Time-independent perturbation theory: degeneracy lifted to the second order

Abstract

Although the fundamental principles of time-independent perturbation theory are well-established and widely applied, the specific treatment of degeneracy lifted to the second order remains largely unexplored. This work fills the gap by presenting a systematic derivation of time-independent perturbation theory that accounts for the added complexity introduced by non-trivial degeneracies. We provide a general procedure and corresponding formulae for calculating state and energy corrections to any order, with degeneracy lifted to the second order. Additionally, we apply these formulae to solve a prototype problem of the dressed quark system. This work not only advances the theoretical framework but also provides practical tools for tackling complex problems in quantum physics.