# WKB and "Cubic-WKB" methods as an adiabatic approximation

## TL;DR

This paper demonstrates that WKB and cubic-WKB wave functions can be formulated as adiabatic expansions, providing a pedagogical link between these approximation methods and addressing divergence issues in WKB.

## Contribution

It introduces a unified adiabatic framework for WKB and cubic-WKB methods, clarifying their relationship and improving understanding of their approximation schemes.

## Key findings

- WKB wave functions can be expressed as adiabatic expansions.
- Cubic-WKB method is also representable as an adiabatic approximation.
- Adjusting parameters in the adiabatic expansion aligns with the cubic-WKB approach.

## Abstract

This paper shows that WKB wave function can be expressed in the form of an adiabatic expansion. To build a bridge between two widely invoked approximation schemes seems pedagogically instructive. Further "cubic-WKB" method that has been devised in order to overcome the divergence problem of WKB can be also presented in the form of an adiabatic approximation: The adiabatic expansion of a wave function contains a certain parameter. When this parameter is adjusted so as to make the next order correction vanish approximately, the adiabatic wave function becomes equivalent to that of the "cubic-WKB".