  One-dimensional quantum systems can be experimentally studied in recent
nano-technology like the carbon nanotube and the nanowire. We have considered
the mathematical model of the one-dimensional Schr\"{o}dinger particle with a
junction and have analyzed the phase factor in the boundary condition of the
junction. We have shown that the phase factor in the tunneling case appears in
the situation of the non-adiabatic transition with the three energy levels in
the exact WKB analysis.
