Resonant states and transmission coefficient oscillations for potential wells and barriers

The oscillatory behavior of the transmission coefficient $T$ as a function of energy is examined for an attractive square well and a rectangular barrier. We calculate $T$ using resonant state boundary conditions and demonstrate that the maxima in $T$ are correlated with the broad resonances generated by these potentials. For barrier potentials the maxima signify resonances occurring at energies above the barrier height. It is shown that the resonance position and width can also be generated from the complex poles of the amplitude of the transmitted plane wave. We also explain the relation between the positions of the resonances generated by the square well and the rectangular barrier to the energy eigenvalues of the corresponding rigid box with the same range. We show for a potential with an attractive well and a repulsive barrier that $T$ exhibits oscillations when the particle energy is below the barrier, implying that in many cases the simple WKB type barrier penetration expression for $T$ is not adequate. These features of $T$ are likely to hold for most attractive potentials and flat repulsive barriers. We also discuss the attractive modified Poschl–Teller type potential for which $T$ does not show oscillations as a function of energy.