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Modern Physics Letters A

ABOVE BARRIER POTENTIAL DIFFUSION

Alex E. Bernardini, Stefano De Leo, PIETRO P. ROTELLI

Published Nov 8, 2004• 42 citations
Publisher

Abstract

The stationary phase method is applied to diffusion by a potential barrier for an incoming wave packet with energies greater than the height of the barrier. It is observed that a direct application leads to paradoxical results. The correct solution, confirmed by numerical calculations is the creation of multiple peaks as a consequence of multiple reflections. Lessons concerning the use of the stationary phase method are drawn.

DOI
10.1142/s0217732304015877

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The stationary phase method is applied to diffusion by a potential barrier for an incoming wave packet with energies greater than the height of the barrier. It is observed that a direct application leads to paradoxical results. The correct solution, confirmed by numerical calculations is the creation of multiple peaks as a consequence of multiple reflections. Lessons concerning the use of the stationary phase method are drawn.
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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.