Rigorous test of the Raleigh-Ritz method for Mexican hat type potentials
A. M. Rodriguez Zarate, T. Thiemann
TL;DR
This work develops a rigorous framework to test the Raleigh-Ritz method by constructing quantum Hamiltonians with exactly known ground states from supersymmetric quantum mechanics, yielding Mexican hat–type potentials as physically meaningful toy models. The authors project the Hamiltonians onto a harmonic-oscillator basis to form finite-dimensional matrices $H^N$ and then diagonalize them to approximate the ground state energy and wavefunction, using $\Omega$ and its closed-form overlaps as benchmarks. They show that the Raleigh-Ritz approximations converge systematically with the matrix size $N$, with faster convergence for smaller $\epsilon$ and lower $k$, and they demonstrate strong agreement with the exact overlaps for the ground state Fourier coefficients. A detailed comparison with stationary perturbation theory reveals that the Raleigh-Ritz method is non-perturbative and often superior in regimes of stronger coupling, albeit at higher computational cost. The results provide a rigorous benchmarking platform and suggest that the method can be extended to broader families of polynomial superpotentials.
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.