Computational efficacy of Hamiltonian moments-based methods for the calculation of non-tractable potentials.


Ian M. Lowry

Date of Award


Document Type



College of Liberal Arts

Degree Name

Bachelor in Arts


Two versions of the connected moments expansion, those commonly known as the CMX-HW and the CMX-LT, as well as Lanczos tri-diagonalization were computed up to 7th order for the ground state and first excited state of the anharmonic oscillator with V = x² + [Lambda]x⁴. There is no exact solution for this potential, and so it can only be approximated numerically. Trial wave functions for each method were pre-conditioned by using variational analysis. The results were compared to previously published figures, and the computation times to each other. Lanczos tri-diagonalization was found to be the most accurate and quickest method, with CMX-LT matching its accuracy but being far slower, and CMX-HW being slightly less accurate and taking similar time to CMX-LT. The first excited state for both connected moments expansions was more accurate and quicker than the ground state, however the first excited state in Lanczos tri-diagonalization had extremely poor accuracy. The preconditioning was found to improve the accuracy of the calculation by more than two orders of magnitude. Additionally, a brief study of trial wave function kurtoses was conducted, which found no systematic effect of kurtosis on the accuracy of the approximation. All calculations were carried out in Mathematica 9.0 run on a laptop computer.