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Journal of Contemporary Physics (Armenian Academy of Sciences)

Quantum Tunneling Time of a Moderately Singular Potential: Regularization Method

A. Zh. Muradyan

Published Sep 1, 2022• 0 citations
Publisher

Abstract

In quantum mechanics, the singularity of the potential is usually removed at first, the desired values are computed for the regularized potential, and then an inverse limit transition is made in the resulting expressions. In this paper, the regularization method is used in the quantum tunneling of a power-law potential of the form $$V(x) = {{V}_{0}}/{{\left| x \right|}^{\alpha }}$$ . Тhe reverse limit transition is possible for the range of power $$0 < \alpha < 1$$ . The tunneling and reflection time of the wave packet is computed. It is shown that the McColl-Hartman paradox, well-known for ordinary potential barriers, is also valid for the moderately singular potential. The mathematical structure of the uncommon formation of paradox is presented.

DOI
10.1134/s1068337222030136

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In quantum mechanics, the singularity of the potential is usually removed at first, the desired values are computed for the regularized potential, and then an inverse limit transition is made in the resulting expressions. In this paper, the regularization method is used in the quantum tunneling of a power-law potential of the form V(x)=V0/∣x∣αV(x) = {{V}_{0}}/{{\left| x \right|}^{\alpha }}V(x)=V0​/∣x∣α . Тhe reverse limit transition is possible for the range of power 0<α<10 < \alpha < 10<α<1 . The tunneling and reflection time of the wave packet is computed. It is shown that the McColl-Hartman paradox, well-known for ordinary potential barriers, is also valid for the moderately singular potential. The mathematical structure of the uncommon formation of paradox is presented.
Abstract To get around the difficulty that the singular nature of the potential function V(x) = V0/|x|α introduces into physics, the regularization methods are used. However, they affect the singular nature of the problem, and so we discuss here how quantum tunneling behaves if the original singular nature of the Schrodinger equation remains unperturbed. For this purpose, I am starting from the precondition that the singular terms are mutually compensated in the current probability density and the current can be considered continuous. As a result, it is obtained that the mild-singular potential (with 0 < α < 1) has finite but unusual tunnel transparency, in particular, a non-zero value at zero energy of the incident particle. The transparency of a Coulomb potential well oscillates infinitely at zero energy, and only the strongly singular potential (with α > 1) repeats the property of the regularized prototypes: to be impenetrable.
We investigate the tunneling time of a wave packet propagating through a non-Hermitian potential Vr − iVi in space-fractional quantum mechanics. By applying the stationary phase method, we derive a closed-form expression for the tunneling time for this system. This study presents the first investigation of tunneling time at the interplay of non-Hermitian quantum mechanics and space-fractional quantum mechanics. The variation in tunneling time as the system transitions from a real to a complex potential is analyzed. We demonstrate that the tunneling time exhibits a dependence on the barrier width d in the limit d → ∞, showing the absence of the Hartman effect. A particularly striking feature of our findings is the potential manifestation of the Hartman effect for a specific combination of the absorption component Vi and the Levy index α. This behavior arises from the fact that the presence of the absorption component Vi leads to a monotonic increase in tunneling time with barrier thickness, whereas the Levy index α reduces the tunneling time. The interplay of these contrasting influences facilitates the emergence of the Hartman effect under a specific combination of Vi and the fractional parameter α.
The problem of quantum tunneling through the singular potential barrier $$V(x) = \left\{ {\begin{array}{*{20}c} {V_0 (b/x - x/a)^2 ,} & {0 < \left| x \right| \leqslant \sqrt {ab} } \\ {0,} & {\left| x \right| > \sqrt {ab} } \\ \end{array} } \right.$$ is discussed on the subject of the possibility to replace the singular behavior of the problem at the point x = 0 by a limiting process at the top of the truncated potential. The validity of such a replacement and, on this basis, the zero transparency of the quantum potential barrier are shown.
We consider the evolution of a wave packet that is made up of a group of the wave functions describing the stationary scattering process and tunnels through a one-dimensional potential of arbitrary form. As the main characteristics of the time difference of the tunnelling process, use is made of the propagation speed of the wave-packet maximum. We show that the known Hartman formula for the tunnelling time corresponds to the wave packet with a wavenumber-uniform spectral composition in the case, when the phase and transmission coefficient modulus dispersions are taken into account only in the linear approximation. The amplitude of the main peak of the transmitted wave intensity is proven to be independent of the tunnelling time and is determined by the transmission coefficient of the spectral component at the carrier frequency and the spectral width of the wave packet. In the limit of an infinitely wide potential barrier the amplitude of the wave-packet maximum is shown to tend to zero slower than the tunnelling time tends to its asymptotic value, i.e., indeed we deal with the paradox of an infinitely large propagation speed of a wave disturbance through the barrier.
I show that the MacColl-Hartman effect, namely, the saturation of the group delay time of sub-barrier quantum tunneling as a function of the barrier width, comes from the saturating behavior of a more fundamental concept - the phase of the stationary wave function. The explanation of saturation is given based on the decomposition of the stationary wave function into the spectrum of wave numbers and formulation of the initial condition for the direction of propagation of the incident matter wave. It is also shown that the saturation plateau of MacColl and Hartman actually doesn't continue indefinitely, but has a finite length. After the plateau, the sub-barrier tunneling time monotonically increases with increasing width of the potential, and this applies both to the maximum's of the wave packet and to the average tunneling time.