A long-standing argument in elementary quantum mechanics concerns the prescription which specifies how long it takes for a particle to tunnel through a classically forbidden region. In this paper, the controversy is reviewed by analysing a selection of principal contenders and comparing with the authors' own analytic expressions and numerical simulations. Their overall conclusion is that the phase-time result originally obtained by Wigner (1955) and Hartman (1962) is the best expression to use for a wide parameter range of barriers, energies and wavepackets.
A long-standing argument in elementary quantum mechanics concerns the prescription which specifies how long it takes for a particle to tunnel through a classically forbidden region. In this paper, the controversy is reviewed by analysing a selection of principal contenders and comparing with the authors' own analytic expressions and numerical simulations. Their overall conclusion is that the phase-time result originally obtained by Wigner (1955) and Hartman (1962) is the best expression to use for a wide parameter range of barriers, energies and wavepackets.
How long does a quantum particle take to traverse a classically forbidden energy barrier? In other words, what is the correct expression for quantum tunneling time? This seemingly simple question has inspired widespread debate in the physics literature. By drawing an analogy with the double-slit experiment, I argue that we should not even expect the standard interpretation of quantum mechanics to provide an expression for quantum tunneling time. I explain how this conclusion connects to time’s special status in quantum mechanics, the meaningfulness of classically inspired concepts in different interpretations of quantum mechanics, and the prospect of constructing experimental tests to distinguish between different interpretations.
Abstract How long does a quantum particle take to traverse a classically forbidden energy barrier? In other words, what is the correct expression for quantum tunnelling time? This seemingly simple question has inspired widespread debate in the physics literature. I argue that we should not expect the orthodox interpretation of quantum mechanics to provide a unique correct expression for quantum tunnelling time, because to do so it would have to provide a unique correct answer to a question whose assumptions are in tension with its core interpretational commitments. I explain how this conclusion connects to time’s special status in quantum mechanics, the meaningfulness of classically inspired concepts in different interpretations of quantum mechanics, the prospect of constructing experimental tests to distinguish between different interpretations, and the status of weak measurement in resolving questions about the histories of subensembles.
Introduction In classical physics it is meaningful to ask the question, ‘How much time does a particle take to pass through a given region?’ The interesting question in quantum mechanics is: does a particle take a definite time to tunnel through a classically forbidden region? The question has been debated ever since the idea that there was such a time in quantum theory was first put forward by MacColl way back in 1932 [253]. A plethora of times has since then been proposed, and the answer seems to depend on the interpretation of quantum mechanics one uses. A reliable answer is clearly of great importance for the design of high-frequency quantum devices, tunnelling phenomena (as, for example, in scanning tunneling microscopy), nuclear and chemical reactions and, of course, for purely conceptual reasons. Most of the controversies centre around simple and intuitive notions in idealized one-dimensional models in a scattering configuration in which a particle (usually represented by a wave packet) is incident on a potential barrier localized in the interval [ a, b ]. Three kinds of time have been defined in this context. One, called the transmission time τ T ( a, b ), is the average time spent within the barrier region by the particles that are eventually transmitted. Similarly, the reflection time τ R ( a, b ) is the average time spent within the barrier region by the particles that are eventually reflected.
When a particle tunnels through a barrier it interacts with that barrier for a time that has been analysed by the authors. An appreciable number of other investigators have, with modest variations, come to the same result. Collins, Lowe and Barker (1987) have criticised this approach and advocated an older answer which depends primarily on the phase changes encountered at the turning points of the barrier, and not on barrier thickness, nor on the exact potential variation within the barrier. This is a rebuttal of their criticism.
The time a quantum particle spends traversing a classically forbidden region has resisted un-ambiguous definition since the earliest analyses of wave-packet tunneling, and the proliferation ofmutually incompatible “clocks”—the Wigner phase time, the Büttiker–Landauer traversal time, theLarmor time, and the path-integral complex time among them—has kept the question open fordecades. The controversy has been sharpened, not settled, by the advent of attosecond angular-streaking metrology, in which competing analyses of tunnel ionization continue to disagree over theexistence of a finite delay. Here we show that these rival definitions are not alternative answersto a single question but distinct cumulants of one traversal-time quasi-distribution, generated bythe exact rectangular-barrier transmission amplitude when the latter is promoted to a cumulantgenerating functional through a barrier-height counting field. The first cumulant unifies the Wignerphase time and the Larmor/Büttiker–Landauer time as the real and imaginary parts of a singlecomplex traversal time, recovering the celebrated Hartman saturation as the width-independentvalue ℏ/V0 of its real part. The second cumulant—which, to our knowledge, has not previouslybeen enlisted as a tunneling-time observable—yields a traversal-time variance that entirely evadesthe Hartman saturation and instead grows linearly with the barrier width d. The standard devia-tion of the tunneling time therefore scales as στ ∝ √d: rather than becoming superluminally sharpfor thick barriers, the tunneling time grows progressively more uncertain, in direct opposition tothe naive superluminal reading of the Hartman effect. This parameter-free scaling, independentof the O(1) coefficient that fixes its magnitude, constitutes a falsifiable prediction: attosecond an-gular streaking, which resolves the full arrival-time distribution of tunneled electrons rather thanmerely its centroid, should register a temporal spread broadening as √d even where the mean delaysaturates. We thereby recast a nearly century-old semantic dispute as a concrete experimentalquestion.
A simple model of a quantum clock is applied to the old and controversial problem of how long a particle takes to tunnel through a quantum barrier. The model has the advantage of yielding sensible results for energy eigenstates and does not require the use of time-dependent wave packets. Although the treatment does not forbid superluminal tunneling velocities, there is no implication of faster-than-light signaling because only the transit duration is measurable, not the absolute time of transit. A comparison is given with the weak-measurement post-selection calculations of Steinberg.