The dependence of the time delay on the shape of the wave packet tunneling through the potential barrier
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THE EUROPEAN PHYSICAL JOURNAL D
Regular Article
The dependence of the time delay on the shape of the wave packet tunneling through the potential barrier Andrey A. Bocharova Rzhanov Institute of Semiconductor Physics SB RAS, 13 Lavrentyev Ave., Novosibirsk 630090, Russia Received 22 January 2020 / Received in final form 12 May 2020 Published online 4 July 2020 c EDP Sciences / Societ`
a Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2020 Abstract. The problem of quasi-monochromatic wave packet tunneling through a potential rectangular barrier is studied. It is shown that the approach to calculating the time delay of a wave packet called “phase time” is not universally correct though well known in the literature. In this paper we discuss the mathematical reasons for this discrepancy, and substantiate another, more correct analytical method for calculating the time delay in the approximation of wide barriers. This time depends on the shape of the envelope complex wave function falling on the barrier. Analytical results for the Gaussian wave packet are presented.
1 Introduction The problem of calculating the time of particle tunneling through a potential barrier has a long history since the 1930s [1,2], and still remains far from its full resolution. The issue was widely disputed after the 1990s (reviews [3–6]). The development of atto-second physics experiments [7] has given a new impetus to the discussion. Such experiments include the process of atom ionization by an ultrashort laser pulse, in which the electron overcomes the barrier formed by the Coulomb potential of the atom and the external laser field [7,8]. These works also make efforts to assess adequacy of the widely used definitions of tunneling time for experimental data. In [9], in view of these experiments, the long-standing question of considering time as a Hermitian operator is raised. In theoretical papers [10–12], the Keldysh theory [13,14] is revised for such experiments. In the present paper, an attempt is made to revise the old problem of a particle tunneling through a rectangular potential barrier, which is a source of ideas for calculating the tunneling time. The term “tunneling time” of a particle through a potential barrier is very vague because of the uncertainty principle. The particle described by the probabilistic wave function can be located in different places of space for both the incident and outgoing wave packets. Therefore, it is necessary to determine the average characteristics of this value. This paper will not discuss the physical meaning of the issue, but instead will formally examine the transformation of the wave packet and the delay time of the packet as a whole at its tunneling through a potential barrier. In addition to the quantummechanical problem of tunneling, this may be useful in a
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studying the passage of classical fields in the problems of frustrated total internal reflection in optics [15,16], plasmonics [17,18] and acoustics [19], when the wave in the clas
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