Quantum transit times have a long history of inequivalent definitions, including phase times, dwell times, and quantum-clock constructions. In this context we revisit the Salecker--Wigner--Peres stationary quantum clock as a phase-sensitive scattering observable, with clock time defined by the energy derivative of the transmission phase shift across the interaction region. For real compactly supported one-dimensional potentials, we show that the raw stationary Peres clock generically contains a universal (1/\sqrt{E}) continuum-edge term whose coefficient is fixed by low-energy scattering data. For the attractive square well, this threshold singularity is inherited from the vanishing exterior momentum and the associated scattering matching, rather than from resonant delay itself. We derive the exact stationary clock time for the square well and introduce a new threshold-subtracted clock observable. Away from exceptional zero-energy tuning, the subtraction removes the universal low-energy term and isolates the resonant contribution. Comparison with the dwell time and the transmission Wigner phase delay shows that the threshold-subtracted clock acquires the expected local Lorentzian form near isolated transmission resonances. Near the continuum edge, if (\varepsilon) denotes the detuning from threshold, the resonant peak grows only as (\varepsilon^{-1/2}), whereas the unsubtracted threshold background grows as (\varepsilon^{-3/2}). A symmetric barrier--well--barrier cavity and a numerical asymmetric two-step attractive well provide complementary controls. The result is a new threshold-subtracted stationary-clock candidate that separates universal threshold kinematics from pole-sensitive resonant delay.
The time-of-flight measurement approach of Peres based on the Salecker-Wigner quantum clock is applied to the one-dimensional scattering of a wave packet from a rectangular barrier. By directly evaluating the expectation value of the clock-time operator in the asymptotic states of wave packet long after the scattering process, we derive an average wave-packet clock time for the barrier interaction, which is expressed as an average of the stationary-state clock time over all possible initial scattering states of the wave packet. We show that the average wave-packet clock time is identical to the average dwell time of a wave packet.
The Salecker-Wigner-Peres (SWP) clock is often used to determine the duration a quantum particle is supposed to spend in a specified region of space $\mathrm{\ensuremath{\Omega}}$. By construction, the result is a real positive number, and the method seems to avoid the difficulty of introducing complex time parameters, which arises in the Feynman paths approach. However, it tells little about the particle's motion. We investigate this matter further, and show that the SWP clock, like any other Larmor clock, correlates the rotation of its angular momentum with the durations $\ensuremath{\tau}$, which the Feynman paths spend in $\mathrm{\ensuremath{\Omega}}$, thereby destroying interference between different durations. An inaccurate weakly coupled clock leaves the interference almost intact, and the need to resolve the resulting ``which way?'' problem is one of the main difficulties at the center of the ``tunnelling time'' controversy. In the absence of a probability distribution for the values of $\ensuremath{\tau}$, the SWP results are expressed in terms of moduli of the ``complex times,'' given by the weighted sums of the corresponding probability amplitudes. It is shown that overinterpretation of these results, by treating the SWP times as physical time intervals, leads to paradoxes and should be avoided. We also analyze various settings of the SWP clock, different calibration procedures, and the relation between the SWP results and the quantum dwell time. The cases of stationary tunneling and tunnel ionization are considered in some detail. Although our detailed analysis addresses only one particular definition of the duration of a tunneling process, it also points towards the impossibility of uniting various time parameters, which may occur in quantum theory, within the concept of a single tunnelling time.
Abstract In this paper, we calculate the quantum time delays for a neutral particle scattering off the Earth gravitational linear potential. The quantum time delays are obtained by subtracting the classical returning time (CRT) from the Wigner time, the dwell time and the Larmor time respectively. Different from the conventional definition, our Larmor time is defined by aligning the magnetic field along particle’s direction of motion, and this definition does give reasonable results for motions through a free region and a square barrier. It is worth noting that in the zero magnetic field limit, the Larmor time coincides well with the CRT, which is due to the special shape of linear barrier, and may have some relevance to the weak equivalence principle. It is also found that the classical forbidden region plays an essential role for the dwell time τ DW to match with the CRT, and the difference between the dwell and the phase times, i.e. the self-interference time delay, is barrier shape sensitive and clearly shows the peculiarity of the linear barrier. All the time delays are on the order of sub-millisecond and exhibit oscillating behaviors, signaling the self-interference of the scattering particle, and the oscillations become evident only when the de Broglie wavelength λ k = 2 π / k is comparable to the characteristic length L c = [ 2 m 2 g / ℏ 2 ] − 1 / 3 . If the time delay measurement is experimentally realizable, it can probe the quantum nature for particle scattering off the gravitational potential in the temporal domain.
