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Physical Review A

Construction of a symmetric potential barrier from tunneling transmission coefficients

C. K. Chan, Pao Lu

Published Nov 1, 1980• 1 citation
Publisher

Abstract

For a symmetric potential barrier with two turning points there exists an inversion procedure which is closely related to the inversion of scattering phase shifts. The inversion formulas correct to first order in ${\ensuremath{\hbar}}^{2}$ are derived in the WKB approximation. To illustrate the applicability and limitation of the proposed procedure, we construct the quartic potential barrier $U(X)=\ensuremath{-}{X}^{4}$, from its transmission coefficients.

DOI
10.1103/physreva.22.1869

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For a symmetric potential barrier with two turning points there exists an inversion procedure which is closely related to the inversion of scattering phase shifts. The inversion formulas correct to first order in \ensuremathℏ2{\ensuremath{\hbar}}^{2}\ensuremathℏ2 are derived in the WKB approximation. To illustrate the applicability and limitation of the proposed procedure, we construct the quartic potential barrier U(X)=\ensuremath−X4U(X)=\ensuremath{-}{X}^{4}U(X)=\ensuremath−X4, from its transmission coefficients.
By calculating the reflection and transmission coefficients for a square potential barrier in terms of a series of scatterings at step potential discontinuities, a multiple scattering approach to potential problems is presented. This approach is applied to WKB potential problems by interpreting the WKB connection formulas in terms of reflection and transmission coefficients at each turning point. An algorithm for the systematic construction of the energy Green’s function is given. The significance of complex coordinate turning points is demonstrated. The equivalence of this approach and that of the Gaussian approximation of the path integral is mentioned briefly.
This work is a contribution to the theory of the quantum tunneling effect. In the literature at least two barriers are studied, for which the third-order contribution to the phase - integral asymptotic approximation of the transmission coefficient do not contribute. These are the parabolic barrier and the inverse Morse barrier. In the present work we will show that with a proper choice of the so called base function there is at least one more barrier in this category namely the Eckart-Epstein potential. The fact that the third - order contribution vanishes is a good indication that we have found an optimal choice of the base function, and the treatment to find an optimal base function may be possible to generalize to other classes of potential barriers. For particles of a low energy compared to the energy near the top of the barrier we obtain a vary low transmission coefficient, which means that the probability for tunneling to occur is very low. There exist some cases, for example that with a double barrier which is transparent, even for certain relatively low energies but no evidence for this kind of transparency for a single barrier has been found. The present work does not give any such evidence. At the same time there are still speculations on cold fusion like effects, which would demand a higher probability for tunneling through for a single barrier.
A method of approximating solutions of the one-dimensional Schr\"odinger equation is presented in this paper. The method closely resembles the usual WKB approximation. Whereas in the ordinary WKB method the exponential function is used as the basis of the approximation, in this paper the solutions of an arbitrary Schr\"odinger equation are used. The general advantage is that by proper choice of the arbitrary equation an improved approximation can be obtained. The method is illustrated by treating the potential well and potential barrier problems when there are two turning points. The approximations to the wave functions are continuous even across the turning points. The barrier transmission problem is treated uniformly for energies above and below the peak of the barrier.
We derive the quantum-mechanical tunneling transmission coefficient for energies near the peak of a general potential barrier, to fourth order in the Wentzel-Kramers-Brillouin (WKB) approximation, using a method based on an eighth-order Taylor expansion of the potential near the peak. The result agrees with contour-integral formulations of WKB tunneling. For the P\"oschl-Teller potential the WKB series converges rapidly to the known exact transmission coefficient. Higher-order WKB corrections may be important in attempts to determine the internuclear potential by inversion using low-energy fusion cross-section data.
In extending the Miller-Good modified WKB approximation to include the higher-order terms, a divergence was introduced. Because of this divergence, the approximation was limited to energies above the potential barrier. With this divergence removed, the modified WKB method is no longer limited to energies above the potential barrier. In order to demonstrate this method, we calculate the transmission coefficients for energies below the peak of the potential barrier and show that the higher-order terms are essential to the approximation.