Role of a Phase Factor in the Boundary Condition of a One-Dimensional Junction

TL;DR

This paper investigates how a phase factor in boundary conditions affects quantum tunneling in one-dimensional systems, using exact WKB analysis to explore non-adiabatic transitions across energy levels.

Contribution

It introduces a detailed analysis of the phase factor in boundary conditions of 1D quantum junctions, linking it to non-adiabatic transitions in tunneling scenarios.

Findings

Phase factor influences tunneling boundary conditions.

Non-adiabatic transitions involve the phase factor.

WKB analysis reveals the role of the phase in energy level transitions.

Abstract

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.