What do we approximate and what are the consequences in perturbation theory?

TL;DR

This paper discusses the implications of using approximate zeroth order Hamiltonians in perturbation theory, showing that exact eigenfunctions and eigenvalues can be obtained for these approximations and exploring their effects on physical quantities like origin independence.

Contribution

It demonstrates that approximate Hamiltonians constructed via common methods still yield exact eigenfunctions and eigenvalues, and clarifies conditions under which physical invariances hold.

Findings

Exact eigenfunctions and eigenvalues are obtainable for approximate Hamiltonians.

Origin independence of intensities can be achieved in both length and velocity gauges under certain conditions.

Numerical demonstration confirms theoretical predictions about origin dependence in second-order intensities.

Abstract

We present a discussion of the consequences in perturbation theory when an exact eigenfunctions and eigenvalues to to the zeroth order Hamiltonian $H_0$ cannot be found. Since the usual approximations such as projecting the wavefunction on to a finite basis set and restricting the particle interaction is a way of constructing an approximate zeroth order Hamiltonian $H_0'$ we will here argue that the exact eigenfunctions and eigenvalues are always found for $H_0'$. We will show that as long as the perturbative expansion does not depend on any intrinsic properties of $H_0$ but only on knowing the exact eigenfunctions and eigenvalues then any perturbative statement, such as origin independence intensities, will be true for any $H_0'$ provided that $H_0'$ has a spectrum. We will use this to show that the origin independence for the intensities is trivially fulfilled in the velocity gauge but also can be fulfilled exactly in the length gauge if an appropriate $H_0$ is chosen. Finally a small numerically demonstration of the origin dependence of the terms for the second-order intensities in both the length and velocity gauge is undertaking to numerically illustrate the theoretical statements.