Path integrals and the quantum traversal time problem

The quantum-mechanical traversal- (tunneling) time problem is reformulated in terms of a square-integrable amplitude distribution (wave function) whose evolution is governed by a generalized Schr\"odinger equation. The method uses an analogy between quantum mechanics and classical Brownian motion. Brownian and quantum traversal-time distributions are evaluated by the direct integration of corresponding equations of motion. The general relation between Feynman and Schr\"odinger quantum mechanics is discussed.The quantum-mechanical traversal- (tunneling) time problem is reformulated in terms of a square-integrable amplitude distribution (wave function) whose evolution is governed by a generalized Schr\"odinger equation. The method uses an analogy between quantum mechanics and classical Brownian motion. Brownian and quantum traversal-time distributions are evaluated by the direct integration of corresponding equations of motion. The general relation between Feynman and Schr\"odinger quantum mechanics is discussed.

Physical Review AJul 1, 199534 citations

Path decomposition and the traversal-time distribution in quantum tunneling

H. A. Fertig

We define, using a path-integral method, an amplitude distribution for the traversal time of a particle tunneling through a barrier. We calculate this without approximation for the case of a square barrier potential. The form of the integrated distribution is shown to be determined by the uncertainty principle: if the particle is known to spend a time \ensuremath{\le}\ensuremath{\tau} inside the barrier, then an effective uncertainty \ensuremath{\Delta}V is introduced in the barrier height, such that \ensuremath{\Delta}V\ensuremath{\tau}\ensuremath{\approxeq}\ensuremath{\Elzxh}. The distribution has a number of interesting properties: it is noisy at very short times, oscillates periodically at intermediate times, and saturates to a constant at long times. The first two properties arise as a result of resonances in the transmission amplitude, which play an important role in the distribution. The characteristic (median) time scale for the distribution is shown to be ${\mathrm{\ensuremath{\tau}}}_{\mathit{Q}}$=2 ${\mathit{md}}^{2}$ /\ensuremath{\Elzxh}. We show that this represents the average traversal time for particle energies that are close to the energy of the barrier height. For very opaque barriers, the oscillations have a very short period. We show that this can lead to anomalously short average traversal times, and resolves the outstanding problem that the average velocity of a tunneling particle may exceed the speed of light. For particles with energies above the height of the barrier (classical traversal), we find that the real part of the distribution is dominated by the classical path, and has a sharp peak around the classical traversal time. It is argued that the form of the distribution found in this work may be applied to general barrier shapes.We define, using a path-integral method, an amplitude distribution for the traversal time of a particle tunneling through a barrier. We calculate this without approximation for the case of a square barrier potential. The form of the integrated distribution is shown to be determined by the uncertainty principle: if the particle is known to spend a time \ensuremath{\le}\ensuremath{\tau} inside the barrier, then an effective uncertainty \ensuremath{\Delta}V is introduced in the barrier height, such that \ensuremath{\Delta}V\ensuremath{\tau}\ensuremath{\approxeq}\ensuremath{\Elzxh}. The distribution has a number of interesting properties: it is noisy at very short times, oscillates periodically at intermediate times, and saturates to a constant at long times. The first two properties arise as a result of resonances in the transmission amplitude, which play an important role in the distribution. The characteristic (median) time scale for the distribution is shown to be ${\mathrm{\ensuremath{\tau}}}_{\mathit{Q}}$=2${\mathit{md}}^{2}$/\ensuremath{\Elzxh}. We show that this represents the average traversal time for particle energies that are close to the energy of the barrier height. For very opaque barriers, the oscillations have a very short period. We show that this can lead to anomalously short average traversal times, and resolves the outstanding problem that the average velocity of a tunneling particle may exceed the speed of light. For particles with energies above the height of the barrier (classical traversal), we find that the real part of the distribution is dominated by the classical path, and has a sharp peak around the classical traversal time. It is argued that the form of the distribution found in this work may be applied to general barrier shapes.

Physical review. B, Condensed matterJan 15, 199333 citations

Path integrals and quantum computing

David B. Mumford

American Mathematical Society eBooksSep 27, 20230 citations

Path Integrals and Quantum Mechanics

World Scientific lecture notes in physicsJun 1, 200635 citations

{\mathit{md}}^{2}

/\ensuremath{\Elzxh}. We show that this represents the average traversal time for particle energies that are close to the energy of the barrier height. For very opaque barriers, the oscillations have a very short period. We show that this can lead to anomalously short average traversal times, and resolves the outstanding problem that the average velocity of a tunneling particle may exceed the speed of light. For particles with energies above the height of the barrier (classical traversal), we find that the real part of the distribution is dominated by the classical path, and has a sharp peak around the classical traversal time. It is argued that the form of the distribution found in this work may be applied to general barrier shapes.