We compare the exact tunneling time with the quasi-classical tunneling time for idealized potentials. We examine three one-dimensional cases where the potential is chosen to have a simple form. In each case, the exact tunneling time and the quasi-classical time differ significantly. In one case, the two differ in magnitude by a factor of about ten. In another case, the two differ not only quantitatively, but qualitatively as well. A discussion is given as to why the two times are significantly different, and suggestions for further inquiries are made. PACS Nos.: 03.65Xp, 03.65-w
For smooth potential barriers, we compare the quasi-classical tunneling time with an expression that gives a fully quantum mechanical tunneling time. The expression we choose for the quantum mechanical tunneling time is one that has heuristic value. We report results wherein this quantum mechanical tunneling time and the quasi-classical time differ significantly, both quantitatively and qualitatively. To determine the reasons for these differences, we compare the trends in the two times that result from varying the potential. Our findings suggest that, for smooth potential barriers, the quasi-classical tunneling time is unreliable for many cases where it is employed. PACS Nos.: 03.65Xp, 03.65-w
The quantum tunneling rate of a macroscopic particle in one-dimensional potentials subject to oscillating external fields is considered using the complex time path integral method and the time-dependent Schrödinger equation. A simple calculation scheme for obtaining the tunneling rate within quasi-classical accuracy is described and applied to two systems. The one is the tunneling motion through an inverted parabola potential and the other is the escaping motion from a cubic potential well. Both systems are subject to small oscillating fields. It is found that both of the rates are exponentially enhanced due to the imaginary time motion of the tunneling particle under oscillating potential barriers, and that time scale is identical to the Büttiker-Landauer traversal time.
We demonstrate that although the well-known analogy between the time-independent solutions for two-dimensional tunneling (e.g., frustrated total internal reflection) and tunneling through a one-dimensional potential barrier cannot, in general, be extended to the time domain, there are certain limits in which the delay times for the two problems obey a simple relationship. In particular, when an effective mass is chosen such that
mc2=\ensuremath{\Elzxh}\ensuremath{\omega}, the ``classical'' traversal times for allowed transmission become identical for a photon of energy \ensuremath{\Elzxh}\ensuremath{\omega} traversing an air gap between regions of index n and for a particle of mass m traversing the analogous square barrier of height
V0 in one dimension. The quantum-mechanical group delays are also identical, given this effective mass, both for E\ensuremath{\approxeq}
V0 (\ensuremath{\theta}\ensuremath{\approxeq}${\mathrm{\ensuremath{\theta}}}_{\mathit{c}}$) and for E\ensuremath{\gg}
V0 (\ensuremath{\theta}\ensuremath{\ll}${\mathrm{\ensuremath{\theta}}}_{\mathit{c}}$). (For a smoothly varying potential or index of refraction, the agreement persists for all values of E where the WKB approximation applies.) The same relation serves to equate the quantum-mechanical ``dwell'' times for any values of E and
V0. On the other hand, in the ``deep tunneling'' limit, E\ensuremath{\ll}
V0 (\ensuremath{\theta}\ensuremath{\approxeq}\ensuremath{\pi}/2), one must choose
mc2=
n2\ensuremath{\Elzxh}\ensuremath{\omega} in order to make the group delays equal for the two problems. These equivalences simplify certain calculations, and the two-dimensional analogy may also be useful for geometrically visualizing the tunneling process and the anomalously small group delays known to occur in the opaque limit. We also demonstrate that the equality of the group delays for transmission and reflection for lossless barriers follows from a simple intuitive argument based on time-reversal invariance, and discuss the extension of the result to the case of lossy barriers.
Various approaches have been used to calculate the quantum-mechanical tunneling time through potential barriers including the phase-delay method first introduced by Bohm and Wigner, Buttiker's analysis of the Larmor clock and its generalization, the dwell time of Smith, and numerical studies of wavepacket propagation through potential barriers among others. Most of those previous estimates have only dealt with the tunneling time through simple obstacles, including delta-potential scatterers and simple rectangular barriers under zero bias condition. Even for these simple cases, the agreement among the various estimates is far from being satisfactory. In this manuscript, the transmission line technique is employed to solve the time-dependent Schrodinger equation and an expression of the quantum-mechanical tunneling time is derived for an arbitrary potential profile under non zero bias condition. An exact analytical expression of the tunneling time through a rectangular barrier is derived and shown to be identical to the one obtained recently by Spiller et. al. using the Bohm's quantum potential approach. The tunneling time through resonant tunneling structures under zero bias is also calculated and is shown to be minimum at the quasi-boundstate energy. Finally, the quantum-mechanical tunneling time through the emitter-base junction of a typical heterojunction bipolar transistor is shown to be larger than its semiclassical counterpart.
We present 1d-qt-ideal-solver, an open-source Python library for simulating one-dimensional quantum tunneling dynamics under idealized coherent conditions. The solver implements the split-operator method with second-order Trotter-Suzuki factorization, utilizing FFT-based spectral differentiation for the kinetic operator and complex absorbing potentials to eliminate boundary reflections. Numba just-in-time compilation achieves performance comparable to compiled languages while maintaining code accessibility. We validate the implementation through two canonical test cases: rectangular barriers modeling field emission through oxide layers and Gaussian barriers approximating scanning tunneling microscopy interactions. Both simulations achieve exceptional numerical fidelity with machine-precision energy conservation over femtosecond-scale propagation. Comparative analysis employing information-theoretic measures and nonparametric hypothesis tests reveals that rectangular barriers exhibit moderately higher transmission coefficients than Gaussian barriers in the over-barrier regime, though Jensen-Shannon divergence analysis indicates modest practical differences between geometries. Phase space analysis confirms complete decoherence when averaged over spatial-temporal domains. The library name reflects its scope: idealized signifies deliberate exclusion of dissipation, environmental coupling, and many-body interactions, limiting applicability to qualitative insights and pedagogical purposes rather than quantitative experimental predictions. Distributed under the MIT License, the library provides a deployable tool for teaching quantum mechanics and preliminary exploration of tunneling dynamics.
The precise definition and physical interpretation of a tunneling time is a fundamental problem in quantum mechanics. The lack of a well-defined time operator in quantum theory precludes the calculation of time in terms of an expectation value. Consequently, model calculations and simulation studies have been proposed and used to determine the time for a particle to traverse a quantum-mechanical barrier. The results offer both qualitatively and quantitatively, disparate predictions. One of the more commonly used approaches is the phase method, which we have applied to several one-dimensional model potentials to show that the tunneling time is characterized by the ratio of the typical decay length in the barrier to the incident velocity, τ∼(1/κ)/v0. We also show that the phase and the spin precession methods are equivalent when the magnetic field is applied throughout the space containing the particle wave function and the tunneling barrier. In principle, the spin precession method can be regarded as an operational definition of the tunneling time. It also has been shown that a tunneling time can be defined operationally for a time-dependent barrier, for which the model calculations predict a tunneling time given by the mean barrier width divided by the magnitude of the imaginary velocity of the particle in the barrier, τ∼d/‖v‖. This pseudokinematic result has been inferred by magnetic field dependent tunneling through heterostructures and computer simulation of wave packet tunneling through one-dimensional square barriers. Other tunneling time theories predict one or the other of the above mentioned characteristic times; several theories predict both. We interpret these different results for τ to mean that a tunneling time is not uniquely defined as a function of particle energy and the barrier shape unless the process in which the tunneling is involved is also specified. That is, the tunneling time is a meaningful and unique concept only for particular tunneling processes or definite operational procedures.