Landau-Zener problem for energies close to potential crossing points

TL;DR

This paper investigates the Landau-Zener problem near potential crossing points, calculating a 4x4 connection matrix that accounts for all coupled states in the intermediate energy region, extending previous models.

Contribution

It introduces a comprehensive 4x4 connection matrix for the Landau-Zener problem near crossing points, addressing a previously overlooked energy region.

Findings

Derived the 4x4 connection matrix for intermediate energies

Matched the matrix with known solutions in tunneling and over-barrier regions

Enhanced understanding of state coupling near crossing points

Abstract

We examine one overlooked in previous investigations aspect of well - known Landau - Zener (LZ) problem, namely, the behavior in the intermediate, i.e. close to a crossing point, energy region, when all four LZ states are coupled and should be taken into account. We calculate the 4 x 4 connection matrix in this intermediate energy region, possessing the same block structure as the known connection matrices for the tunneling and in the over-barrier regions of the energy, and continously matching those in the corresponding energy regions.