Physics – Quantum Physics
Scientific paper
2011-02-15
Physics
Quantum Physics
23 pages, 10 figures, REVTeX 4 preprint style, to be submitted shortly to Am. J. Phys
Scientific paper
Following the general scheme presented in [Am. J. Phys. 78 (8), 842--850 (2010)] we analyze the quantum-mechanical behavior of a system described by a one-dimensional asymmetric potential constituted by a step plus i) a linear barrier or ii) an exponential barrier. We explicitly solve the energy eigenvalue equation by means of the integral representation method, which allows us to classify the independent solutions as equivalence classes of homotopic paths in the complex plane. We discuss the structure of the bound states as function of the height U_0 of the step and we study the propagation of a sharp-peaked wave packet reflected by the barrier. For both the linear and the exponential barrier we provide an explicit formula for the delay time tau(E) as a function of the peak energy E. We display the resonant behavior of tau(E) at energies close to U_0. By analyzing the asymptotic behavior for large energies of the eigenfunctions of the continuous spectrum we also show that, as expected, tau(E) approaches the classical value for large values of E, thus diverging for the step-linear case and vanishing for the step-exponential one.
Cacciatori Sergio L.
Gorini Vittorio
Piattella Oliver F.
Rizzi Luca
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