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Journal of Physics A Mathematical Nuclear and General

The approximate use of the complex gamma function in some wave propagation problems

John Heading

Published Jul 1, 1973• 10 citations
Publisher

Abstract

After a historical survey of the problem of the potential barrier, and an examination of the types of approximations, introduced and the reason for them, certain fallacious processes are discussed whereby the complex gamma function is given false approximate representations by means of exponential functions. Two kinds of problems involving complex barriers are then examined; firstly the phenomenon of resonant tunnelling through two complex potential barriers, and secondly, reflection phenomena arising from a single complex barrier The common fallacies are also examined, and the proper treatment of these problems shows when the complex gamma function must be retained and when it can be dispensed with.

DOI
10.1088/0305-4470/6/7/016

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The Journal of Chemical Physics•Oct 27, 2015•0 citations
After a historical survey of the problem of the potential barrier, and an examination of the types of approximations, introduced and the reason for them, certain fallacious processes are discussed whereby the complex gamma function is given false approximate representations by means of exponential functions. Two kinds of problems involving complex barriers are then examined; firstly the phenomenon of resonant tunnelling through two complex potential barriers, and secondly, reflection phenomena arising from a single complex barrier The common fallacies are also examined, and the proper treatment of these problems shows when the complex gamma function must be retained and when it can be dispensed with.
Abstract There is developed the theory of the propagation of waves through a system of potential barriers. Many properties are derived and discussed, mainly by the use of matrix theory arising from the description of each barrier by a transmission matrix. The extent to which certain previously used approximate methods are applicable and valid is also examined. Two distinct kinds of resonant phenomena are discussed – first when there is no incident wave and secondly when there is no reflected wave. This latter case of perfect transmission is investigated in detail, an examination being given of both why and when it occurs and the time taken to establish such exceptional fields within the barriers.
Abstract The success of approximate methods in the calculation of transmission coefficients of waves penetrating potential barriers rests, not so much on any satisfactory approximation theory, but on a comparison with a particular barrier, namely the parabolic barrier. The paucity of exact analytical solutions of barrier problems has prevented an explicit detailed analysis of the errors involved, but two recently devised barriers with analytical solutions are investigated here, in order to compare the exact values of the transmission coefficients (both as to their mathematical structure and numerically) with their approximate forms. An estimate is also made of the effect of neglecting poles existing in the complex propagation plane.
Various applications of the real trajectories in the complex-time method are discussed. It is shown that the proper way of summation over semiclassical orbits removes existing doubts and leads to the more general formulas than the standard WKB-type one in the energy eigenvalue problem. This method is also used in the description of explicitly time-dependent phenomena. In the problem of escaping from a metastable state, the estimation of a lifetime is in a good agreement with the inverse width of a false ground state energy.
The time independent semiclassical treatment of barrier tunneling has been understood for a very long time. Several semiclassical approaches to time dependent tunneling through barriers have also been presented. These typically involve trajectories for which the position variable is a complex function of time. In this paper, a method is presented that uses only real valued trajectories, thus avoiding the complications that can arise when complex trajectories are employed. This is accomplished by expressing the time dependent wave packet as an integration over momentum. The action function in the exponent in this expression is expanded to second order in the momentum. The expansion is around the momentum, p0(*) , at which the derivative of the real part of the action is zero. The resulting Gaussian integral is then taken. The stationary phase approximation requires that the derivative of the full action is zero at the expansion point, and this leads to a complex initial momentum and complex tunneling trajectories. The "pseudo-stationary phase" approximation employed in this work results in real values for the initial momentum and real valued trajectories. The transmission probabilities obtained are found to be in good agreement with exact quantum results.