the classical concept of spatiality is transcended during tunneling, since one cannot describe the process as a crossing of the potential barrier. This means that quantum tunneling strongly indicates that quantum non-locality should be understood as an aspect of quantum non-spatiality. It is also emphasized that according to the Conceptuality Interpretation of quantum mechanics, a non-spatial state should be understood as an abstract state of a conceptual-like entity, which only when it reaches its maximum degree of concreteness, during the wave-function collapse, can enter the spatiotemporal layer of our physical reality.

Figures (2)

• Figure 1: A potential barrier whose region of constant height $V_0$ has width $2a$, while the left and right transition regions have each width $b$, hence $V(x)$ has its support in the interval $[-a-b,a+b]$, contained in a larger interval $[-R,R]$, $R>a+b$, which is the one considered for the calculation of the time-delay. If the energy $E={1\over 2}mv^2$ of a classical particle coming from the left is strictly below the barrier height, $E<V_0$, it can only reach point $x=-a-s_0$, with $V(-a-s_0)-E=0$, before being reflected back.
• Figure 2: Unlike the reference entity that evolves freely, the potential region for which $E-V(x)<0$ remains inaccessible both to the reflected and the transmitted (tunneling) entities. Outside of this inaccessible region, the scattering entities interact for the same amount of time, on average, with the transition regions, whether they are ultimately reflected or transmitted. In the first case, they interact twice with the left transition region (assuming they come from the left), in the second case they interact once with both transition regions, which are here identical being the barrier symmetric. For non-symmetric barriers, the same applies if one considers an average over the processes where the entity approaches the barrier from the left and from the right, respectively, as per (\ref{['time-relation']}).