We show the equivalence of the functions ${G}_{\mathrm{p}}(t)$ and $|\ensuremath{\Psi}(d,t){|}^{2}$ for the ``passage time'' in tunneling. The former, obtained within the framework of the real-time Feynman histories approach to the tunneling time problem, uses the Gell-Mann and Hartle's decoherence functional, and the latter involves an exact analytical solution to the time-dependent Schr\"odinger equation for cutoff initial waves.We show the equivalence of the functions ${G}_{\mathrm{p}}(t)$ and $|\ensuremath{\Psi}(d,t){|}^{2}$ for the ``passage time'' in tunneling. The former, obtained within the framework of the real-time Feynman histories approach to the tunneling time problem, uses the Gell-Mann and Hartle's decoherence functional, and the latter involves an exact analytical solution to the time-dependent Schr\"odinger equation for cutoff initial waves.

arXiv:quant-ph/0302096Physical Review AMay 27, 200327 citations

Decoherent histories approach to tunneling times and its implication

Norifumi Yamada

Decoherent histories approach to quantum mechanics assigns probabilities to events that are not restricted to a single moment of time, provided that the decoherence condition holds for the set of events. The condition is however hardly satisfied in general, which has been considered to limit the usefulness of the histories approach. The present paper applies the histories approach to the tunneling time problem to show that, although the decoherence condition does not hold, the histories approach is still useful in predicting the central tendency and the dispersion of resident times. We consider such quantities that would become the average and the standard deviation of resident times if the decoherence condition held. Since the condition does not actually hold, they are not regarded as the true average and the true standard deviation. Nevertheless, a quantum clock approach allows us to interpret these quantities as a central value and a dispersion of resident times within the time resolution of the clock.Decoherent histories approach to quantum mechanics assigns probabilities to events that are not restricted to a single moment of time, provided that the decoherence condition holds for the set of events. The condition is however hardly satisfied in general, which has been considered to limit the usefulness of the histories approach. The present paper applies the histories approach to the tunneling time problem to show that, although the decoherence condition does not hold, the histories approach is still useful in predicting the central tendency and the dispersion of resident times. We consider such quantities that would become the average and the standard deviation of resident times if the decoherence condition held. Since the condition does not actually hold, they are not regarded as the true average and the true standard deviation. Nevertheless, a quantum clock approach allows us to interpret these quantities as a central value and a dispersion of resident times within the time resolution of the clock.

Journal of Physics Conference SeriesMay 1, 20070 citations

Time dependent quantum tunnelling

G J Papadopoulos

Springer eBooksApr 7, 20080 citations

Quantum Theory of Tunneling

Mohsen Razavy

A Brief History of Quantum Tunneling Some Basic Questions Concerning Quantum Tunneling Simple Solvable Problems Time-Dependence of the Wave Function in One-Dimensional Tunneling Semiclassical Approximations Generalization of the Bohr - Sommerfeld Quantization Rule and its Application to Quantum Tunneling Gamow's Theory, Complex Eigenvalues, and the Wave Function of a Decaying State Tunneling in Symmetric and Asymmetric Local Potentials and Tunneling in Nonlocal and Quasi-Solvable Barriers Classical Descriptions of Tunneling Tunneling in Time-Dependent Barriers Decay Width and Scattering Theory The Method of Variable Reflection Amplitude Applied to Solve Multichannel Tunneling Problems Path Integral and Its Semi-Classical Approximation in Quantum Tunneling Heisenberg's Equations of Motion for Tunneling Wigner Distribution Function in Quantum Tunneling Decay Widths of Siegert States, Complex Scaling and Dilatation Transformation Multidimensional Quantum Tunneling Group and Signal Velocities Time-Delay, Reflection Time Operator and Minimum Tunneling Time More about Tunneling Time Tunneling of a System with Internal Degrees of Freedom Motion of a Particle in a Waveguide with Variable Cross Section and in a Space Bounded by a Dumbbell-Shaped Object Relativistic Formulation of Quantum Tunneling Inverse Problems of Quantum Tunneling Some Examples of Quantum Tunneling in Atomic and Molecular Physics Some Examples from Condensed Matter Physics Alpha Decay.A Brief History of Quantum Tunneling Some Basic Questions Concerning Quantum Tunneling Simple Solvable Problems Time-Dependence of the Wave Function in One-Dimensional Tunneling Semiclassical Approximations Generalization of the Bohr - Sommerfeld Quantization Rule and its Application to Quantum Tunneling Gamow's Theory, Complex Eigenvalues, and the Wave Function of a Decaying State Tunneling in Symmetric and Asymmetric Local Potentials and Tunneling in Nonlocal and Quasi-Solvable Barriers Classical Descriptions of Tunneling Tunneling in Time-Dependent Barriers Decay Width and Scattering Theory The Method of Variable Reflection Amplitude Applied to Solve Multichannel Tunneling Problems Path Integral and Its Semi-Classical Approximation in Quantum Tunneling Heisenberg's Equations of Motion for Tunneling Wigner Distribution Function in Quantum Tunneling Decay Widths of Siegert States, Complex Scaling and Dilatation Transformation Multidimensional Quantum Tunneling Group and Signal Velocities Time-Delay, Reflection Time Operator and Minimum Tunneling Time More about Tunneling Time Tunneling of a System with Internal Degrees of Freedom Motion of a Particle in a Waveguide with Variable Cross Section and in a Space Bounded by a Dumbbell-Shaped Object Relativistic Formulation of Quantum Tunneling Inverse Problems of Quantum Tunneling Some Examples of Quantum Tunneling in Atomic and Molecular Physics Some Examples from Condensed Matter Physics Alpha Decay.

WORLD SCIENTIFIC eBooksJan 1, 2003259 citations

We show the equivalence of the functions

{G}_{\mathrm{p}}(t)

and $|\ensuremath{\Psi}(d,t){|}^{2}$ for the ``passage time'' in tunneling. The former, obtained within the framework of the real-time Feynman histories approach to the tunneling time problem, uses the Gell-Mann and Hartle's decoherence functional, and the latter involves an exact analytical solution to the time-dependent Schr\"odinger equation for cutoff initial waves.