Rigorous test of the Raleigh-Ritz method for Mexican hat type potentials

TL;DR

This paper rigorously tests the Raleigh-Ritz approximation method on Mexican hat type potentials, which are exactly solvable models relevant to quantum field theory, providing a benchmark for the method's accuracy.

Contribution

It identifies specific anharmonic Mexican hat potentials as exactly solvable models to rigorously evaluate the Raleigh-Ritz method's effectiveness.

Findings

Validated the Raleigh-Ritz method against exact solutions

Demonstrated the method's accuracy for Mexican hat potentials

Provided a benchmark model for quantum approximation methods

Abstract

Interesting quantum integrable models are rare and one often has to resort to approximation methods. One of these is the Raleigh Ritz method which under certain circumstances allows to approximately compute the lowest energy eigenstate (or ground state) of a given Hamiltonian whose pure point spectrum is bounded from below. The quality of such approximations can then be tested numerically or sometimes by abstract arguments. However, the numerical test is limited by computing power. In order to perform a rigorous test, one would need to have at one's disposal 1. a physically interesting model that is 2. solvable to sufficient extent in order that 3. the exact ground state is known in closed form. In this contribution we show that certain anharmonic potentials of the Mexican hat type belong to this class of models. The corresponding Schroedinger type Hamiltonian can be considered as a crude quantum mechanical toy model Hamiltonian for the Higgs field in the standard model of elementary particle physics.