We reexamine the Salecker-Wigner-Peres clock formalism and show that it can be directly applied to the phenomenon of tunneling. Then we apply this formalism to the determination of the tunneling time of a nonrelativistic wave packet, sharply concentrated around a tunneling energy, incident on a symmetric double-barrier potential. In order to deepen the discussion about the generalized Hartman effect, we consider the case in which the clock runs only when the particle can be found inside the region between the barriers and show that, whenever the probability to find the particle in this region is non-negligible, the corresponding time (which in this case turns out to be a dwell time) increases with barrier spacing.
The Larmor clock extracts time information from quantum scattering through a weak spin-dependent phase readout. We apply this formulation to the transmitted Larmor time in trapezoidal barriers and examine its geometry-dependent threshold response. The main quantity is the dimensionless peak coordinate xp=Ep/V0, where Ep is the energy of the dominant transmitted-time peak. For a right trapezoidal barrier consisting of a linear ramp and a flat segment, the peak moves from the classically forbidden side (xp<1) to the allowed side (xp>1) as the flat part becomes more important. The crossover is characterized by rc=(l-a)/a, defined through xp=1. For a=2 nm and V0=2 μeV, we obtain rc≃0.428. A symmetric triangle-rectangle-triangle barrier shows a corresponding crossover at sc=(d-a)/(2a)≃0.220. These results demonstrate that the strongest transmitted-channel clock response is controlled by the spatial distribution of the sloped matching regions and the flat phase-accumulating region. The peak coordinate therefore provides a phase-sensitive marker of how this response shifts across the nominal threshold as the barrier geometry is varied.
Recent generalizations of B\"uttiker's analysis of the Larmor clock to any region
z1\ensuremath{\le}z\ensuremath{\le}
z2 within an arbitrary one-dimensional barrier lead to two local transmission ``times'' for an incident electron of energy E=${\ensuremath{\Elzxh}}^{2}$
k2/2m. These are conveniently regarded as the real and (minus) the imaginary parts of a complex time ${\ensuremath{\tau}}_{T}^{V}$(k;
z1,
z2). In this paper the properties of ${\ensuremath{\tau}}_{T}^{V}$(k;
z1,
z2) are investigated for the double-rectangular-barrier potential V(z)=
V1\ensuremath{\Theta}(z)\ensuremath{\Theta}(a-z) +
V2\ensuremath{\Theta}(z-b)\ensuremath{\Theta}(d-z). Results are presented for the dependence of the real and imaginary parts of the Larmor-clock transmission time on incident energy and barrier and well widths. The local transmission time ${\ensuremath{\tau}}_{T}^{V}$(k;
z1,
z2) is real for symmetric double barriers when the energy of the incident electron is exactly on resonance. Remarkably, the behavior of the real quantity ${\ensuremath{\tau}}_{T}^{V}$(k;0, z\ensuremath{\ll}a) for this special case provides further evidence for the importance of the imaginary part of the local Larmor-clock transmission time for the corresponding (isolated) single barrier V(z)=
V1\ensuremath{\Theta}(z)\ensuremath{\Theta}(a-z). At resonance, for symmetric double barriers, (real) local transmission speeds
vT(z) can be very much in excess of the speed of light in the well region for z very close to a quasinode of the Schr\"odinger stationary-state wave function. It is proven that
vT(z)\ensuremath{\le}c for a symmetric double barrier at resonance when the Dirac equation is used in place of the Schr\"odinger equation. On the other hand, it is shown that application of the Dirac equation to a single opaque rectangular barrier does not alter the well-known difficulty that a/Re[${\ensuremath{\tau}}_{T}^{V}$(k;0,a)] exceeds c for sufficiently large barrier width a.