Transmission through a non-overlapping well adjacent to a finite barrier

Abstract: We point out that a non-overlapping well (at negative energies) adjacent to a finite barrier (at positive energies) is a simple potential which is generally missed out while discussing the one-dimensional potentials in the textbooks of quantum mechanics. We show that these systems present interesting situations wherein transmitivity $(T_b(E))$ of a finite barrier can be changed both quantitatively and qualitatively by varying the depth or width of the well or by changing the distance between the well and the barrier. Using delta (thin) well near a delta (thin) barrier we show that the well induces energy oscillations riding over $T_b(E)$ in the transmitivity $T(E)$ at both the energies below and above the barrier. More generally we show that a thick well separated from a thick barrier also gives rise to energy oscillations in $T(E)$. A well joining a barrier discontinuously (a finite jump) reduces $T(E)$ (as compared to $T_b(E))$ over all energies. When the well and barrier are joined continuously, $T(E)$ increases and then decreases at energies below the barrier. At energy above the the barrier the changes are inappreciable. In these two cases if we separate the well and the barrier by a distance, $T(E)$ again acquires oscillations. Paradoxically, it turns out that a distant well induces more energy oscillations in $T(E)$ than when it is near the barrier.