Determination of the modes in two types of closed circuits with quantum tunneling

Others have solved the Schr\"odinger equation for a one-dimensional model having a square potential barrier in free-space by requiring an incident and a reflected wave in the semi-infinite pre-barrier region, two opposing waves in the square barrier, and a transmitted wave in the semi-infinite post-barrier region. Now we model a pre-barrier region having finite length that is shunted by the barrier to form a closed circuit. We use the boundary condition that the wavefunction and its derivative are continuous at the both ends of this model to obtain a homogeneous matrix equation. Thus, the determinant must be zero for a non-trivial solution. All but one of the following four parameters are specified and the remaining one is varied to bring the determinant to zero for a solution: (1) the electron energy, (2) the barrier length, (3) the barrier height, and (4) the pre-barrier length. The solutions with a square barrier are sets of non-intersecting S-shaped lines in the four-parameter space. The solutions with a triangular barrier have the product of the propagation constant and the length of the pre-barrier region as integer multiples of two-pi radians. Only static solutions are considered, but this method could be applied to time-dependent cases under quasistatic conditions. Suggestions are given for the design and testing of prototypes.