Instanton versus traditional WKB approach to Landau - Zener problem

TL;DR

This paper compares the instanton and traditional WKB approaches to the Landau-Zener problem, highlighting the instanton method's ability to accurately handle all transition points in level crossing scenarios.

Contribution

It introduces a unified instanton-based semiclassical approach for the 1D Landau-Zener problem, improving the treatment of all transition points over standard WKB methods.

Findings

Instanton approach accurately treats all four transition points.

WKB approach only accounts for two transition points.

Results applicable to physics, chemistry, and biology systems.

Abstract

Different theoretical approaches to the famous two state Landau - Zener problem are briefly discussed. Apart from traditional methods of the adiabatic perturbation theory, Born - Oppenheimer approximation with geometric phase effects, two-level approach, and momentum space representation, the problem is treated semiclassically also in the coordinate space. Within the framework of the instanton approach we present a full and unified description of 1D Landau-Zener problem of level crossing. The method enables us to treat accurately all four transition points (appearing at two levels crossing), while the standard WKB approach takes into account only two of them. The latter approximation is adequate for calculating of the transition probability or for studying of scattering processes, however it does not work for finding corresponding chemical reactions rates, where very often for typical range of parameters all four transition points can be relevant. Applications of the method and of the results may concern the various systems in physics, chemistry and biology.