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

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