Born-Oppenheimer Approximation for a Singular System

TL;DR

This paper analyzes a one-dimensional singular system with two heavy particles and a light particle, applying the Born-Oppenheimer approximation and a Fock space approach to estimate errors and develop systematic corrections, including relativistic modifications.

Contribution

It introduces a detailed application of the Born-Oppenheimer approximation to a simple singular system, with systematic error estimation and higher-order correction methods, including a relativistic extension.

Findings

Error terms can be estimated self-consistently.

Systematic expansion allows higher order corrections.

Relativistic modifications are feasible with careful second quantization.

Abstract

We discuss a simple singular system in one dimension, two heavy particles interacting with a light particle via an attractive contact interaction. It is natural to apply Born-Oppenheimer approximation to this problem. We present a detailed discussion of this approach, the advantage of this simple model is that one can estimate the error terms self-consistently. Moreover, a Fock space approach to this problem is presented where a systematic expansion can be proposed to get higher order corrections. A slight modification of the same problem in which the light particle is relativistic is discussed in a later section. Here, the second quantized description is more challenging but with some care one can recover the first order term as well as introducing a more systematic approach to higher orders.