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Quantum Information and Computation

Quantum complexity associated with tunneling

Ofir Flom, Asher Yahalom A, Haggai Zilberberg, Lawrence Paul Horwitz, Jacob Levitan

Published Mar 1, 2019• 0 citations
Publisher

Abstract

We use a one dimensional model of a square barrier embedded in an infinite potential well to demonstrate that tunneling leads to a complex behavior of the wave function and that the degree of complexity may be quantified by use of a locally defined spatial entropy function defined by S=-\int |\Psi(x,t)|^2 \ln |\Psi(x,t)|^2 dx . We show that changing the square barrier by increasing the height or width of the barrier not only decreases the tunneling but also slows down the rapid rise of the entropy function, indicating that the locally defined entropy growth is an essentially quantum effect.

DOI
10.26421/qic19.3-4-3

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We use a one dimensional model of a square barrier embedded in an infinite potential well to demonstrate that tunneling leads to a complex behavior of the wave function and that the degree of complexity may be quantified by use of a locally defined spatial entropy function defined by S=-\int |\Psi(x,t)|^2 \ln |\Psi(x,t)|^2 dx . We show that changing the square barrier by increasing the height or width of the barrier not only decreases the tunneling but also slows down the rapid rise of the entropy function, indicating that the locally defined entropy growth is an essentially quantum effect.
We use an one dimensional model of a square barrier embedded in an infinite potential well to demonstrate that tunneling leads to a complex behavior of the wave function and that the degree of complexity may be quantified by use of the spatial entropy function defined by S = -\int |Ψ(x,t)|^2 ln |Ψ(x,t)|^2 dx. There is no classical counterpart to tunneling, but a decrease in the tunneling in a short time interval may be interpreted as an approach of a quantum system to a classical system. We show that changing the square barrier by increasing the height/width do not only decrease the tunneling but also slows down the rapid rise of the entropy function, indicating that the entropy growth is an essentially quantum effect.
It is demonstrated in the context of the simple one-dimensional example of a barrier in an infinite well that highly complex behavior of the time evolution of a wave function is associated with the near degeneracy of levels in the process of tunneling. Degenerate conditions are obtained by shifting the position of the barrier. The complexity strength depends on the number of almost degenerate levels which depend on geometrical symmetry. The presence of complex behavior is studied to establish correlation with spectral degeneracy.
Abstract We theoretically study the tunneling time by investigating a wave packet of Bose-condensed atoms passing through a square barrier. We find that the tunneling time exhibits different scaling laws in different energy regimes. For negative incident energy of the wave packet, counterintuitively, the tunneling time decreases very rapidly with decreasing incident velocity. In contrast, for positive incident energy smaller than the barrier height, the tunneling time increases slowly and then reaches a maximum, which is in agreement with the Larmor clock experiments. The effect of the barrier width related to the uncertainty principle on the maximum tunneling time is also addressed. Our work provides a general scenario of tunneling time that can be used to understand and explain the controversy over tunneling time.
The working principle of a large number of modern scientific equipment has been applied to the quantum tunneling effect. Although the explanations for the relevant principles of this effect are clarified, the calculation process is very complex and difficult to understand. Previous studies combine specific applications without theoretical model references. This study adopts the method of using the time-independent Schrodinger equation to solve the wave function, which is more convenient to obtain the expression of transmission probability. Then, one discusses the relationship between transmission probability and potential barrier width, potential barrier spacing, and incident particle energy. The experimental results show that the transmission probability decreases exponentially from 1 to 0 as the potential barrier width gradually increases. As the potential barrier width increases, the transmission probability exhibits a clear periodic oscillation relationship, the oscillation period remains unchanged, and this paper derives the relationship between the oscillation period and the potential barrier width. As the incident energy of particles gradually increases, the transmission probability exhibits periodic changes, and the minimum value of transmission probability increases with the periodic changes. Based on the analysis, the smaller the potential barrier width, the greater the incident energy, and the more likely quantum tunneling occurs.
Quantum tunneling is a phenomenon where particles have a probability of penetrating a potential barrier despite having total energy llower than the barrier height. This study analyzes the tunneling effect by solving the one-dimensional (1D) Schrodinger equation using the finite difference method to obtain the wave function evolution for various potential barrier configurations. The solution is implemented in a Graphical User Interface (GUI) MATLAB to facilitate analysis and visualization, allowing users to interactively adjust potential parameters, energy, and other conditions. Simualation results demonstratehow transmission probability depends on energy, height and width of potential barrier. This GUI provides an intiuitive tool for exploring quantun tunneling, making it valuable for both education and research in quantum physics.