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arXiv:quant-ph/0510065 · Modern Physics Letters B

THE STATIONARY PHASE CONDITION APPLICABILITY TO THE STUDY OF TUNNEL EFFECT

A. E. BERNARDINI

Published Aug 10, 2005• 1 citation
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Abstract

Some recent theoretical studies tended to employ analytically-continuous gaussian, or infinite-bandwidth step pulses to examine tunneling process. The stationary phase method is often employed to this aim. However, gaussian functions do not have a well-defined front, such that their speed of propagation becomes ambiguous. Also, infinite bandwidth signals cannot propagate through any real physical medium (whose transfer function is therefore finite) without pulse distortion, which also leads to ambiguities in determining propagation velocity during the tunneling process. In this manuscript, we appoint some incompatibilities with the application of the method of stationary phase in deriving tunneling times which, eventually, can ruin the superluminal interpretation of transition times.

DOI
10.1142/s0217984905008906

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Some recent theoretical studies tended to employ analytically-continuous gaussian, or infinite-bandwidth step pulses to examine tunneling process. The stationary phase method is often employed to this aim. However, gaussian functions do not have a well-defined front, such that their speed of propagation becomes ambiguous. Also, infinite bandwidth signals cannot propagate through any real physical medium (whose transfer function is therefore finite) without pulse distortion, which also leads to ambiguities in determining propagation velocity during the tunneling process. In this manuscript, we appoint some incompatibilities with the application of the method of stationary phase in deriving tunneling times which, eventually, can ruin the superluminal interpretation of transition times.
Some recent theoretical studies have tended to employ analytically-continuous {\\em gaussian}, or infinite-bandwidth step pulses to examine tunneling process. The stationary phase method is often employed to this aim. However, {\\em gaussian} functions do not have a well-defined front, such that their speed of propagation becomes ambiguous. Also, infinite bandwidth signals cannot propagate through any real physical medium (whose transfer function is therefore finite) without pulse distortion, which also leads to ambiguities in determining propagation velocity during the tunneling process. In this manuscript, we appoint some incompatibilities with the application of the method of stationary phase in deriving tunneling times which, eventually, can ruin the {\\em superluminal} interpretation of transition times.
The stationary phase method is often employed for computing tunneling phase times of analytically-continuous gaussian or infinite-bandwidth step pulses which collide with a potential barrier. The indiscriminate utilization of this method without considering the barrier boundary effects leads to some misconceptions in the interpretation of the phase times. After re-examining the above barrier diffusion problem where we notice the wave packet collision necessarily leads to the possibility of multiple reflected and transmitted wave packets, we study the phase times for tunneling/reflecting particles in a framework where an idea of multiple wave packet decomposition is recovered. To partially overcome the analytical incongruities which rise up when tunneling phase time expressions are obtained, we present a theoretical exercise involving a symmetrical collision between two identical wave packets and a one-dimensional squared potential barrier, where the scattered wave packets can be recomposed by summing the amplitudes of simultaneously reflected and transmitted waves.
In the context of superluminal propagation of wave packets through potential barriers, the tunneling speed is usually characterized by the Wigner velocity. We propose an alternative speed that takes into account the interference between the incoming and the reflected waves and leads to a better estimation of arrival time for a wave packet entering the tunneling region. This arrival time is derived by an extrapolation from inside the barrier. The analytical theory is based on the stationary phase approximation whose validity is justified by a comparison with the numerical solution of the time-dependent Dirac equation.
Evanescent modes or the tunnelling process are characterized by an imaginary wave number. These special solutions of the Helmholtz and of the Schrödinger equations first noticed in connection with the total reflection were said to have no physical meaning about 200 years ago. Last century the tunnelling problem in quantum mechanics has been described by the phase time approach. The phase time approach yields the group velocity of a wave packet (see e.g. Hartman, 1962). Nowadays the phase time approach is used in network analyzers to determine the group velocity of an electromagnetic wave in devices. We have reported about superluminal signal velocities in photonic tunnelling. It was observed that the superluminal photonic tunnelling time data measured in the time domain are in agreement with the calculated phase time data. This result is also in agreement with the definition of the barrier transition time given in the Federal Standard 1037 C (NTIA, USA 2000). Thus we have added further credibility to the assumption that the photonic tunnelling results are representative for quantum mechanical tunnelling of particles.