#### Abstract

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 ${\mathit{mc}}^{2}$=ħω, the ‘‘classical’’ traversal times for allowed transmission become identical for a photon of energy ħω 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*≊${\mathit{V}}_{0}$ (θ≊${\mathrm{\theta}}_{\mathit{c}}$) and for *E*≫${\mathit{V}}_{0}$ (θ≪${\mathrm{\theta}}_{\mathit{c}}$). (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*≪${\mathit{V}}_{0}$ (θ≊π/2), one must choose ${\mathit{mc}}^{2}$=${\mathit{n}}^{2}$ħω 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.

DOI: http://dx.doi.org/10.1103/PhysRevA.49.3283

- Received 18 May 1993
- Published in the issue dated May 1994

© 1994 The American Physical Society