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Journal of Guizhou University

Research on Tunneling Time of Rectangle Complex Potential and Delta Complex Potential

Yujin Li

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

Rectangle potential and delta potential are different,while the width of the rectangle potential tends to be zero,the rectangle will become delta potential.In this paper,phase time of the two potentials was researched,and the relationship between phase time and the potential's property including real part and imaginary part were analyzed.At last,it is found that both of the complex potentials have the same condition of differential quotient.

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First,resolving the Schroinger equation for the boundary condition,we got the tunneling time of one and double delta complex potentials.Then,we also analyzed the effection of the related aspects to the phase time by using the Matlab and Origin.The research we got that,the phase time under the two conditions,was different for the effection of the real and imaginary part,as well as the projectile energy. Reserching the signal delta complex potentials,we got the maximum,while the double delta complex potentials having the minimum.Lastly,the two different potentials were simulated and analyzed.

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Rectangle potential and delta potential are different,while the width of the rectangle potential tends to be zero,the rectangle will become delta potential.In this paper,phase time of the two potentials was researched,and the relationship between phase time and the potential's property including real part and imaginary part were analyzed.At last,it is found that both of the complex potentials have the same condition of differential quotient.
First,resolving the Schroinger equation for the boundary condition,we got the tunneling time of one and double delta complex potentials.Then,we also analyzed the effection of the related aspects to the phase time by using the Matlab and Origin.The research we got that,the phase time under the two conditions,was different for the effection of the real and imaginary part,as well as the projectile energy. Reserching the signal delta complex potentials,we got the maximum,while the double delta complex potentials having the minimum.Lastly,the two different potentials were simulated and analyzed.
The electron tunnelling phase time τp and dwell time τD through an associated delta potential barrier U(x) = ξδ(x) are calculated and both are in the order of 10−17−10−16s. The results show that the dependence of the phase time on the delta barrier parameter ξ can be described by the characteristic length lc = 2/meξ and the characteristic energy Ec = meξ2/2 of the delta barrier, where me is the electron mass, lc and Ec are assumed to be the effective width and height of the delta barrier with lcEc = ξ, respectively. It is found that τD reaches its maximum and τD = τp as the energy of the tunnelling electron is equal to Ec/2, i.e. as lc = DB, DB is de Broglie wave length of the electron.
For gaining further insight into the nature of the eigenspectra of a complex octic potential [say], we investigate the quasi exact solutions of the Schr?dinger equation in an extended complex phase space characterized by . The analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalues and eigenfunction of a system. Explicit expressions of eigenvalues and eigenfunctions for the ground state as well as for the first excited state of a complex octic potential and its variant are worked out. It is found that imaginary part of the eigenvalue turns out to be zero for real coupling parameters, whereas it becomes non-zero for complex coupling parameters. However, the PT-symmetric version of a non-hermitian Hamiltonian possesses the real eigenvalue even if coupling parameters in the potential are complex.
In this paper we calculate the analytic expression of the phase time for the scattering of an electron off a complex square barrier. As is well known the (negative) imaginary part of the potential takes into account, phenomenologically, the absorption. We investigate the so-called Hartman-Fletcher effect, and find that it is suppressed by the presence of a (not negligible) imaginary potential. In fact, when a sufficiently large absorption is present, the asymptotical transmission speed is finite. Actually, the tunnelling time does increase linearly with the barrier width. A recent optical experiment seems to be in agreement with our theoretical previsions.
We develop a Green's function formalism and with it we are able to obtain close expressions for the tunnelling and for the reflection times. For each problem, there appear two characteristic times which correspond to the real and imaginary parts of the integral of the Green's function at coinciding coordinates. The time related to the imaginary part represents the minimum uncertainty of the measurement. A strong analogy between the results of the different existing approaches is established, and we show that their main differences are due to finite-size effects.