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 ${\mathit{mc}}^{2}$ =\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 ${\mathit{V}}_{0}$ in one dimension. The quantum-mechanical group delays are also identical, given this effective mass, both for E\ensuremath{\approxeq} ${\mathit{V}}_{0}$ (\ensuremath{\theta}\ensuremath{\approxeq}${\mathrm{\ensuremath{\theta}}}_{\mathit{c}}$) and for E\ensuremath{\gg} ${\mathit{V}}_{0}$ (\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 ${\mathit{V}}_{0}$ . On the other hand, in the ``deep tunneling'' limit, E\ensuremath{\ll} ${\mathit{V}}_{0}$ (\ensuremath{\theta}\ensuremath{\approxeq}\ensuremath{\pi}/2), one must choose ${\mathit{mc}}^{2}$ = ${\mathit{n}}^{2}$ \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.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 ${\mathit{mc}}^{2}$=\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 ${\mathit{V}}_{0}$ in one dimension. The quantum-mechanical group delays are also identical, given this effective mass, both for E\ensuremath{\approxeq}${\mathit{V}}_{0}$ (\ensuremath{\theta}\ensuremath{\approxeq}${\mathrm{\ensuremath{\theta}}}_{\mathit{c}}$) and for E\ensuremath{\gg}${\mathit{V}}_{0}$ (\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 ${\mathit{V}}_{0}$. On the other hand, in the ``deep tunneling'' limit, E\ensuremath{\ll}${\mathit{V}}_{0}$ (\ensuremath{\theta}\ensuremath{\approxeq}\ensuremath{\pi}/2), one must choose ${\mathit{mc}}^{2}$=${\mathit{n}}^{2}$\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.

Physical Review AMay 1, 1994210 citations

Quantum reflection and tunneling in multi-quantum barrier

二三夫小山, FUMIO KOYAMA, KENICHI IGA

Jun 1, 19920 citations

New approach to the quantum tunneling process: Characteristic times for transmission and reflection

Nikolay Leonidovich Chuprikov

Russian Physics JournalMar 1, 20067 citations

Transmission and Reflection of Wave Packets by Asymmetric Semi-Harmonic Potential Barriers

Alfonso Jaimes-Nájera

We study the scattering of a particle by a square barrier potential that has a parabolic hole in between. The barrier is parameterized in such a form that the hole can be partially or completely removed. This can be also chosen to be symmetric or asymmetric with respect to the hole. Some expressions for the phase time are given and the time spent by the particle in the interaction zone of the barrier is calculated. It is shown that the related time delay depends on the symmetry operations that one can do on the potential.We study the scattering of a particle by a square barrier potential that has a parabolic hole in between. The barrier is parameterized in such a form that the hole can be partially or completely removed. This can be also chosen to be symmetric or asymmetric with respect to the hole. Some expressions for the phase time are given and the time spent by the particle in the interaction zone of the barrier is calculated. It is shown that the related time delay depends on the symmetry operations that one can do on the potential.

Journal of Physics Conference SeriesMar 1, 20163 citations

{\mathit{mc}}^{2}

=\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

{\mathit{V}}_{0}

in one dimension. The quantum-mechanical group delays are also identical, given this effective mass, both for E\ensuremath{\approxeq}

{\mathit{V}}_{0}

(\ensuremath{\theta}\ensuremath{\approxeq}${\mathrm{\ensuremath{\theta}}}_{\mathit{c}}$) and for E\ensuremath{\gg}

{\mathit{V}}_{0}

(\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

{\mathit{V}}_{0}

. On the other hand, in the ``deep tunneling'' limit, E\ensuremath{\ll}

{\mathit{V}}_{0}

(\ensuremath{\theta}\ensuremath{\approxeq}\ensuremath{\pi}/2), one must choose

{\mathit{mc}}^{2}

{\mathit{n}}^{2}

\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.